# Complexity reduction in the 3D Kuramoto model

Ana Elisa D. Barioni  Marcus A. M. de Aguiar  Instituto de Física ‘Gleb
Wataghin’, Universidade Estadual de Campinas, Unicamp 13083-970, Campinas, SP,
Brazil

###### Abstract

The dynamics of large systems of coupled oscillators is a subject of
increasing importance with prominent applications in several areas such as
physics and biology. The Kuramoto model, where a set of oscillators move
around a circle representing their phases, is a paradigm in this field,
exhibiting a continuous transition between disordered and synchronous motion.
Reinterpreting the oscillators as rotating unit vectors, the model was
extended to allow vectors to move on the surface of D-dimensional spheres,
with $D=2$ corresponding to the original model. It was shown that the
transition to synchronous dynamics was discontinuous for odd D, raising a lot
of interest. Inspired by results in 2D, Ott et al Chandra et al. [2019a]
proposed an ansatz for density function describing the oscillators and derived
equations for the ansatz parameters, effectively reducing the dimensionality
of the system. Here we take a different approach for the 3D system and
construct an ansatz based on spherical harmonics decomposition of the
distribution function. Our result differs significantly from that proposed in
Chandra et al. [2019a] and leads to similar but simpler equations determining
the dynamics of the order parameter. We derive the phase diagram of
equilibrium solutions for several distributions of natural frequencies and
find excellent agreement with simulations. We also compare the dynamics of the
order parameter with numerical simulations and with the previously derived
equations, finding good agreement in all cases. We believe our approach can be
generalized to higher dimensions and help to achieve complexity reduction in
other systems of equations.

##  I Introduction

The study of synchronization between elements of a system has a long history
that dates back to Huygens Oliveira and Melo [2014]. Over the past years
several examples of synchronization have been found in different physical
systems, from coupled metronomes Pantaleone [2002] and electochemical
oscillators Kiss et al. [2008] to neural networks Novikov and Benderskaya
[2014], motivating the development of new mathematical methods to understand
their global behavior. One aspect of great importance is the transition from
disordered to synchronized motion as the coupling intensity between the
elements increases.

In 1975 Yoshiki Kuramoto proposed a simple model of $N$ coupled oscillators
that could be solved analytically in the limit where $N$ goes to infinity
Kuramoto [1975, 1984]. The oscillators (or particles) were described by phases
$\theta_{i}$ and natural frequencies $\omega_{i}$, selected from a symmetric
distribution $g(\omega)$. The oscillators interact according to the equations

$\dot{\phi}_{i}=\omega_{i}+\frac{K}{N}\sum_{j=1}^{N}\sin{(\phi_{j}-\phi_{i})}.$
(1)

where $K$ is the coupling strength and $i=1,...,N$. The complex order
parameter

$z=pe^{i\psi}\equiv\frac{1}{N}\sum_{i=1}^{N}e^{i\theta_{i}}$ (2)

measures the degree of phase synchronization of the particles: disordered
motion implies $p\approx 0$ and coherent motion $p\approx 1$. Kuramoto showed
that the onset of sychronization could be described, in equilibrium, as a
second order phase transition, where $p$ remains very small for $0<K<K_{c}$
and increases as $p=c\sqrt{K-K_{c}}$ for $K>K_{c}$. These results opened the
way to large number of modifications and generalizations of his model,
including different types of coupling functions Hong and Strogatz [2011],
Yeung and Strogatz [1999], Breakspear et al. [2010], introduction of networks
of connections (so that not all oscillators are connected to each other)
Rodrigues et al. [2016], Climaco and Saa [2019], different distributions of
the oscillator’s natural frequencies (including frequencies proportional to
the number of connections, leading to explosive synchronization) Gomez-
Gardenes et al. [2011], Ji et al. [2013], inertial terms Acebrón et al.
[2005], Dörfler and Bullo [2011], Olmi et al. [2014] and external periodic
driving forces Childs and Strogatz [2008], Moreira and de Aguiar [2019a, b].
More recently, interest has been shifted to understand oscillations in larger
dimensions. It has been shown, in particular, that the natural extension of
the Kuramoto model to more dimensions exhibit first order phase transitions in
odd dimensions and second order transitions in even dimensions Chandra et al.
[2019b].

In terms of mathematical methods, a breakthrough insight was given by Ott and
Antonsen Ott and Antonsen [2008], who considered the continuity equation
satisfied by the oscillators and proposed an ansatz for their distribution on
the unit circle involving only two time dependent parameters. In the special
case where the natural frequencies of the oscillators were given by a
Lorentzian distribution, the differential equations for the ansatz parameters
could be translated into equations for the order parameter $z$, allowing the
system to be studied through two simple equations. This dimensional reduction
permitted, in particular, the construction of complete bifurcation diagrams
when the oscillators are driven by periodic forces Childs and Strogatz [2008].
External driving forces acting only on subsets of oscillators Moreira and de
Aguiar [2019a] were also used to probe modular structures in neural networks
Moreira and de Aguiar [2019b].

The possibility of describing a large system of $N$ coupled particles by a
small set of time-dependent parameters was pointed out by Watanabe and
Strogatz Watanabe and Strogatz [1994], Goebel [1995] and further developed in
Marvel et al. [2009]. They showed that the trajectory of the i-th Kuramoto
oscillator can be described by the Moebius transformation

$z_{i}(t)=e^{i\sigma(t)}\frac{a(t)-z_{i}(0)}{1-a^{*}(t)z_{i}(0)}$ (3)

where $z_{i}=e^{i\phi_{i}}$, $\sigma$ is real and $a$ complex. The Moebius
transformation was generalized by Tanaka Tanaka [2014] who extended it to
multidimensional real of complex variables. For complex variables, attractive
coupling between particles (similar to the Kuramoto model) and multivariate
Lorentzian distribution of natural frequencies, equations of motion for the
order parameter similar to the Stuart-Landau equation could be derived. For
real variables, corresponding, for instance, to the Kuramoto model in 3
dimensions, an equation for the order parameter could only be obtained for the
case of identical oscillators.

In this work we consider the 3-dimensional Kuramoto model proposed in Chandra
et al. [2019b]. In this model the phases of the 2D oscillators are re-
interpreted as unit vectors rotating on the surface of a circle and extended
to unit vectors moving on the surface of the sphere. Chandra et al Chandra et
al. [2019b] showed that for $N$ odd the onset of synchronization happens
through a first order phase transition. They also proposed an extension of the
Ott-Antonsen ansatz to the N-dimensional system Chandra et al. [2019a] by
modifying slightly the original function proposed in Ott and Antonsen [2008].
Here we take a different approach to construct an ansatz for the system:
instead of guessing the form of the distribution of oscillators on the sphere,
we expand the distribution in spherical harmonics and make an ansatz for the
expansion coefficients, similar to the construction of the 2D case. We obtain
a function that differs from the one derived in Chandra et al. [2019b] and
leads to a simpler relation between the ansatz and the order parameter. We
derive a differential equation for the ansatz parameters and the phase
transition diagram for three types of distributions of natural frequencies.
Our results agree well with numerical simulations using $N=10000$ oscillators.

This paper is organized as follows: in section II we re-derive the reduced
equations for the 2D Kuramoto model using unit vectors. We show that Ott and
Antonsen’s ansatz describes the phase transition for any distribution of
frequencies, not only the Lorentzian. In section III we briefly review the 3D
Kuramoto model, propose our ansatz and derive the differential equations for
the ansatz parameters using the continuity equation. We then consider the case
of identical oscillators and then the equilibrium solutions for general
distributions of natural frequencies to study the behavior of the order
parameter in terms of the coupling intensity, where we find a first order
phase transition in dimension 3. In section IV we discuss the differences and
similarities between our reduced system and that proposed in Chandra et al.
[2019a].

##  II 2D Kuramoto model

In this section we rewrite the original Kuramoto model in vector form and show
how the Ott-Antonsen ansatz can be used to characterize the behavior of the
order parameter as a function of the coupling constant in equilibrium.

###  II.1 Vector formulation

Following Chandra et al, we describe the oscillators by unit vectors
$\vec{\sigma_{i}}=(\cos{\phi_{i}},\sin{\phi_{i}}).$ It is easy to show that if
$\phi_{i}$ satisfies Kuramoto’s equation (1) then

$\frac{d\vec{\sigma_{i}}}{dt}=\mathbf{W}_{i}\vec{\sigma_{i}}+\frac{K}{N}\sum_{j}[\vec{\sigma_{j}}-(\vec{\sigma_{i}}\cdot\vec{\sigma_{j}})\vec{\sigma_{i}}]$
(4)

where $\mathbf{W}_{i}$ is an anti-symmetric matrix containing the natural
frequency $\omega_{i}$:

$\mathbf{W}_{i}=\left(\begin{array}[]{cc}0&-\omega_{i}\\\
\omega_{i}&0\end{array}\right).$ (5)

The complex order parameter $z$, Eq.(2), is replaced by the vector

$\vec{p}=\frac{1}{N}\sum_{i}\vec{\sigma_{i}}$ (6)

describing the center of mass of the system.

We also consider an extension of the model where the coupling constant $K$ is
replaced be a $2\times 2$ matrix ${\mathbf{K}}_{i}$ with elements
$K_{ij}=K_{ij}(\phi_{i})$ that might depend on $\phi_{i}$. It is convenient to
define an auxiliary vector $\vec{q}_{i}={\mathbf{K}}_{i}\,\vec{p}$ and rewrite
the generalization of the dynamical equations as

$\frac{d\vec{\sigma_{i}}}{dt}=\mathbf{W}_{i}\vec{\sigma_{i}}+[\vec{q}_{i}-(\vec{\sigma_{i}}\cdot\vec{q}_{i})\vec{\sigma_{i}}].$
(7)

Norm conservation, $|\vec{\sigma_{i}}|=1$, is guaranteed for any set of
regular matrices ${\mathbf{K}}_{i}$ (as can be seen by taking the scalar
product of Eq.(8) with $\vec{\sigma_{i}}$). Moreover, the equations of motion
are invariant under the transformation
$\vec{q}_{i}\rightarrow\vec{q}_{i}+a_{i}\vec{\sigma_{i}}$, for any function
$a_{i}(\phi_{i})$. It is also convenient to add a third (artificial) dimension
and write $\vec{\sigma_{i}}=(\cos{\phi_{i}},\sin{\phi_{i},0})$ and
$\vec{\omega}_{i}=(0,0,\omega_{i}).$ With these definitions we obtain

$\frac{d\vec{\sigma_{i}}}{dt}=\vec{\omega}_{i}\times\vec{\sigma_{i}}+[\vec{q}_{i}-(\vec{\sigma_{i}}\cdot\vec{q}_{i})\vec{\sigma_{i}}].$
(8)

###  II.2 Continuity equation

In the limit of infinitely many oscillators we define $f(\omega,\phi,t)$ as
the density of oscillators with natural frequency $\omega$ at position $\phi$
in time $t$. It satisfies

$\int_{0}^{2\pi}d\phi f(\omega,\phi,t)=g(\omega),$ (9)

where $g(\omega)$ is the distribution of natural frequencies and

$\int_{0}^{2\pi}d\phi\int d\omega f(\omega,\phi,t)=\int g(\omega)d\omega=1.$
(10)

Conservation of oscillator number implies the continuity equation

$\frac{\partial f}{\partial t}+\frac{\partial(fv_{\phi})}{\partial\phi}=0$
(11)

with velocity field

$\vec{v}=\omega\times\hat{r}+\vec{q}-(\hat{r}\cdot\vec{q})\hat{r}=(\omega+q_{\phi})\hat{\phi}\equiv
v_{\phi}\hat{\phi}.$ (12)

Equation (11) can be written explicitly as

$\frac{\partial f}{\partial t}+(\omega+q_{\phi})\frac{\partial
f}{\partial\phi}-fq_{r}=0$ (13)

where $q_{r}=\vec{q}\cdot\hat{r}$, $q_{\phi}=\vec{q}\cdot\hat{\phi}$,
$\hat{r}=(\cos\phi,\sin\phi)$ and $\hat{\phi}=(-\sin\phi,\cos\phi)$ (we shall
omit the artificial third dimension whenever possible). We also note that
$\frac{\partial q_{\phi}}{\partial\phi}=-q_{r}$. Eq.(6) for the order
parameter becomes

$\vec{p}(t)=\int\hat{r}(\phi)f(\omega,\phi,t)d\phi\,d\omega.$ (14)

###  II.3 Ansatz for density function

The density of oscillators is a periodic function of $\phi$ and, therefore,
can be expanded in Fourier series:

$\displaystyle
f(\omega,\phi,t)=\frac{g(\omega)}{2\pi}\sum_{m=-\infty}^{\infty}a_{m}(\omega,t)e^{im\phi}.$
(15)

Ott & Antonsen Ott and Antonsen [2008] showed that the ansatz
$a_{m}=\rho^{|m|}e^{-im\Phi}$ is self-consistent, in the sense that it
preserves this form at all times, remaining in this restricted subset of
density functions. The ansatz is, therefore, parametrized by $\rho(\omega,t)$
and $\Phi(\omega,t)$:

$\displaystyle
f(\omega,\phi,t)=\frac{g(\omega)}{2\pi}\sum_{m=-\infty}^{\infty}\rho^{|m|}e^{im(\phi-\Phi)}.$
(16)

Summing the geometric series we obtain

$\displaystyle
f(\omega,\phi,t)=\frac{g(\omega)}{2\pi}\frac{(1-\rho^{2})}{1+\rho^{2}-2\rho\cos\xi}$
(17)

where $\xi=\phi-\Phi$. It is convenient to define the vector
$\vec{\rho}=\rho(\cos\Phi,\sin\Phi,0)$ so that
$\cos\xi=\cos(\phi-\Phi)=\hat{r}\cdot\hat{\rho}$.

Substituting Eq.(16) into (14) we see that only the terms with $m=1$ and
$m=-1$ contribute to the integral:

$\displaystyle\vec{p}(t)$ $\displaystyle=$
$\displaystyle\int\frac{g(\omega)}{2\pi}\hat{r}(\phi)\rho\left(e^{i\xi}+e^{-i\xi}\right)d\phi\,d\omega=\int\frac{g(\omega)}{\pi}\left[\int_{0}^{2\pi}d\phi\,\hat{r}(\phi)\hat{r}^{T}(\phi)\right]\vec{\rho}\,d\omega$

where the dyadic product $\hat{r}(\phi)\hat{r}^{T}(\phi)$ is the projector in
the $\hat{r}$ direction. It is easy to check that the integral between
brackets results $\pi{\mathbf{1}}$ and

$\vec{p}(t)=\int\vec{\rho}(\omega,t)g(\omega)\,d\omega.$ (18)

This is a key equation, that connects the parameters of the ansatz with the
order parameter. The specific relation depends on the form of $g(\omega)$. We
will also use polar variables for the order parameter,
$\vec{p}=p(\cos\Psi,\sin\Psi,0)$ and $\nu=\phi-\Psi$.

###  II.4 Equations of motion for ansatz parameters

The derivatives of $f$ are

$\frac{\partial f}{\partial\phi}=-\frac{\partial
f}{\partial\Phi}=\frac{2\rho_{\phi}(1-\rho^{2})}{D^{2}},$ (19) $\frac{\partial
f}{\partial\rho}=-\frac{2[2\rho-(1+\rho^{2})\cos\xi]}{D^{2}},$ (20)

where we defined $D=1+\rho^{2}-2\rho\cos\xi.$

We can write Eq.(13) explicitly as

$\frac{\partial f}{\partial\rho}\dot{\rho}+\frac{\partial
f}{\partial\Phi}\dot{\Phi}+(\omega+q_{\phi})\frac{\partial
f}{\partial\phi}-fq_{r}=0$ (21)

Replacing the derivatives we find, after some simplifications presented in
Appendix A, the following equation for the dynamics of $\vec{\rho}$

$\displaystyle\dot{\vec{\rho}}=\vec{\omega}\times\vec{\rho}+\frac{1}{2}(1+\rho^{2}){\mathbf{K}}\vec{p}-({\mathbf{K}}\vec{p}\cdot\vec{\rho})\vec{\rho}.$
(22)

Taking the scalar product with $\hat{\rho}$ we also get

$\displaystyle\dot{\rho}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1-\rho^{2})({\mathbf{K}}\vec{p}\cdot\hat{\rho}).$
(23)

###  II.5 Equilibrium solutions

In this subsection we particularize for the case ${\mathbf{K}}=K{\mathbf{1}}$.
If $K=0$ the only equilibrium solution is $\rho=p=0$. For $K\neq 0$ we can
solve Eq.(22) with $\dot{\vec{\rho}}=0$ for a given $\vec{p}$ and check that
the solution allows the determination of $\vec{p}$ via Eq.(18) self-
consistently. If $\rho\neq 0$, then either $\rho=1$, and we only need to find
the direction $\hat{\rho}$, or $\vec{\rho}$ is perpendicular to $\vec{p}$ (see
Eq.(23)). Fixing $\hat{x}$ as the direction of $\vec{p}$, so that
$\vec{p}=p\hat{x}$ and $\vec{p}\cdot\hat{\rho}=p\cos\Phi$, and setting
$\rho=1$ we find for the $\hat{x}$ and $\hat{y}$ components of Eq.(22)

$\displaystyle-\omega\sin\Phi+Kp-Kp\cos^{2}\Phi=0$ (24)
$\displaystyle\omega\cos\Phi-Kp\cos\Phi\sin\Phi=0.$ (25)

The solution is $\sin\Phi=\frac{\omega}{Kp}$ if $|\omega|\leq Kp$. For
$|\omega|>Kp$ we set $\vec{\rho}=\rho\hat{y}$ and get
$\rho=\omega/Kp[1\pm\sqrt{1-K^{2}p^{2}/\omega^{2}}]$. One can check that the
stable solution is the one with the minus sign. The final solution, therefore
is

$\vec{\rho}(\omega)=\frac{\omega}{Kp}\left\\{\begin{array}[]{l}\sqrt{\frac{K^{2}p^{2}}{\omega^{2}}-1}\,\hat{x}\,+\,\hat{y}\qquad\mbox{for}\quad|\omega|\leq
Kp\\\ \\\
\left[1-\sqrt{1-\frac{K^{2}p^{2}}{\omega^{2}}}\right]\hat{y}\qquad\mbox{for}\quad|\omega|>Kp.\\\
\end{array}\right.$ (26)

Figure 1 illustrates the dependence of $\vec{\rho}$ on $\omega$.

Figure 1: Solution $\vec{\rho}(\omega)$ for $\omega>0$ (red). For
$0<\omega<Kp$ $\vec{\rho}$ stays on the unit circle, starting at $\Phi=0$ and
moving towards $\Phi=\pi/2$. For $\omega>Kp$ $\vec{\rho}$ points in the
y-direction and decreases to zero as $\omega\rightarrow\infty$. For $\omega<0$
(blue) the solution is reflected on the x-axis.

The order parameter can now be computed for any distribution function
$g(\omega)$. Assuming $g(\omega)$ to be symmetric around $\omega=0$ we obtain

$\displaystyle p=Kp\int_{-\pi/2}^{\pi/2}\cos^{2}(\Phi)g(Kp\sin\Phi)d\Phi$ (27)

which is the result found by Kuramoto. Notice that this was obtained from the
ansatz, not from the exact equations. The trivial solution is $p=0$. The non-
trivial solution is obtained canceling $p$ on each side and solving for $p$.
The critical value of $K$, where the non-trivial solution starts, is obtained
by setting $p=0$ in the resulting equation, leading to $K_{c}=2/\pi g(0)$.
Expanding $g$ around $p=0$ to second order we find

$p=\sqrt{\frac{16}{\pi K^{3}g^{\prime\prime}(0)}(1-K/K_{c})},$ (28)

which is exactly what was found by Kuramoto.

The complete $p\times K$ phase diagram can be constructed for specific
distributions $g(\omega)$ Hu et al. [2014]. Noting that the right hand side of
Eq.(27) is a function $F(Kp)$ the parametric plot $p=p(K)$ can be obtained as
$(K,p)=(x/F(x),F(x))$ for $x\in(0,\infty)$. For Gaussian distribution with
unit variance we get

$F(x)=\sqrt{\frac{\pi}{8}}\,xe^{-x^{2}/4}[I_{0}(x^{2}/4)+I_{1}(x^{2}/4)]$ (29)

where $I_{n}(x)$ is the modified Bessel function of the first kind. For the
Lorentzian distribution with unit width we get

$F(x)=\frac{1}{x}\left[\sqrt{1+x^{2}}-1\right].$ (30)

In both cases $K_{c}$ can be obtained as $\lim_{x\rightarrow 0}x/F(x)$ which
results $K_{c}=\sqrt{8/\pi}$ and $K_{c}=2$ for the Gaussian and Lorentzian
distributions respectively. Figure 2 compares these results with numerical
simulations. We note that numerically solving the pair of Eqs.(8) is much
faster than the original Eq.(1).

Figure 2: Order parameter as a function of the coupling constant for the
Gaussian (a) and Lorentzian (b) distribution of natural frequencies. Lines
show the analytical result and points the result of simulations with $N=1000$
oscillators.

##  III 3D Kuramoto model

The 3D model is a direct extension of the vector 2D model and represents unit
vectors $\vec{\sigma_{i}}$ rotating on the surface of a sphere. The equations
are Chandra et al. [2019b]

$\frac{d\vec{\sigma_{i}}}{dt}=\mathbf{W}_{i}\vec{\sigma_{i}}+\frac{1}{N}\sum_{j}[\mathbf{K}_{i}\vec{\sigma_{j}}-(\vec{\sigma_{i}}\cdot\mathbf{K}_{i}\vec{\sigma_{j}})\vec{\sigma_{i}}]$
(31)

where $\mathbf{W}_{i}$ is a $3\times 3$ anti-symmetric matrix

$\mathbf{W}_{i}=\left(\begin{array}[]{ccc}0&-\omega_{3i}&\omega_{2i}\\\
\omega_{3i}&0&-\omega_{1i}\\\ -\omega_{2i}&\omega_{1i}&0\end{array}\right)$
(32)

and $\mathbf{K}_{i}$ is the $3\times 3$ coupling matrix. There are now three
independent frequencies for each oscillator (or unit vector) that are taken
from a normalized distribution $G(\vec{\omega})$. The order parameter is
defined as in the 2D case as the system’s center of mass

$\vec{p}=\frac{1}{N}\sum_{i}\vec{\sigma_{i}}.$ (33)

Defining the 3D vector $\vec{w}_{i}=(\omega_{1i},\omega_{2i},\omega_{3i})$,
renaming $\vec{\sigma}\equiv\hat{r}$ and defining

$\displaystyle{\mathbf{K}}_{i}\,\vec{p}\equiv\vec{q}_{i}$ (34)

we obtain the dynamical equations

$\displaystyle\dot{\hat{r}}_{i}=\vec{w}_{i}\times\hat{r}_{i}+\vec{q}_{i}-(\hat{r}_{i}\cdot\vec{q}_{i})\hat{r}_{i}.$
(35)

As in the 2D case, the equations of motion are invariant under the
transformation $\vec{q}_{i}\rightarrow\vec{q}_{i}+a_{i}\hat{r}_{i}$ and we
shall use this freedom later.

###  III.1 Continuity equation

The continuity equation on the sphere is

$\frac{\partial f}{\partial
t}+\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}(f\sin\theta
v_{\theta})+\frac{1}{\sin\theta}\frac{\partial}{\partial\phi}(fv_{\phi})=0$
(36)

with velocity field given by Eq. (35):

$\displaystyle\vec{v}=\vec{\omega}\times\hat{r}+\vec{q}-(\hat{r}\cdot\vec{q})\hat{r}.$
(37)

Computing the partial derivatives the continuity equation takes the form

$\displaystyle\frac{\partial f}{\partial
t}-2fq_{r}+(\omega_{\phi}+q_{\theta})\frac{\partial
f}{\partial\theta}+\frac{1}{\sin\theta}(-\omega_{\theta}+q_{\phi})\frac{\partial
f}{\partial\phi}=0$ (38)

and the order parameter becomes:

$\displaystyle\vec{p}(t)=\int\hat{r}(\theta^{\prime},\phi^{\prime})f(\theta^{\prime},\phi^{\prime},\vec{\omega},t)d\Omega^{\prime}d^{3}\omega$
(39)

###  III.2 Ansatz for the density function

In 3D we can expand the density function in spherical harmonics:

$\displaystyle
f(\vec{\omega},\theta,\phi,t)=G(\vec{\omega})\sum_{l=0}^{\infty}\sum_{m=-l}^{l}f_{lm}Y_{lm}(\theta,\phi).$
(40)

Now we propose an ansatz analogous to the 2D case, but with three parameters
$\rho$, $\Theta$, $\Phi$, with the form
$f_{lm}=\rho^{l}Y^{*}_{lm}(\Theta,\Phi)$. The parameters define a vector
$\vec{\rho}(\vec{\omega},t)=\rho(\sin\Theta\cos\Phi,\sin\Theta\sin\Phi,\cos\Theta)$.
We write

$\displaystyle
f(\vec{\omega},\theta,\phi,t)=\frac{G(\vec{\omega})}{4\pi}\left[1+4\pi\sum_{l=1}^{\infty}\sum_{m=-l}^{l}\rho^{l}Y^{*}_{lm}(\Theta,\Phi)Y_{lm}(\theta,\phi)\right].$
(41)

Note that for $\rho=1$ we obtain
$f(\omega,\theta,\phi,t)=G(\vec{\omega})\frac{\delta(\theta-\Theta)\delta(\phi-\Phi)}{\sin\theta}$
indicating full synchrony. For $\rho=0$, on the other hand,
$f(\omega,\theta,\phi,t)=\frac{G(\vec{\omega})}{4\pi}$ and the oscillators are
uniformly spread over the sphere.

To sum the series and calculate the distribution $f$ explicitly we use the
relations

$\displaystyle\frac{4\pi}{2l+1}\sum_{m=-l}^{l}Y_{lm}(\hat{y})Y^{*}_{lm}(\hat{x})=P_{l}(\hat{x}\cdot\hat{y})$
(42)
$\displaystyle\sum_{l=0}^{\infty}y^{l}P_{l}(x)=\frac{1}{\sqrt{1+y^{2}-2xy}}$
(43)

Applying these relations to (41) we find

$f(\vec{\omega},\theta,\phi,t)=\frac{G(\vec{\omega})(1-\rho^{2})}{4\pi(1+\rho^{2}-2\rho\hat{r}\cdot\hat{\rho})^{{\frac{3}{2}}}}.$
(44)

Notice that $f$ depends on $\Theta$ and $\Phi$ only through

$\displaystyle\hat{r}\cdot\hat{\rho}\equiv\cos\xi=\sin\theta\sin\Theta\cos(\phi-\Phi)+\cos\theta\cos\Theta.$
(45)

In addition, using the relation

$\displaystyle\frac{4\pi}{3}\sum_{m=-1}^{1}Y_{1m}(\hat{y})Y^{*}_{1m}(\hat{x})=\hat{x}\cdot\hat{y}$
(46)

and the orthogonality of the spherical harmonics we can write the order
parameter as

$\vec{p}=\int
G(\vec{\omega})\rho\frac{3}{4\pi}\left[\int\hat{r}^{\prime}\hat{r}^{\prime
T}d\Omega^{\prime}\right]\cdot\hat{\rho}d^{3}\omega$ (47)

The angular integral of the dyadic matrix can be easily performed and results
$(4\pi/3)\mathbf{1}$. We obtain

$\displaystyle\vec{p}=\int G(\vec{\omega})\,\vec{\rho}\,d^{3}\omega.$ (48)

This is formally identical to the 2D case (see Eq. (18)), and much simpler
than the relation found in Chandra et al. [2019a]. Note that this
simplification was only possible because we applied orthogonality properties
of the spherical harmonics. Also, Eq.(44) should be contrasted with the
expression derived in Chandra et al. [2019a], where the denominator is raised
to the power 2, instead of $3/2$, and the numerator $(1-\rho^{2})$ to power 2,
instead of 1. These differences will have consequences for the derivation of
the equations of motion.

###  III.3 Continuity equation for ansatz distribution

The partial derivatives of $f$ are

$\displaystyle\frac{\partial
f}{\partial\rho}=\frac{\cos\xi(3+\rho^{2})-\rho(5-\rho^{2})}{D^{\frac{5}{2}}}$
(49)

and

$\displaystyle\frac{\partial f}{\partial x}=\frac{\partial
f}{\partial\cos\xi}\frac{\partial\cos\xi}{\partial
x}=\frac{3\rho(1-\rho^{2})}{D^{\frac{5}{2}}}\frac{\partial\cos\xi}{\partial
x},$ (50)

for $x=\theta,\phi,\Theta,\Phi$ where $D\equiv 1+\rho^{2}-2\rho\cos\xi$ and

$\begin{split}\frac{\partial\cos\xi}{\partial\theta}&=\frac{\rho_{\theta}}{\rho}\qquad\qquad\quad\frac{\partial\cos\xi}{\partial\phi}=\sin\theta\frac{\rho_{\phi}}{\rho}\\\
\qquad\frac{\partial\cos\xi}{\partial\Theta}&=\hat{r}\cdot\hat{\Theta}\qquad\qquad\frac{\partial\cos\xi}{\partial\Phi}=\sin\Theta\hat{r}\cdot\hat{\Phi}\end{split}$
(51)

Writing the time derivative of $f$ as

$\displaystyle\frac{\partial f}{\partial t}=\frac{\partial
f}{\partial\rho}\dot{\rho}+\frac{\partial
f}{\partial\Theta}\dot{\Theta}+\frac{\partial f}{\partial\Phi}\dot{\Phi}$ (52)

we obtain

$\begin{split}D^{\frac{5}{2}}\frac{\partial f}{\partial
t}=\left[\cos\xi(3+\rho^{2})-\rho(5-\rho^{2})\right]\dot{\rho}+3\rho(1-\rho^{2})\hat{r}\cdot[\dot{\Theta}\hat{\Theta}+\sin\Theta\dot{\Phi}\hat{\Phi}].\end{split}$
(53)

Using eqs.(38) and (49)-(51) we can also write (see Appendix B)

$\begin{split}D^{\frac{5}{2}}\nabla(f\vec{v})&=3\rho(1-\rho^{2})\hat{r}\cdot(\hat{\rho}\times\vec{\omega})+(1-\rho^{2})\left[-2q_{r}(1+\rho^{2})+4q_{r}\rho_{r}+3q_{\theta}\rho_{\theta}+3q_{\phi}\rho_{\phi}\right].\end{split}$
(54)

###  III.4 Compatibility conditions

At this point we need to specify the vector $\vec{q}$ that will describe the
coupling between the oscillators. The natural choice
$\vec{q}={\mathbf{K}}\vec{p}$, does not work. To see this consider, for
instance ${\mathbf{K}}=K{\mathbf{I}}$ or $\vec{q}=K\vec{p}$. In this case the
last three terms in Eq.(54) become

$4\rho_{r}p_{r}+3\rho_{\theta}p_{\theta}+3\rho_{\phi}p_{\phi}=3\vec{\rho}\cdot\vec{p}+\rho_{r}p_{r}=3\vec{\rho}\cdot\vec{p}+\rho
p\cos\xi\cos\nu$ (55)

and the term $\cos\xi\cos\nu$ has no counterpart in the continuity equation.
Chandra et al Chandra et al. [2019a] have a similar problem, that they solve
by choosing the exponent of their tentative ansatz function. Here we do not
have this freedom. However, there is still a way out, using the invariance of
the exact equations under changes in the radial part of $\vec{q}$. We can
cancel the unwanted terms without altering the exact equations of motion if we
choose

$\vec{q}=\left[\mathbf{1}-\frac{1}{4}\hat{r}\hat{r}^{\top}\right]{\mathbf{K}}\vec{p}+\beta\hat{r}$
(56)

where $\hat{r}\hat{r}^{\top}$ is the projection in the radial direction and
the coefficient $\beta$ will be chosen later to ensure consistency of the
equations of motion. With this choice Eq.(54) becomes

$\begin{split}D^{\frac{5}{2}}\nabla(f\vec{v})&=3\rho(1-\rho^{2})\hat{r}\cdot(\hat{\rho}\times\vec{\omega})\\\
&+(1-\rho^{2})\left[-2\beta(1+\rho^{2})+3\vec{\rho}\cdot({\mathbf{K}}\vec{p})-\frac{3}{2}(1+\rho^{2})\hat{r}\cdot({\mathbf{K}}\vec{p})+4\beta\hat{r}\cdot\vec{\rho}\right].\end{split}$
(57)

###  III.5 Equations of motion

The continuity equation for the ansatz function, after multiplying by
$D^{5/2}$, is given by

$\begin{split}\left[\cos\xi(3+\rho^{2})-\rho(5-\rho^{2})\right]\dot{\rho}+3\rho(1-\rho^{2})\hat{r}\cdot[\dot{\Theta}\hat{\Theta}+\sin\Theta\dot{\Phi}\hat{\Phi}]+3\rho(1-\rho^{2})\hat{r}\cdot(\hat{\rho}\times\vec{\omega})\\\
-(1-\rho^{2})\left[-2\beta(1+\rho^{2})+3\vec{\rho}({\mathbf{K}}\vec{p})-\frac{3}{2}(1+\rho^{2})\hat{r}\cdot({\mathbf{K}}\vec{p})+4\beta\hat{r}\cdot\vec{\rho}\right]=0\end{split}$
(58)

We simplify this equation in Appendix C. This leads us to choose

$\beta=\frac{\vec{\rho}\cdot({\mathbf{K}}\vec{p})}{4}.$ (59)

and the final equations of motion become

$\begin{split}\dot{\vec{\rho}}=\vec{\omega}\times\vec{\rho}+\frac{1}{2}(1+\rho^{2})({\mathbf{K}}\vec{p})-[\vec{\rho}\cdot({\mathbf{K}}\vec{p})]\vec{\rho}.\end{split}$
(60)

From this equation it also follows that

$\begin{split}\dot{\rho}=\frac{1}{2}(1-\rho^{2})[\hat{\rho}\cdot({\mathbf{K}}\vec{p})].\end{split}$
(61)

The connection between $\vec{p}$ and $\vec{\rho}$ is given by Eq.(48). Eq.(60)
is identical to the equation obtained in Chandra et al. [2019a], but Eq.(48)
is not. In our formulation the order parameter $\vec{p}(t)$ is the average
over the distribution of natural frequencies of the ansatz parameter
$\vec{\rho}(\omega,t)$, just like the 2-dimensional case, whereas the
expression in Chandra et al. [2019a] includes an extra integral that depends
on the dimension of system. For $D=2$ both formulations recover the original
result in Ott and Antonsen [2008].

###  III.6 Identical oscillators with symmetric coupling and external forces

If all oscillators have identical natural frequencies,
$G(\vec{\omega})=\delta(\vec{\omega}-\vec{\omega}_{0})$, then
$\vec{p}(t)=\vec{\rho}(\vec{\omega}_{0},t)$. Moreover, if the coupling matrix
is diagonal, $\mathbf{K}=K\mathbf{1}$, the order parameter satisfies

$\begin{split}\dot{\vec{p}}=\vec{\omega}_{0}\times\vec{p}+\frac{K}{2}(1-p^{2})\vec{p}\quad\quad\textrm{and}\quad\quad\dot{p}=\frac{K}{2}(1-p^{2})p.\end{split}$
(62)

The equilibrium points are $p=0$, which is stable for $K<0$ and $p=1$, stable
for $K>0$. Figure 3 shows the behavior of the order parameter comparing the
exact solution, obtained by a simulation with 10000 oscillators, the solution
based on our ansatz and the solution for the ansatz proposed in Chandra et al.
[2019a], which we call Chandra’s solution for short. In this case it is clear
that our solution is delayed with respect to the exact (numerical)
integration, whereas Chandra’s solution is only slightly advanced. Both reach
the same equilibrium $p=1$.

Figure 3: Time evolution of the order parameter for identical oscillators with
$\vec{\omega}=\hat{z}$ and $K=1$. (a) module $p(t)$; (b) evolution on the
$p_{x}-p_{y}-p_{z}$ space. Thick black line shows the result of simulations
with $N=10000$ oscillators. Thin red and blue line shows the ansatz solution
using our approach and that from Chandra et al. [2019a], respectively.

Time dependent external forces can be included making
${\mathbf{K}}\cdot\vec{p}\rightarrow{\mathbf{K}}\cdot\vec{p}+\vec{F}$ in Eqs.
(56) and (59)-(61). For the case of diagonal coupling matrix this gives

$\begin{split}\dot{\vec{p}}=\vec{\omega}_{0}\times\vec{p}+\left[\frac{K}{2}(1-p^{2})-\vec{p}\cdot\vec{F}\right]\vec{p}+\frac{1}{2}(1+p^{2})\vec{F}\end{split}$
(63)

and

$\begin{split}\dot{p}=\frac{1}{2}(1-p^{2})(Kp+\vec{F}\cdot\hat{p}).\end{split}$
(64)

###  III.7 Equilibrium solutions for symmetric coupling

Here we consider equilibrium solutions for $\mathbf{K}=K\mathbf{1}$ but
general distributions of natural frequencies. In this case Eqs. (60) and (61)
simplify to:

$\begin{split}\dot{\vec{\rho}}=\vec{\omega}\times\vec{\rho}+\frac{K}{2}[(1+\rho^{2})\vec{p}-2(\vec{\rho}\cdot\vec{p})\vec{\rho}]\quad\quad\textrm{and}\quad\quad\dot{\rho}=\frac{K}{2}(1-\rho^{2})(\hat{\rho}\cdot\vec{p}).\end{split}$
(65)

We first consider the limit $K\rightarrow 0$, or $K=0^{+}$. According to Eqs.
(65), for $\vec{p}\neq 0$ equilibrium requires $\rho=1$ and $\hat{\rho}$
parallel to $\hat{\omega}$, i.e., $\vec{\rho}=\pm\hat{\omega}$. However, only
one of these solutions is stable. To find out which we make
$\vec{\rho}=\pm\hat{\omega}+\delta\vec{\rho}$, or $\rho=1+\delta\rho$ and
expand the module Eq.(65) to first order to obtain

$\delta\dot{\rho}=-\frac{K}{2}\delta\rho(\vec{p}\cdot\hat{\rho}).$ (66)

Therefore, for each $\vec{\omega}$ the stable solution is the one for which
$\vec{p}\cdot\hat{\rho}>0$, i.e., the one in the hemisphere defined by the
direction of $\vec{p}$. Setting $\vec{p}=p\hat{z}$ we have for
$G(\vec{\omega})=g(\omega)/4\pi$

$\displaystyle p=\int
G(\vec{\omega})\rho_{z}d^{3}\omega=\frac{1}{4\pi}\int_{0}^{2\pi}d\Phi\int_{0}^{\pi/2}2\sin\Theta\cos\Theta
d\Theta=\frac{1}{2},$ (67)

where the factor 2 comes from the fact that, in the upper hemisphere, the
stable solution is $\hat{\rho}=\hat{\omega}$ and in the lower hemisphere it is
$\hat{\rho}=-\hat{\omega}$. So when we cross from one hemisphere to another
both $\cos(\theta)$ and $\hat{\rho}$ changes signal, making the integral over
$0<\theta<\pi$ symmetrical with respect to $\pi/2$.

For $K\neq 0$ we define the orthonormal set of vectors

$\displaystyle\hat{\xi}_{1}$ $\displaystyle=$
$\displaystyle(\cos\theta\cos\phi,\cos\theta\sin\phi,-\sin\theta)$
$\displaystyle\hat{\xi}_{2}$ $\displaystyle=$
$\displaystyle(-\sin\phi,\cos\phi,0)$ (68) $\displaystyle\hat{\omega}$
$\displaystyle=$
$\displaystyle(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$

and expand
$\vec{\rho}=\alpha\hat{\xi}_{1}+\beta\hat{\xi}_{2}+\gamma\hat{\omega}.$

For $K=0$ we use $\alpha=\beta=0$ and $\gamma=1$, with $0<\theta<\pi/2$. We
again set $\vec{p}=p\hat{z}$ and obtain

$\vec{p}=-p\sin\theta\,\hat{\xi}_{1}+p\cos\theta\,\hat{\omega}.$ (69)

Projecting Eq.(60) with $\dot{\vec{\rho}}=0$ in these directions we obtain

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\beta-\bar{K}\sin\theta(1+\rho^{2})/2-\bar{K}\alpha(-\alpha\sin\theta+\gamma\cos\theta)$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\alpha-\bar{K}\beta(-\alpha\sin\theta+\gamma\cos\theta)$ (70)
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\bar{K}\cos\theta(1+\rho^{2})/2-\bar{K}\gamma(-\alpha\sin\theta+\gamma\cos\theta).$

where $\rho^{2}=\alpha^{2}+\beta^{2}+\gamma^{2}$ and $\bar{K}=Kp/\omega$.
These equations can be solved analytically:

$\displaystyle\beta$ $\displaystyle=$
$\displaystyle-\frac{(1+\bar{K}^{2})}{2\bar{K}\sin\theta}\left[1-\sqrt{1-\frac{4\bar{K}^{2}\sin^{2}\theta}{(1+\bar{K}^{2})^{2}}}\right]$
(71) $\displaystyle\alpha$ $\displaystyle=$
$\displaystyle-\sqrt{1+\beta\left(\frac{1+\bar{K}^{2}\cos^{2}\theta}{\bar{K}\sin\theta}\right)}$
(72) $\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\frac{\bar{K}\beta}{\alpha}\cos\theta$ (73)

and satisfy $\rho^{2}=\alpha^{2}+\beta^{2}+\gamma^{2}=1$.

For a delta function distribution
$G(\vec{\omega})=\frac{\delta(\omega-\omega_{0})}{4\pi\omega^{2}}$ we find

$p=\int_{0}^{\pi/2}\rho_{z}(\bar{K},\theta)\sin\theta d\theta\equiv
F(\bar{K})$ (74)

where $\rho_{z}=-\alpha\sin\theta+\gamma\cos\theta$ depends only on
$\bar{K}=Kp/\omega_{0}$ and $\theta$. Once the function $F(\bar{K})$ is
calculated numerically, $p(K)$ can be computed in parametric form for
$\omega_{0}=1$ as $(p=F(x),K=x/F(x))$. The result is shown in Fig. 4.
Equilibrium is found only for $K>1.5$. For $K<1.5$ the system converges to a
limit cycle, which is well described both by the exact and reduced equations
of motion. For $K=2$ our solution is slightly closer to the exact, whereas
Chandra’s is advanced, reaching equilibrium before the exact. In both cases
the solutions present fluctuations that depend on the (random) initial
conditions: every time the equations are integrated a slightly different curve
is obtained.

Figure 4: (a) Order parameter as a function of the coupling constant for the
uniform angular distribution of natural frequencies and
$|\vec{\omega}_{i}|=1$. The continuous curve shows the numerical integration
from Eq. (74) and the dots the value obtained by integrating the exact
equations of motion (31). For $K<1.5$ the system converges to a limit cycle.
The dashed line is the quadratic approximation, Eq.(75). (b) $p(t)$ for $K=1$;
(c) $p(t)$ for $K=2$. Thin red and blue line shows the ansatz solution using
our approach and that from Chandra et al. [2019a], respectively.

For small $K$ we can obtain analytical expressions expanding Eqs. (73) to
second order. We find $\alpha\approx-\bar{K}^{2}\sin\theta\cos\theta$,
$\beta\approx-\bar{K}\sin\theta$ and $\gamma\approx
1-(\bar{K}^{2}/2)\sin^{2}\theta$. This gives

$p=\frac{1}{2}+\frac{K^{2}p^{2}}{8\omega_{0}^{2}}\approx\frac{1}{2}+\frac{K^{2}}{32\omega_{0}^{2}}.$
(75)

In the limit of very large coupling $\bar{K}\rightarrow\infty$, on the other
hand, we get $\alpha\approx-\sin\theta$, $\beta\approx-\sin\theta/\bar{K}$ and
$\gamma\approx\cos\theta$, which gives

$p=\int_{0}^{\pi/2}(-\alpha\sin\theta+\gamma\cos\theta)\sin\theta
d\theta=\int_{0}^{\pi/2}\sin\theta d\theta=1.$ (76)

For the Gaussian distribution
$G(\vec{\omega})=\frac{\sqrt{2}}{\sqrt{\pi}\Delta^{3}4\pi}e^{-\omega^{2}/2\Delta^{2}}$
we have

$p=\frac{\sqrt{2}}{\sqrt{\pi}\Delta^{3}}\int_{0}^{\pi/2}d\theta\,\sin\theta\int_{0}^{\infty}d\omega\,\omega^{2}\,e^{-\omega^{2}/2\Delta^{2}}\rho_{z}(Kp/\omega,\theta).$
(77)

Changing the integration variable to $\Omega=\omega/Kp$ and setting $\Delta=1$
we obtain

$p=\frac{\sqrt{2}}{\sqrt{\pi}}K^{3}p^{3}\int_{0}^{\pi/2}d\theta\,\sin\theta\int_{0}^{\infty}d\Omega\,\Omega^{2}\,e^{-\Omega^{2}p^{2}K^{2}/2}\rho_{z}(1/\Omega,\theta)\equiv
H(Kp).$ (78)

Once again the function $H(x)$ can be computed numerically and the parametric
curve is given by $(p=H(x),K=x/H(x))$. The result is shown in Fig. 5. For
small $K$ we obtain

$p=\frac{1}{2}+\frac{K^{2}p^{2}}{8\Delta^{2}}\approx\frac{1}{2}+\frac{K^{2}}{32\Delta^{2}}.$
(79)

Figure 5: (a) Order parameter as a function of the coupling constant for the
Gaussian distribution of natural frequencies with $\Delta=1$. The continuous
curve shows the numerical integration from Eq. (78) and the dots the value
obtained by integrating the exact equations of motion (31). The dashed line is
the quadratic approximation, Eq.(79). (b) $p(t)$ for $K=2$. Thin red and blue
line shows the ansatz solution using our approach and that from Chandra et al.
[2019a], respectively.

##  IV Conclusions

In this paper we studied the problem of dimensional reduction for the Kuramoto
model in 2 and 3 dimensions. In 2D we used the ansatz proposed by Ott and
Antonsen Ott and Antonsen [2008] and solved the equations for the ansatz
parameters at equilibrium for arbitrary distributions of natural frequencies,
re-deriving Kuramoto expressions near the bifurcation point.

We extended the ansatz to 3D expanding the distribution function in spherical
harmonics. The resulting function differs from that obtained by Chandra et al
Chandra et al. [2019a] and satisfies the continuity equation only if the
coupling term is modified. Our ansatz is connected to a especific choice of
the coupling between the oscillators. This modification, however, is only in
its radial component, which does not affect the exact equations of motion. The
interpretation of the approximate density function is very similar to that in
2D: for $\rho\rightarrow 0$ it describes a uniform distribution over the
sphere and for $\rho\rightarrow 1$ it gives a delta function at position
$(\Theta,\Phi)$. The continuity equation leads to a vector equation for the
ansatz parameters that is exactly that obtained in Chandra et al. [2019a].
However, the connection between the ansatz and the order parameter differs and
it is simpler in our approach. The relationship we find is the natural
extension of the 2D case, where $\vec{p}$ is the integral of $\vec{\rho}$ over
$G(\vec{\omega})$, whereas that found in Chandra et al. [2019a] is more
complicated. This might facilitate the analysis of more general systems where
the coupling is a full matrix ${\mathbf{K}}$ or in the presence of external
forces.

As applications we consider two types of $G(\vec{\omega})$ consisting of
uniform angular distributions with delta or Gaussian distribution of frequency
module. In both cases the ansatz equations describe very well the full system
and show the first order transition behavior as expected. Curiously, the case
of identical oscillators, although agrees exactly at equilibrium, shows a
delay in the dynamics, as shown in figure 2. Among the examples we explored,
this was the only case where the ansatz in Chandra et al. [2019a] was more
accurate than ours.

The complete dimensional reduction is only achieved for the case of identical
oscillators. For all other cases we have to solve Eq.(60) together with (48).
However, as pointed out in Chandra et al. [2019a], the latter can be solved by
Monte Carlo methods, sampling the distribution $G(\vec{\omega})$, which
converges fast for most distributions of interest.

Finally, we remark that our ansatz points to an alternative way to treat the
Kuramoto model in higher dimensions. It allows us to easily obtain the
dynamics of the order parameter and study the behavior at equilibrium for
different distributions of oscillator’s natural frequencies.

###### Acknowledgements.

This work was partly supported by FAPESP, grants 2019/2027-5 (MAMA),
2019/24068-0 (ANDB), 2016/01343‐7 (ICTP‐SAIFR FAPESP) and CNPq, grant
301082/2019‐7 (MAMA). We would like to thank Alberto Saa and Jose A. Brum for
suggestions and careful reading of this manuscript.

##  Appendix A Derivation of the dynamics of $\rho$ in 2D

Rewriting Eq.(21) explicitly as

$\frac{\partial f}{\partial\rho}\dot{\rho}+\frac{\partial
f}{\partial\Phi}\dot{\Phi}+(\omega+q_{\phi})\frac{\partial
f}{\partial\phi}-fq_{r}=0$ (80)

and replacing the derivatives we find, after multiplying by $D^{2}$,

$\displaystyle-2[2\rho-(1+\rho^{2})\cos\xi]\,\dot{\rho}-2\rho_{\phi}(1-\rho^{2})\,\dot{\Phi}+$
$\displaystyle
2\omega(1-\rho^{2})\rho_{\phi}+2(1-\rho^{2})(\rho_{\phi}q_{\phi}+\rho_{r}q_{r})-(1-\rho^{4})q_{r}=0$
(81)

Using $\vec{q}={\mathbf{K}}\vec{p}$ and the relations
$\rho_{\phi}=\vec{\rho}\cdot\hat{\phi}=-\rho\hat{r}\cdot\hat{\Phi}$,
$\rho_{r}=\vec{\rho}\cdot\hat{r}$ and
$\omega\rho_{\phi}=-\hat{r}\cdot(\vec{\omega}\times\vec{\rho})$ we can greatly
simplify Eq.(81) to

$\displaystyle
2(1-\rho^{2})\hat{r}\cdot\left[\vec{\rho}\times\vec{\omega}+\rho\dot{\Phi}\hat{\Phi}+\dot{\rho}\hat{\rho}\frac{(1+\rho^{2})}{1-\rho^{2}}-\frac{1}{2}(1+\rho^{2}){\mathbf{K}}\vec{p}\right]-4\rho\dot{\rho}+2(1-\rho^{2})\vec{\rho}\cdot{\mathbf{K}}\vec{p}=0.$

Adding and subtracting $\dot{\rho}\hat{\rho}$ inside the square brackets we
obtain

$\displaystyle
2(1-\rho^{2})\hat{r}\cdot\left[\vec{\rho}\times\vec{\omega}+\dot{\vec{\rho}}+\dot{\rho}\hat{\rho}\frac{(2\rho^{2})}{1-\rho^{2}}-\frac{1}{2}(1+\rho^{2}){\mathbf{K}}\vec{p}\right]+4\left[\frac{1}{2}(1-\rho^{2})({\mathbf{K}}\vec{p}\cdot\vec{\rho})-\rho\dot{\rho}\right]=0.$

Since this equation must hold for all values of $\hat{r}$ each term in the
brackets must be zero:

$\displaystyle\dot{\vec{\rho}}$ $\displaystyle=$
$\displaystyle\vec{\omega}\times\vec{\rho}-\dot{\rho}\hat{\rho}\frac{(2\rho^{2})}{1-\rho^{2}}+\frac{1}{2}(1+\rho^{2}){\mathbf{K}}\vec{p}$
(82) $\displaystyle\dot{\rho}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1-\rho^{2})({\mathbf{K}}\vec{p}\cdot\hat{\rho}).$
(83)

Taking the scalar product of the first of these equations with $\vec{\rho}$
and noting that $\vec{\rho}\cdot\dot{\vec{\rho}}=\rho\dot{\rho}$ we can check
that the second equation is recovered. Therefore the equations are compatible
and we can replace $\dot{\rho}$ given by the second into the first equation to
finally obtain

$\displaystyle\dot{\vec{\rho}}=\vec{\omega}\times\vec{\rho}+\frac{1}{2}(1+\rho^{2}){\mathbf{K}}\vec{p}-({\mathbf{K}}\vec{p}\cdot\vec{\rho})\vec{\rho}.$
(84)

##  Appendix B Derivation of Eq.(54)

From Eq.(52) we obtain

$\begin{split}D^{\frac{5}{2}}\frac{\partial f}{\partial
t}=\left[\cos\xi(3+\rho^{2})-\rho(5-\rho^{2})\right]\dot{\rho}+3\rho(1-\rho^{2})\hat{r}\cdot[\dot{\Theta}\hat{\Theta}+\sin\Theta\dot{\Phi}\hat{\Phi}]\end{split}$
(85)

Using eq.(38) and eqs. (49)-(51) we write

$\begin{split}D^{\frac{5}{2}}\nabla(f\vec{v})&=-2(1-\rho^{2})q_{r}D+(\omega_{\phi}+q_{\theta})3\rho(1-\rho^{2})\frac{\partial\cos\xi}{\partial\theta}+\frac{1}{\sin\theta}(-\omega_{\theta}+q_{\phi})3\rho(1-\rho^{2})\frac{\partial\cos\xi}{\partial\phi}\\\
&=3\rho(1-\rho^{2})\left[\omega_{\phi}\frac{\partial\cos\xi}{\partial\theta}-\frac{\omega_{\theta}}{\sin\theta}\frac{\partial\cos\xi}{\partial\phi}\right]\\\
&+\left[-2(1-\rho^{2})q_{r}D+3\rho(1-\rho^{2})q_{\theta}\frac{\partial\cos\xi}{\partial\theta}+3\rho(1-\rho^{2})\frac{q_{\phi}}{\sin\theta}\frac{\partial\cos\xi}{\partial\phi}\right].\end{split}$
(86)

The first line contains terms that are independent of the coupling vector
$\vec{q}$. They can be simplified as follows:

$\begin{split}\omega_{\phi}\frac{\partial\cos\xi}{\partial\theta}-\frac{\omega_{\theta}}{\sin\theta}\frac{\partial\cos\xi}{\partial\phi}=\frac{1}{\rho}(\omega_{\phi}\rho_{\theta}-\omega_{\theta}\rho_{\phi})=\hat{r}\cdot(\hat{\rho}\times\vec{\omega}).\end{split}$
(87)

Finally, using eqs. (49)-(51), the terms containing $\vec{q}$ can be written
as

$\begin{split}-2(1-\rho^{2})q_{r}D+3(1-\rho^{2})(q_{\theta}\rho_{\phi}+q_{\phi}\rho_{\theta})\\\
=(1-\rho^{2})\left[-2q_{r}(1+\rho^{2})+4q_{r}\rho_{r}+3q_{\theta}\rho_{\theta}+3q_{\phi}\rho_{\phi}\right].\end{split}$
(88)

##  Appendix C Derivation of the dynamics of $\rho$ in 3D

To simplify Eq.(58) we first add $\dot{\rho}\hat{\rho}$ to the middle term of
the first line to obtain $3\rho(1-\rho^{2})\hat{r}\cdot\dot{\vec{\rho}}$. We
also subtract an equal term to keep the equation unaltered. This gives

$\begin{split}3(1-\rho^{2})\hat{r}\cdot[\vec{\omega}\times\vec{\rho}-\dot{\vec{\rho}}]+\dot{\rho}\left[\rho(5-\rho^{2})-4\rho^{2}\cos\xi\right]\\\
-(1-\rho^{2})\left[-2\beta(1+\rho^{2})+3\vec{\rho}({\mathbf{K}}\cdot\vec{p})-\frac{3}{2}(1+\rho^{2})\hat{r}\cdot({\mathbf{K}}\cdot\vec{p})+4\beta\hat{r}\cdot\vec{\rho}\right]=0\end{split}$
(89)

Next we put all terms containing the angular coordinates $\hat{r}$ together.
We obtain

$\begin{split}3(1-\rho^{2})\hat{r}\cdot[\vec{\omega}\times\vec{\rho}-\dot{\vec{\rho}}+\frac{1}{2}(1+\rho^{2})({\mathbf{K}}\cdot\vec{p})-\frac{4\beta}{3}\vec{\rho}-\frac{4}{3(1-\rho^{2})}\rho\dot{\rho}\vec{\rho}]\\\
+\dot{\rho}\rho(5-\rho^{2})-(1-\rho^{2})\left[3\vec{\rho}\cdot({\mathbf{K}}\cdot\vec{p})-2\beta(1+\rho^{2})\right]=0\end{split}$
(90)

This implies that the following equations must be satisfied simultaneously:

$\begin{split}\dot{\vec{\rho}}=\vec{\omega}\times\vec{\rho}+\frac{1}{2}(1+\rho^{2})({\mathbf{K}}\cdot\vec{p})-\frac{4\beta}{3}\vec{\rho}-\frac{4}{3(1-\rho^{2})}\rho\dot{\rho}\vec{\rho}\end{split}$
(91)

and

$\begin{split}\dot{\rho}\rho(5-\rho^{2})=(1-\rho^{2})\left[3\vec{\rho}\cdot({\mathbf{K}}\cdot\vec{p})-2\beta(1+\rho^{2})\right]\end{split}$
(92)

Taking the scalar product of (91) with $\vec{\rho}$ and noting that
$\vec{\rho}\cdot\dot{\vec{\rho}}=\rho\dot{\rho}$ we find that equations (91)
and (92) are compatible only if

$\beta=\frac{\vec{\rho}\cdot({\mathbf{K}}\cdot\vec{p})}{4}.$ (93)

With this condition (91) and (92) become

$\begin{split}\dot{\vec{\rho}}=\vec{\omega}\times\vec{\rho}+\frac{1}{2}(1+\rho^{2})({\mathbf{K}}\cdot\vec{p})-[\vec{\rho}\cdot({\mathbf{K}}\cdot\vec{p})]\vec{\rho}\end{split}$
(94)

and

$\begin{split}\dot{\rho}=\frac{1}{2}(1-\rho^{2})[\hat{\rho}\cdot({\mathbf{K}}\cdot\vec{p})].\end{split}$
(95)

## References

  * Chandra et al. [2019a] Sarthak Chandra, Michelle Girvan, and Edward Ott.  Complexity reduction ansatz for systems of interacting orientable agents: Beyond the kuramoto model.  _Chaos: An Interdisciplinary Journal of Nonlinear Science_ , 29(5):053107, 2019a.
  * Oliveira and Melo [2014] Henrique M Oliveira and Luís V Melo.  Huygens synchronization of two pendulum clocks.  _arXiv preprint arXiv:1410.7926_ , 2014.
  * Pantaleone [2002] James Pantaleone.  Synchronization of metronomes.  _American Journal of Physics_ , 70(10):992–1000, 2002.  doi: 10.1119/1.1501118.  URL https://doi.org/10.1119/1.1501118.
  * Kiss et al. [2008] István Z. Kiss, Yumei Zhai, and John L. Hudson.  Resonance clustering in globally coupled electrochemical oscillators with external forcing.  _Phys. Rev. E_ , 77:046204, Apr 2008.  doi: 10.1103/PhysRevE.77.046204.  URL https://link.aps.org/doi/10.1103/PhysRevE.77.046204.
  * Novikov and Benderskaya [2014] a. V. Novikov and E. N. Benderskaya.  Oscillatory neural networks based on the Kuramoto model for cluster analysis.  _Pattern Recognition and Image Analysis_ , 24(3):365–371, 2014.  ISSN 1054-6618.  doi: 10.1134/S1054661814030146.  URL http://link.springer.com/10.1134/S1054661814030146.
  * Kuramoto [1975] Yoshiki Kuramoto.  Self-entrainment of a population of coupled non-linear oscillators.  In _International Symposium on Mathematical Problems in Theoretical Physics_ , pages 420–422. Springer-Verlag, Berlin/Heidelberg, 1975\.  doi: 10.1007/BFb0013365.  URL http://www.springerlink.com/index/10.1007/BFb0013365.
  * Kuramoto [1984] Yoshiki Kuramoto.  Chemical Waves.  In _Chemical Oscillations, Waves, and Turbulence_ , pages 89–110. Springer Berlin Heidelberg, 1984.  doi: 10.1007/978-3-642-69689-3˙6.  URL http://www.springerlink.com/index/10.1007/978-3-642-69689-3{_}6.
  * Hong and Strogatz [2011] Hyunsuk Hong and Steven H Strogatz.  Kuramoto model of coupled oscillators with positive and negative coupling parameters: an example of conformist and contrarian oscillators.  _Physical Review Letters_ , 106(5):054102, 2011\.
  * Yeung and Strogatz [1999] MK Stephen Yeung and Steven H Strogatz.  Time delay in the kuramoto model of coupled oscillators.  _Physical Review Letters_ , 82(3):648, 1999.
  * Breakspear et al. [2010] Michael Breakspear, Stewart Heitmann, and Andreas Daffertshofer.  Generative models of cortical oscillations: neurobiological implications of the kuramoto model.  _Frontiers in human neuroscience_ , 4:190, 2010.
  * Rodrigues et al. [2016] Francisco A. Rodrigues, Thomas K D M Peron, Peng Ji, and J??rgen Kurths.  The Kuramoto model in complex networks.  _Physics Reports_ , 610:1–98, 2016.  ISSN 03701573.  doi: 10.1016/j.physrep.2015.10.008.  URL http://dx.doi.org/10.1016/j.physrep.2015.10.008.
  * Climaco and Saa [2019] Joyce S. Climaco and Alberto Saa.  Optimal global synchronization of partially forced kuramoto oscillators.  _Chaos: An Interdisciplinary Journal of Nonlinear Science_ , 29(7):073115, 2019.  doi: 10.1063/1.5097847.  URL https://doi.org/10.1063/1.5097847.
  * Gomez-Gardenes et al. [2011] Jesus Gomez-Gardenes, Sergio Gomez, Alex Arenas, and Yamir Moreno.  Explosive synchronization transitions in scale-free networks.  _Physical Review Letters_ , 106(12):1–4, 2011\.  ISSN 00319007.  doi: 10.1103/PhysRevLett.106.128701.
  * Ji et al. [2013] Peng Ji, Thomas K Dm Peron, Peter J. Menck, Francisco A. Rodrigues, and J??rgen Kurths.  Cluster explosive synchronization in complex networks.  _Physical Review Letters_ , 110(21):1–5, 2013\.  ISSN 00319007.  doi: 10.1103/PhysRevLett.110.218701.
  * Acebrón et al. [2005] Juan A. Acebrón, L. L. Bonilla, Conrad J Pérez Vicente, Félix Ritort, and Renato Spigler.  The Kuramoto model: A simple paradigm for synchronization phenomena.  _Reviews of Modern Physics_ , 77(1):137–185, 2005\.  ISSN 00346861.  doi: 10.1103/RevModPhys.77.137.
  * Dörfler and Bullo [2011] Florian Dörfler and Francesco Bullo.  On the critical coupling for kuramoto oscillators.  _SIAM Journal on Applied Dynamical Systems_ , 10(3):1070–1099, 2011.
  * Olmi et al. [2014] Simona Olmi, Adrian Navas, Stefano Boccaletti, and Alessandro Torcini.  Hysteretic transitions in the kuramoto model with inertia.  _Physical Review E_ , 90(4):042905, 2014.
  * Childs and Strogatz [2008] Lauren M. Childs and Steven H. Strogatz.  Stability diagram for the forced Kuramoto model.  _Chaos_ , 18(4):1–9, 2008.  ISSN 10541500.  doi: 10.1063/1.3049136.
  * Moreira and de Aguiar [2019a] Carolina A Moreira and Marcus AM de Aguiar.  Global synchronization of partially forced kuramoto oscillators on networks.  _Physica A: Statistical Mechanics and its Applications_ , 514:487–496, 2019a.
  * Moreira and de Aguiar [2019b] Carolina A Moreira and Marcus AM de Aguiar.  Modular structure in c. elegans neural network and its response to external localized stimuli.  _Physica A: Statistical Mechanics and its Applications_ , 533:122051, 2019b.
  * Chandra et al. [2019b] Sarthak Chandra, Michelle Girvan, and Edward Ott.  Continuous versus discontinuous transitions in the d-dimensional generalized kuramoto model: Odd d is different.  _Physical Review X_ , 9(1):011002, 2019b.
  * Ott and Antonsen [2008] Edward Ott and Thomas M. Antonsen.  Low dimensional behavior of large systems of globally coupled oscillators.  _Chaos_ , 18(3):1–6, 2008.  ISSN 10541500.  doi: 10.1063/1.2930766.
  * Watanabe and Strogatz [1994] Shinya Watanabe and Steven H Strogatz.  Constants of motion for superconducting josephson arrays.  _Physica D: Nonlinear Phenomena_ , 74(3-4):197–253, 1994.
  * Goebel [1995] Charles J. Goebel.  Comment on “constants of motion for superconductor arrays”.  _Physica D: Nonlinear Phenomena_ , 80(1):18 – 20, 1995.  ISSN 0167-2789.  doi: https://doi.org/10.1016/0167-2789(95)90049-7.  URL http://www.sciencedirect.com/science/article/pii/0167278995900497.
  * Marvel et al. [2009] Seth A. Marvel, Renato E. Mirollo, and Steven H. Strogatz.  Identical phase oscillators with global sinusoidal coupling evolve by möbius group action.  _Chaos: An Interdisciplinary Journal of Nonlinear Science_ , 19(4):043104, 2009.  doi: 10.1063/1.3247089.  URL https://doi.org/10.1063/1.3247089.
  * Tanaka [2014] Takuma Tanaka.  Solvable model of the collective motion of heterogeneous particles interacting on a sphere.  _New Journal of Physics_ , 16, 01 2014.  doi: 10.1088/1367-2630/16/2/023016.
  * Hu et al. [2014] Xin Hu, S Boccaletti, Wenwen Huang, Xiyun Zhang, Zonghua Liu, Shuguang Guan, and Choy-Heng Lai.  Exact solution for first-order synchronization transition in a generalized kuramoto model.  _Scientific reports_ , 4(1):1–6, 2014.


# New Chaotic dynamics for Yitang Zhang latest results on Landau-Siegel zero
††thanks: Citation: Authors. Title. Pages…. DOI:000000/11111.

Zeraoulia Rafik
University of batna2 Algeria
Departement of mathematics
Yabous, Khenchela
{Zeraoulia<EMAIL_ADDRESS>

###### Abstract

The first part of this paper is about Consequences resulting from Yitang
Zhang’s latest claimed results on Landau-Siegel zero posted by some
mathematicians in Mathoverflow ,For the second part we are able to derive new
Chaotic dynamics for Yitang Zhang on Landau-Siegel zero such that the behavior
of the new dynamics has been discussed ,Lyaponove Exponents has been computed
and bifurcation diagram has been achieved ,The number of limit cycle and
orbits are predicted.The behavior of this new dynamics roughly proves the
validity of Yitang latest results .

_K_ eywords New discret dynamics $\cdot$ Yitang Zhang latest result $\cdot$
Landau-Siegel zeros

##  1 Introduction

In mathematics, more specifically in the field of analytic number theory, a
Landau–Siegel zero or simply Siegel zero (also known as exceptional zero [1],
named after Edmund Landau and Carl Ludwig Siegel, is a type of potential
counterexample to the generalized Riemann hypothesis, on the zeros of
Dirichlet L-functions associated to quadratic number fields. Roughly speaking,
these are possible zeros very near in a quantifiable sense to $s=1$ ,The way
in which Siegel zeros appear in the theory of Dirichlet L-functions is as
potential exceptions to the classical zero-free regions, which can only occur
when the L-function is associated to a real Dirichlet character. Recently,
Yitang Zhang just gave a virtual talk about his work on Landau-Siegel zeros at
Shandong University on the 5th of November’s morning in China. He will also
give a talk on 8th November at Peking University.He published his preprint on
arXiv . This paper [2] shows that for a real primitive character $\chi$ to the
modulus $D$,

$L(1,\chi)>c_{1}(\log D)^{-2022}$

where $c_{1}>0$ is an absolute, effectively computable constant.

Assuming this result is correct, what are some significant number theoretical
consequences that would follow? ,Namely ,what would be the impact on PNT error
estimates, arithmetic progressions, and other related problems? Does these
results add somethings to the side of dynamical system and chaos theory ? .
more specifically,what is then the form of Yitang Zhang dynamics ? Is it
possible to compute the entropy of this dynamics ?

##  2 First Answer by Travor Liu regarding implications on the error term of
the PNT for arithmetic progressions

It has significant implications on the error term of the PNT for arithmetic
progressions.[3]

PNT and Siegel-Walfisz theorem

Let $\psi(x;q,a)$ be the sum of $\Lambda(n)$ over $n\leq x$ and $n\equiv
a\pmod{q}$. Then the PNT states that for fixed $q$ there is

$\psi(x;q,a)\sim{x\over\varphi(q)}.$ (1)

When $q$ is not fixed, Page (1935) proved the following general result:

###### Theorem 1

(page) There exists some absolute and effective $c_{0}>0$ such that for all
$(a,q)=1$:*

$\psi(x;q,a)={x\over\varphi(q)}-\color[rgb]{0,0,1}{{\chi(a)x^{\beta}}\over\varphi(q)\beta}+O\\{xe^{-c_{0}\sqrt{\log
x}}\\}$ (2)

where $\chi$ denotes the exceptional character and $\beta$ denotes the Siegel
zero. The blue term would be dropped if there are no exceptional characters
modulo $q$.*

To unify the error terms, we require a result due to Siegel (1935):

###### Theorem 2

(Siegel): For all $\varepsilon>0$ there exists some $A_{\varepsilon}>0$ such
that $1-\beta>A_{\varepsilon}q^{-\varepsilon}$.*

Plugging this result into the blue term of (2), we have

$x^{\beta}\ll xe^{-A_{\varepsilon}q^{-\varepsilon}\log x}.$ (3)

If $q\leq(\log x)^{2/\varepsilon}$, then the right hand side becomes $\ll
xe^{-A_{\varepsilon}\sqrt{\log x}}$. Combining this with (2) gives us the
result of Walfisz (1936)

###### Theorem 3

(Siegel-Walfisz):

For any $M>0$ there exists some $C_{M}$ such that for all $q\leq(\log x)^{M}$
and $(a,q)=1$ there is*

$\psi(x;q,a)={x\over\varphi(q)}+O\\{e^{-C_{M}\sqrt{\log x}}\\},$ (4)

where the O-constant is absolute.

Due to the drawback in the proof of Siegel’s theorem, $A_{\varepsilon}$ and
$C_{M}$ are not effectively computable.

Improvements due to Zhang:

However, we can significantly obtain a stronger and effective improvement of
Siegel-Walfisz theorem if Zhang’s result is used. That is

###### Theorem 4

(Zhang) There exists $A>0$ and effective $C_{1}>0$ such that
$L(1,\chi)>C_{1}(\log q)^{-A}$.

Zhang proved this result for $A=2022$, but I choose not to plug it in for
generality. Let $\beta$ be the rightmost real zero of $L(s,\chi)$ for some
real $\chi$ modulo $q$ such that $1-\beta\gg(\log q)^{-1}$. Then it follows
from the mean value theorem that there exists some $1-\beta<\sigma<1$ such
that $1-\beta=L(1,\chi)/L^{\prime}(\sigma,\chi)$. Applying the classical bound
$L^{\prime}(\sigma,\chi)=O(\log^{2}q)$ and Zhang’s result gives us the zero-
free region that

$1-\beta>C_{2}(\log q)^{-A-2},$

where $C_{2}>0$ is effectively computable, which indicates that the blue term
in (2) is dominated by

$x^{\beta}\ll xe^{-C_{2}(\log x)(\log q)^{-A-2}}.$

If $(\log q)^{A+2}\leq\sqrt{\log x}$, then the right hand side becomes $\ll
xe^{-C_{2}\sqrt{\log x}}$, which allows Theorem 1 to be improved
significantly:

###### Theorem 5

Let $A$ be as in Theorem 4. There exists some absolute $c_{0}>0$ such that for
all $q\leq e^{(\log x)^{1/(2A+4)}}$ and $(a,q)=1$, we have*

$\psi(x;q,a)={x\over\varphi(q)}+O\\{e^{-c_{0}\sqrt{\log x}}\\}.$ (5)

Asymptotic formulas valid for all $q\geq 1$.

Although Theorem 5 is much stronger than Theorem 3, it is difficult to compare
them to Theorem 1 without the blue term, so this section is dedicated to
deduce asymptotic formula valid for all $q\geq 1$ and $(a,q)=1$ so that a
better comparison can be made.

Since $\Lambda(n)\leq\log n$, we know trivially that

$\psi(x;q,a)\leq\sum_{\begin{subarray}{c}n\leq x\\\ n\equiv
a(q)\end{subarray}}\log x\ll{x\log x\over q}.$ (6)

Combining this with (3), we see that Theorem 3 indicates that

$\psi(x;q,a)={x\over\varphi(q)}+O_{N}\\{x(\log x)^{-N}\\}\quad(N>0).$ (7)

If the trivial upper bound is juxtaposed with (4), then we see that there
exists some absolute and effective $c_{0}>0$ such that

$\psi(x;q,a)={x\over\varphi(q)}+O\\{xe^{-c_{0}(\log x)^{1/(2A+4)}}\\},$ (8)

which has a substantially better error term than (5).

##  3 Second Answer by Stopple regarding classification of discriminants of
binary quadratic forms

There will be many important consequences of Zhang’s result, if correct. One
specific result is that it will reduce one of the last open problem from the
era of Gauss and Euler to a finite amount of computation, namely the
classification of discriminants of binary quadratic forms with one class per
genus. The congruence class of a prime number $p$ modulo $d$ determines which
form of discriminant $-d<0$ represents $p$ if and only if there is one class
per genus.

Such discriminants which are congruent to $0$ modulo $4$ are Euler’s *numeri
idonei* or idoneal numbers. Euler expected there would be infinitely many such
discriminants. [**Edit**: Apparently I’m mis-remembering this. See the remarks
of KConrad below.] It was Gauss who conjectured that the only such
discriminants are the 65 examples (not necessarily fundamental) known to
Euler. There are also 65 known fundamental discriminants (not necessarily
even) with one class per genus. The existence of a 66th is still an open
problem. By genus theory we know that for discriminants with one class per
genus, the class group satisfies

$\mathcal{C}(-d)\cong\left(\mathbb{Z}/2\right)^{g-1},$

where $g$ is the number of prime divisors of $d$. Obviously $d$ is bigger than
the absolute value of the smallest fundamental discriminant with $g$ prime
divisors,

$d_{g}\overset{\text{def.}}{=}3\cdot 4\cdot 5\cdot 7\cdots p_{g}.$

From lower bounds on the size of $p_{g}$, the $g$-th prime, and on
$\theta(x)=\sum_{p\leq x}\log(p)$, one can show that

$d_{g}>g^{g}.$

Since $2^{g-1}\ll\sqrt{g^{g}},$ lower bounds for the class number which we
expect to be true rule out the possibility for one class per genus for large
$g$.

In 1973, Peter Weinberger showed that on GRH, no fundamental discriminant
$-d<-5460$ has one class per genus, and unconditionally there is at most one
more such $d$.

In contrast, Oesterle explicitly observed that the lower bound due to
Goldfeld-Gross-Zagier is not strong enough to finish the classification of
discriminants with one class per genus: $\log(g^{g})$ is $\ll 2^{g-1}$.
Iwaniec and Kowalski observed that even the full strength of the Birch
Swinnerton-Dyer conjecture, "the best effective lower bounds which current
technology allows us to hope for" would not suffice, as $\log(g^{g})^{r}$ is
$\ll 2^{g-1}$ for any $r$. In fact, the outlook is still more bleak: Watkins
observed that if the discriminant $-d$ is divisible by all the primes up to
$(\log\log d)^{3}$ (as $d_{g}$ certainly is), the product over primes dividing
$d$ in the Goldfeld-Gross-Zagier lower bound is so small the resulting bound
is worse than the trivial bound.

If the implied constant is made explicit, Zhang’s result would eliminate the
possibility of one class per genus for discriminants above some bound. For
example, neglecting the constant, the lower bound is discriminants with more
than 6007 prime divisors. This works out to $d>3\cdot 10^{25734}$.

##  4 Chaotic dynamics for Yitang Zhang latest results on Landau-Siegel zeros

In this section we may give an explicit dynamics [4] for Yitang Zhang latest
results on Landau-Siegel zeros using both of Dirichlet definition which uses
odd and even Characters $\chi$ mod $m$and Theorem 4 of Zhang also we shall
discuss its behavior and we show that dynamics has a weak transition to chaos
for some values and parameters of that discret new dynamics.

###### Corollary 6

For special values $s=1$, let $\chi$ be a primitive real character mod $m$,
put $\xi_{m}=\exp\bigg{(}\frac{2\pi i}{m}\bigg{)}$, put
$K=\mathbb{Q}\bigg{(}\sqrt{\chi(-1)m}\bigg{)}$, $h$ is its class number of
roots of unity in it and $\epsilon$ is its fundamental unit ,Dirichelet L
Function may be defined as :

$L(1,\chi)=\begin{cases}\displaystyle\frac{2\pi h}{w\sqrt{m}}\quad\textit{if}\
\chi(-1)=-1\\\
\displaystyle\frac{2h\log{|\epsilon|}}{w\sqrt{m}}\quad\textit{if}\
\chi(-1)=1\\\ \end{cases}$ (9)

Yitang discrete dynamics maybe derived from the definition of Dirichlet L
function [5] in (9) and Theorem 4 , we may start with the first case
$chi(-1)=-1$ in (9),Using both of Theorem 4 and (9) the 1-D discrete dynamics
can be reformulated as :

$x_{n+1}=\frac{\beta}{\sqrt{x_{n}}}+c\log(x_{n})^{-\alpha},\quad\chi(-1)=-1$
(10)

with $x_{n}=m,n=0,1\cdots$ and $\beta=\frac{2\pi h}{w}$ , $c$ is the constant
of Yitang defined in Theorem 4.

For the second case :$\chi(-1)=1$, Yitang Dynamics can be written as :

$x_{n+1}=\frac{\beta\log(|\epsilon|)}{\pi\sqrt{x_{n}}}+c\log(x_{n})^{-\alpha},\quad\chi(-1)=1$
(11)

For Chaotic behavior the iteration of dynamic 1 (10) and dynamics 2 (11) 50000
times gives the similar behavior as we will show in the following section .

##  5 Analysis and discussion

For $\alpha$ in the range $[0;10]$ and $c$ in the range $[0.0005;0.007]$ ,All
Lyaponove exponents
$(\lambda_{1}<\lambda_{2}<\lambda_{3},\cdots>\lambda_{10}=-0.348<0$) of the
both dynamics are negative lead to the Yitang dynamics being non Chaotic
,However we noticed weak transition to Chaos in the range $\alpha\in(3;4)$ ,
see Figure 1

Figure 1: Lyponov exponents for the dynamics 1 with
$c\in[0.0005;0.007]$,$\alpha\in[0;10]$

Increasing more values of $\alpha$ up to $2022$ and $c\in[0.0005;0.007]$, we
see that the Lyponov exponents are constants approximately equals to $-0.39$
,see Figure 2

Figure 2: Lyponov exponents for the dynamics 1 with
$c\in[0.0005;0.007]$,$\alpha\in[0;2022]$

Transition to Chaos appear strongly for $c\in(10.007;11.07)$ and
$\alpha\in(3;5)$ such that Lyaponove exponents values are positive
$(0\leq\lambda\leq 0.21)$ , the max of Lyaponove exponents up to
$\lambda=0.21$ see Figure 3

Figure 3: Lyponov exponents for the dynamics 1 with
$c\in[10.007;11.007]$,$\alpha\in(0;5)$

Increasing more values of $c$ one can obtain large range for transition to
chaos for Yitang dynamics .The bifurcation diagram of 1.D Yitang dynamics for
some values of $c$ has been plotted ,see Figure 4

Figure 4: Bifurcation diagram for Yitang dynamics

We have discovered period doubling bifurcations by varying the parameter
$\alpha$ in the chaotic range and plotting $x_{n},y_{n}$ versus the iterations
$n$ of the map and then print below the values of the points in order to
distinguish period cycles.

For one dimensional systems of the form $x_{n+1}=f(x_{n})$ the period-$2$
cycles happen when the system:

$x_{1}=f(x_{2}),\quad x_{2}=f(x_{1})$

has a unique solution and it seems that $0$ is the fixed point for Yitang
dynamics (see Table1) with initial point $x_{0}=0.4$. In other words, the
trajetory "jumps" between $x_{1}$ and $x_{2}$ so while iterating the map they
keep appearing (until a certain value of the parameter of the system). see
Figure 5

Figure 5: Yitang dynamics Fixed Points Plot versus
Iterations,$c=100$,$\mu=\alpha\in(0;5)$
$\left(\begin{array}[]{cc}\text{point}&\text{x}\\\ 86&3.55687\\\
87&0.265123\\\ 88&0.971062\\\ 89&4.05719\times 10^{76}\\\
90&\text{2.4823161470320785$\grave{}$*${}^{\wedge}$-39}\\\ 91&1.00356\times
10^{19}\\\ 92&\text{1.5783353198977849$\grave{}$*${}^{\wedge}$-10}\\\
93&39798.8\\\ 94&0.00250631\\\ 95&9.9874\\\ 96&0.158214\\\ 97&1.25704\\\
98&1.07455\times 10^{32}\\\
99&\text{4.823440210703131$\grave{}$*${}^{\wedge}$-17}\\\ 100&7.19932\times
10^{7}\\\ 101&0.0000589283\\\ \end{array}\right)$ Table 1:  Fixed Points
values of Yitang dynamics with iteration of $500$ times

We computed the Lyapunove Exponents of Yitang Dynamics by varying alpha from 0
to 600 we got the following results in table 2 and Plot:

Some lyapunov exponents for alpha in the range $(0,600)$ [7] has been computed
using mathematica as shwon in the following separated tables with plot:

$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
0.&-1.15226\\\ 0.01&-1.11974\\\ 0.02&-1.08835\\\ 0.03&-1.058\\\
0.04&-1.02864\\\ 0.05&-1.00021\\\ 0.06&-0.972639\\\ 0.07&-0.945889\\\
0.08&-0.91991\\\ 0.09&-0.894661\\\ 0.1&-0.870103\\\ 0.11&-0.8462\\\
0.12&-0.822918\\\ 0.13&-0.800227\\\ 0.14&-0.778098\\\ 0.15&-0.756505\\\
0.16&-0.735422\\\ 0.17&-0.714826\\\ 0.18&-0.694697\\\ 0.19&-0.675013\\\
0.2&-0.655756\\\ 0.21&-0.636908\\\ 0.22&-0.618452\\\ 0.23&-0.600373\\\
0.24&-0.582655\\\ 0.25&-0.565286\\\ 0.26&-0.54825\\\ 0.27&-0.531537\\\
0.28&-0.515134\\\ 0.29&-0.499031\\\ 0.3&-0.483215\\\ 0.31&-0.467679\\\
0.32&-0.452411\\\ 0.33&-0.437404\\\ 0.34&-0.422647\\\ 0.35&-0.408134\\\
0.36&-0.393857\\\ 0.37&-0.379807\\\ 0.38&-0.365978\\\ 0.39&-0.352363\\\
0.4&-0.338955\\\ 0.41&-0.325749\\\ 0.42&-0.312739\\\ 0.43&-0.299918\\\
0.44&-0.287281\\\ 0.45&-0.274824\\\ 0.46&-0.262541\\\ 0.47&-0.250427\\\
0.48&-0.238478\\\ 0.49&-0.22669\\\ 0.5&-0.215057\\\ 0.51&-0.203577\\\
0.52&-0.192245\\\ 0.53&-0.181057\\\ 0.54&-0.17001\\\ 0.55&-0.1591\\\
0.56&-0.148324\\\ 0.57&-0.137679\\\ 0.58&-0.127162\\\ 0.59&-0.116769\\\
0.6&-0.106497\\\ 0.61&-0.0963444\\\ 0.62&-0.0863079\\\ 0.63&-0.0763849\\\
\end{array}\right)$ Table 2: Table1
$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
0.64&-0.066573\\\ 0.65&-0.0568697\\\ 0.66&-0.0472728\\\ 0.67&-0.0377802\\\
0.68&-0.02839\\\ 0.69&-0.0191008\\\ 0.7&-0.00991274\\\ 0.71&-0.000848175\\\
0.72&-0.016495\\\ 0.73&-0.0342835\\\ 0.74&-0.0518883\\\ 0.75&-0.0693237\\\
0.76&-0.0866054\\\ 0.77&-0.103749\\\ 0.78&-0.12077\\\ 0.79&-0.137683\\\
0.8&-0.154503\\\ 0.81&-0.171244\\\ 0.82&-0.187921\\\ 0.83&-0.204547\\\
0.84&-0.221137\\\ 0.85&-0.237705\\\ 0.86&-0.254262\\\ 0.87&-0.270825\\\
0.88&-0.287405\\\ 0.89&-0.304016\\\ 0.9&-0.320673\\\ 0.91&-0.337389\\\
0.92&-0.354177\\\ 0.93&-0.371052\\\ 0.94&-0.388027\\\ 0.95&-0.405117\\\
0.96&-0.422336\\\ 0.97&-0.439701\\\ 0.98&-0.457226\\\ 0.99&-0.474931\\\
\end{array}\right)$
Table 3:  $\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents
values}\\\ 1&-0.49284\\\ 1.&-0.49284\\\ 1.01&-0.510944\\\ 1.02&-0.529281\\\
1.03&-0.547868\\\ 1.04&-0.566722\\\ 1.05&-0.58586\\\ 1.06&-0.605304\\\
1.07&-0.625074\\\ 1.08&-0.645192\\\ 1.09&-0.665681\\\ 1.1&-0.686564\\\
1.11&-0.707865\\\ 1.12&-0.729611\\\ 1.13&-0.751828\\\ 1.14&-0.774542\\\
1.15&-0.797783\\\ 1.16&-0.821578\\\ 1.17&-0.845957\\\ 1.18&-0.870949\\\
1.19&-0.896582\\\ 1.2&-0.922885\\\ 1.21&-0.949883\\\ 1.22&-0.9776\\\
1.23&-1.00605\\\ 1.24&-1.03526\\\ 1.25&-1.06522\\\ 1.26&-1.09592\\\
1.27&-1.12735\\\ 1.28&-1.15944\\\ 1.29&-1.19213\\\ 1.3&-1.2253\\\
1.31&-1.25878\\\ 1.32&-1.29232\\\ 1.33&-1.32562\\\ 1.34&-1.35824\\\
1.35&-1.38964\\\ 1.36&-1.41912\\\ 1.37&-1.44583\\\ 1.38&-1.46875\\\
1.39&-1.48674\\\ 1.4&-1.49858\\\ 1.41&-1.50307\\\ 1.42&-1.49913\\\
1.43&-1.48595\\\ 1.44&-1.46309\\\ 1.45&-1.43055\\\ 1.46&-1.38873\\\
1.47&-1.33835\\\ 1.48&-1.28031\\\ 1.49&-1.21558\\\ 1.5&-1.14504\\\
1.51&-1.0694\\\ 1.52&-0.989104\\\ 1.53&-0.904293\\\ 1.54&-0.814681\\\
1.55&-0.719402\\\ 1.56&-0.616619\\\ 1.57&-0.502533\\\ 1.58&-0.367932\\\
1.59&-0.174509\\\ \end{array}\right)$ Table 4:
$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
1.6&0.0881717\\\ 1.61&0.138893\\\ 1.62&0.165551\\\ 1.63&0.202492\\\
1.64&0.222027\\\ 1.65&0.242128\\\ 1.66&0.262391\\\ 1.67&0.262112\\\
1.68&0.291839\\\ 1.69&0.306873\\\ 1.7&0.301594\\\ 1.71&0.332126\\\
1.72&0.326654\\\ 1.73&0.335829\\\ 1.74&0.332551\\\ 1.75&0.309413\\\
1.76&0.349316\\\ 1.77&0.368101\\\ 1.78&0.391869\\\ 1.79&0.3934\\\
1.8&0.387423\\\ 1.81&0.382245\\\ 1.82&0.382253\\\ 1.83&0.38575\\\
1.84&0.386156\\\ 1.85&0.38766\\\ 1.86&0.396544\\\ 1.87&0.405967\\\
1.88&0.415302\\\ 1.89&0.424401\\\ 1.9&0.428945\\\ 1.91&0.431023\\\
1.92&0.435248\\\ 1.93&0.432161\\\ 1.94&0.430948\\\ 1.95&0.428278\\\
1.96&0.432447\\\ 1.97&0.433208\\\ 1.98&0.433657\\\ 1.99&0.437374\\\
2&0.349501\\\ 2.&0.349501\\\ 2.01&0.435275\\\ 2.02&0.435097\\\
2.03&0.438616\\\ 2.04&0.437307\\\ \end{array}\right)$ Table 5:
$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
2.05&0.440017\\\ 2.06&0.440217\\\ 2.07&0.435724\\\ 2.08&0.441471\\\
2.09&0.44623\\\ 2.1&0.447963\\\ 2.11&0.442867\\\ 2.12&0.44119\\\
2.13&0.439389\\\ 2.14&0.443069\\\ 2.15&0.445086\\\ 2.16&0.442277\\\
2.17&0.444541\\\ 2.18&0.443888\\\ 2.19&0.439353\\\ 2.2&0.435632\\\
2.21&0.43046\\\ 2.22&0.423262\\\ 2.23&0.417051\\\ 2.24&0.41479\\\
2.25&0.412113\\\ 2.26&0.413576\\\ 2.27&0.412673\\\ 2.28&0.413888\\\
2.29&0.413859\\\ 2.3&0.414319\\\ 2.31&0.417688\\\ 2.32&0.42042\\\
2.33&0.416738\\\ 2.34&0.413714\\\ 2.35&0.42057\\\ 2.36&0.41715\\\
2.37&0.412618\\\ 2.38&0.412854\\\ 2.39&0.412535\\\ 2.4&0.414816\\\
2.41&0.414834\\\ 2.42&0.413606\\\ 2.43&0.416818\\\ 2.44&0.41411\\\
2.45&0.417853\\\ 2.46&0.416911\\\ \end{array}\right)$ Table 6:
$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
2.47&0.421352\\\ 2.48&0.419916\\\ 2.49&0.424899\\\ 2.5&0.430376\\\
2.51&0.433199\\\ 2.52&0.432542\\\ 2.53&0.428365\\\ 2.54&0.428214\\\
2.55&0.425867\\\ 2.56&0.423466\\\ 2.57&0.421258\\\ 2.58&0.420319\\\
2.59&0.421149\\\ 2.6&0.423574\\\ 2.61&0.421921\\\ 2.62&0.424841\\\
2.63&0.423726\\\ 2.64&0.422148\\\ 2.65&0.420218\\\ 2.66&0.418454\\\
2.67&0.419379\\\ 2.68&0.419718\\\ 2.69&0.41695\\\ 2.7&0.41443\\\
2.71&0.41608\\\ 2.72&0.41517\\\ 2.73&0.410443\\\ 2.74&0.409386\\\
2.75&0.409979\\\ 2.76&0.410878\\\ 2.77&0.407457\\\ 2.78&0.406276\\\
2.79&0.406546\\\ 2.8&0.40539\\\ 2.81&0.413248\\\ 2.82&0.41515\\\
2.83&0.419008\\\ \end{array}\right)$ Table 7:
$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
2.84&0.412071\\\ 2.85&0.411538\\\ 2.86&0.399653\\\ 2.87&0.389743\\\
2.88&0.358811\\\ 2.89&0.271317\\\ 2.9&-0.0709225\\\ 2.91&0.204658\\\
2.92&0.339491\\\ 2.93&0.375873\\\ 2.94&0.392489\\\ 2.95&0.405362\\\
2.96&0.415369\\\ 2.97&0.420442\\\ 2.98&0.423161\\\ 2.99&0.427594\\\
3&0.4321\\\ 3.&0.4321\\\ 3.01&0.435507\\\ 3.02&0.441413\\\ 3.03&0.444938\\\
3.04&0.445524\\\ 3.05&0.448535\\\ 3.06&0.45017\\\ 3.07&0.452593\\\
3.08&0.456965\\\ 3.09&0.454544\\\ 3.1&0.457091\\\ 3.11&0.453177\\\
3.12&0.456127\\\ 3.13&0.45441\\\ 3.14&0.45515\\\ 3.15&0.456125\\\
3.16&0.452111\\\ 3.17&0.4553\\\ 3.18&0.458551\\\ 3.19&0.457554\\\
3.2&0.455247\\\ 3.21&0.45498\\\ 3.22&0.454235\\\ 3.23&0.455603\\\
3.24&0.462065\\\ 3.25&0.469304\\\ 3.26&0.465601\\\ 3.27&0.466889\\\
3.28&0.465339\\\ 3.29&0.465456\\\ 3.3&0.458958\\\ 3.31&0.459289\\\
3.32&0.461778\\\ 3.33&0.463729\\\ 3.34&0.460103\\\ 3.35&0.4594\\\
3.36&0.458898\\\ 3.37&0.462396\\\ 3.38&0.465787\\\ 3.39&0.459454\\\
3.4&0.461186\\\ 3.41&0.461277\\\ 3.42&0.462188\\\ 3.43&0.458111\\\
3.44&0.456439\\\ 3.45&0.426665\\\ 3.46&0.451274\\\ \end{array}\right)$ Table
8:  $\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents
values}\\\ 3.47&0.464362\\\ 3.48&0.466476\\\ 3.49&0.469612\\\ 3.5&0.472629\\\
3.51&0.479088\\\ 3.52&0.479697\\\ 3.53&0.485185\\\ 3.54&0.492687\\\
3.55&0.500078\\\ 3.56&0.502497\\\ 3.57&0.502803\\\ 3.58&0.505774\\\
3.59&0.508774\\\ 3.6&0.508004\\\ 3.61&0.515266\\\ 3.62&0.517609\\\
3.63&0.517886\\\ 3.64&0.523134\\\ 3.65&0.527244\\\ 3.66&0.533924\\\
3.67&0.539092\\\ 3.68&0.54134\\\ 3.69&0.538396\\\ 3.7&0.544749\\\
3.71&0.547773\\\ 3.72&0.549363\\\ 3.73&0.551562\\\ 3.74&0.553626\\\
3.75&0.556052\\\ 3.76&0.557295\\\ 3.77&0.56001\\\ 3.78&0.567793\\\
3.79&0.568275\\\ 3.8&0.569386\\\ 3.81&0.571111\\\ 3.82&0.572617\\\
3.83&0.575607\\\ 3.84&0.578082\\\ 3.85&0.58003\\\ 3.86&0.577699\\\
3.87&0.578298\\\ 3.88&0.579757\\\ 3.89&0.579385\\\ 3.9&0.580589\\\
3.91&0.580969\\\ 3.92&0.582456\\\ 3.93&0.576281\\\ 3.94&0.576627\\\
\end{array}\right)$ Table 9:
$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
3.95&0.573432\\\ 3.96&0.574313\\\ 3.97&0.57521\\\ 3.98&0.57762\\\
3.99&0.576215\\\ 4&0.507632\\\ 4.&0.507632\\\ 4.01&0.576928\\\
4.02&0.579918\\\ 4.03&0.583196\\\ 4.04&0.580391\\\ 4.05&0.580994\\\
4.06&0.582307\\\ 4.07&0.582366\\\ 4.08&0.583606\\\ 4.09&0.585411\\\
4.1&0.584807\\\ 4.11&0.584631\\\ 4.12&0.58586\\\ 4.13&0.586655\\\
4.14&0.590236\\\ 4.15&0.591261\\\ 4.16&0.591472\\\ 4.17&0.591893\\\
4.18&0.589417\\\ \end{array}\right)$ Table 10:
$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
4.19&0.58858\\\ 4.2&0.587385\\\ 4.21&0.586986\\\ 4.22&0.586748\\\
4.23&0.588275\\\ 4.24&0.584422\\\ 4.25&0.585339\\\ 4.26&0.583006\\\
4.27&0.582097\\\ 4.28&0.581886\\\ 4.29&0.582088\\\ 4.3&0.581241\\\
4.31&0.581622\\\ 4.32&0.581417\\\ 4.33&0.580996\\\ 4.34&0.579168\\\
4.35&0.581486\\\ 4.36&0.580001\\\ 4.37&0.575849\\\ 4.38&0.569656\\\
4.39&0.569377\\\ 4.4&0.569891\\\ 4.41&0.567187\\\ 4.42&0.570189\\\
4.43&0.570313\\\ 4.44&0.571483\\\ 4.45&0.571226\\\ 4.46&0.574592\\\
4.47&0.573314\\\ 4.48&0.571783\\\ 4.49&0.571822\\\ 4.5&0.569118\\\
4.51&0.568564\\\ 4.52&0.567916\\\ 4.53&0.568978\\\ 4.54&0.570392\\\
4.55&0.572938\\\ 4.56&0.571468\\\ 4.57&0.572372\\\ 4.58&0.571267\\\
4.59&0.572128\\\ 4.6&0.569822\\\ 4.61&0.571771\\\ 4.62&0.574242\\\
4.63&0.576563\\\ 4.64&0.578604\\\ 4.65&0.582042\\\ 4.66&0.581638\\\
4.67&0.58133\\\ 4.68&0.583655\\\ 4.69&0.584481\\\ 4.7&0.586956\\\
4.71&0.58891\\\ 4.72&0.58939\\\ 4.73&0.589091\\\ 4.74&0.590923\\\
4.75&0.591047\\\ 4.76&0.588998\\\ 4.77&0.590917\\\ 4.78&0.588484\\\
4.79&0.589111\\\ \end{array}\right)$ Table 11:
$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
4.8&0.590603\\\ 4.81&0.590248\\\ 4.82&0.587849\\\ 4.83&0.586583\\\
4.84&0.584447\\\ 4.85&0.579494\\\ 4.86&0.579748\\\ 4.87&0.57695\\\
4.88&0.5767\\\ 4.89&0.571527\\\ 4.9&0.575459\\\ 4.91&0.569518\\\
4.92&0.57061\\\ 4.93&0.569778\\\ 4.94&0.570774\\\ 4.95&0.571756\\\
4.96&0.572801\\\ 4.97&0.570044\\\ 4.98&0.568464\\\ 4.99&0.567003\\\
5&0.564626\\\ 6&0.510146\\\ 7&0.556004\\\ 8&-0.0719482\\\ 9&0.534278\\\
10&0.522649\\\ 11&0.517388\\\ 12&0.472097\\\ 13&0.492431\\\ 14&0.446779\\\
15&0.460998\\\ 16&0.459563\\\ 17&0.458949\\\ 18&0.447865\\\ 19&0.405761\\\
20&-0.0390125\\\ 21&0.388358\\\ 22&0.365367\\\ 23&0.379947\\\ 24&0.337853\\\
25&0.385089\\\ 26&0.377315\\\ 27&0.393322\\\ 28&0.384273\\\ 29&0.404729\\\
30&0.394806\\\ 31&0.392084\\\ 32&0.35981\\\ 33&0.361309\\\ 34&0.335672\\\
35&0.325758\\\ 36&0.332982\\\ 37&0.326919\\\ 38&0.334096\\\ 39&0.325021\\\
40&0.31735\\\ 41&0.314913\\\ 42&0.315999\\\ 43&0.304623\\\ 44&0.297582\\\
45&0.310218\\\ \end{array}\right)$ Table 12:
$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
46&0.31658\\\ 47&0.312856\\\ 48&0.304048\\\ 49&0.319614\\\ 50&0.318441\\\
51&0.326547\\\ 52&0.326991\\\ 53&0.333725\\\ 54&0.336305\\\ 55&0.325113\\\
56&0.319471\\\ 57&0.284135\\\ 58&-0.0551551\\\ 59&0.250207\\\ 60&0.052976\\\
61&0.268896\\\ 62&0.243243\\\ 63&0.277856\\\ 64&0.276556\\\ 65&0.272771\\\
66&0.261744\\\ 67&0.268161\\\ 68&0.268744\\\ 69&0.261705\\\ 70&0.250056\\\
71&0.261992\\\ 72&0.262499\\\ 73&0.250503\\\ 74&0.252945\\\ 75&0.249279\\\
76&0.248871\\\ 77&0.246984\\\ 78&0.253233\\\ 79&0.243082\\\ 80&0.222819\\\
81&0.242301\\\ 82&0.235616\\\ 83&0.244237\\\ 84&0.261487\\\ 85&0.25179\\\
86&0.25768\\\ 87&0.253262\\\ 88&0.258194\\\ 89&0.253025\\\ 90&0.244901\\\
91&0.252182\\\ 92&0.261088\\\ 93&0.26153\\\ 94&0.283992\\\ 95&0.268048\\\
96&0.278632\\\ 97&0.269725\\\ 98&0.265437\\\ 99&0.278103\\\
\end{array}\right)$ Table 13:
$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
100&0.294507\\\ 101&0.280263\\\ 102&0.338833\\\ 103&0.282768\\\
104&0.351269\\\ 105&0.271341\\\ 106&0.28702\\\ 107&0.258217\\\ 108&0.298877\\\
109&0.237881\\\ 110&0.267863\\\ 111&0.243872\\\ 112&0.312845\\\
113&0.243733\\\ 114&0.341542\\\ 115&0.248787\\\ 116&0.317934\\\
117&0.247616\\\ 118&0.361362\\\ 119&0.245811\\\ 120&0.270793\\\
121&0.238476\\\ 122&0.264539\\\ 123&0.240243\\\ 124&0.241082\\\
125&0.237228\\\ 126&0.306439\\\ 127&0.240514\\\ 128&0.253423\\\
129&0.233118\\\ 130&0.363916\\\ 131&0.235455\\\ 132&0.236588\\\
133&0.232666\\\ 134&0.239603\\\ 135&0.228082\\\ 136&-0.000682791\\\
137&0.221024\\\ 138&0.248629\\\ 139&0.233006\\\ 140&0.249864\\\
141&0.218172\\\ 142&0.246957\\\ 143&0.222976\\\ 144&0.330647\\\
145&0.222045\\\ 146&0.227534\\\ 147&0.209626\\\ 148&0.224004\\\
149&0.209111\\\ 150&0.238807\\\ 151&0.207961\\\ 152&0.23251\\\ 153&0.204123\\\
154&0.258405\\\ 155&0.203154\\\ 156&0.261367\\\ \end{array}\right)$ Table 14:
$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
157&0.200024\\\ 158&0.276927\\\ 159&0.212927\\\ 160&0.227649\\\
161&0.205282\\\ 162&0.244713\\\ 163&0.215555\\\ 164&0.224634\\\
165&0.210837\\\ 166&0.218134\\\ 167&0.213816\\\ 168&0.25477\\\ 169&0.209253\\\
170&0.338271\\\ 171&0.219416\\\ 172&0.318408\\\ 173&0.209499\\\
174&0.274409\\\ 175&0.207832\\\ 176&0.278909\\\ 177&0.219375\\\
178&0.219368\\\ 179&0.21559\\\ 180&0.266607\\\ 181&0.217853\\\ 182&0.228786\\\
183&0.225598\\\ 184&0.23511\\\ 185&0.231376\\\ 186&0.256866\\\ 187&0.245295\\\
188&0.230966\\\ 189&0.244057\\\ \end{array}\right)$ Table 15:
$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
190&0.267777\\\ 191&0.250117\\\ 192&0.302975\\\ 193&0.255108\\\
194&0.285148\\\ 195&0.257763\\\ 196&0.28807\\\ 197&0.243462\\\ 198&0.318595\\\
199&0.203865\\\ 200&-0.000811814\\\ 201&0.0151214\\\ 202&-0.000277031\\\
203&0.200983\\\ 204&0.0421974\\\ 205&0.246805\\\ 206&0.238059\\\
207&0.22248\\\ 208&0.259055\\\ 209&0.240073\\\ 210&0.244049\\\ 211&0.235992\\\
212&0.214536\\\ 213&0.23179\\\ 214&0.227318\\\ 215&0.224653\\\ 216&0.265822\\\
217&0.213538\\\ 218&0.222347\\\ 219&0.238976\\\ 220&0.261832\\\
221&0.234598\\\ 222&0.266934\\\ 223&0.229075\\\ 224&0.204892\\\
225&0.243175\\\ 226&0.256176\\\ 227&0.216407\\\ 228&0.311056\\\
229&0.212253\\\ 230&0.245005\\\ 231&0.212124\\\ 232&0.240131\\\
233&0.222259\\\ 234&0.254617\\\ 235&0.199404\\\ 236&0.212442\\\
237&0.220351\\\ 238&0.260493\\\ 239&0.221093\\\ 240&0.219656\\\
241&0.205173\\\ \end{array}\right)$ Table 16:
$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
242&0.2216\\\ 243&0.192342\\\ 244&0.250546\\\ 245&0.204687\\\ 246&0.262024\\\
247&0.191431\\\ 248&0.268765\\\ 249&0.204364\\\ 250&0.239521\\\
251&0.198227\\\ 252&0.24595\\\ 253&0.207629\\\ 254&0.213018\\\ 255&0.203958\\\
256&0.230494\\\ 257&0.202481\\\ 258&0.235047\\\ 259&0.196655\\\
260&0.206387\\\ 261&0.236002\\\ 262&0.26797\\\ 263&0.213241\\\ 264&0.235129\\\
265&0.226122\\\ 266&0.220672\\\ 267&0.182704\\\ 268&0.21489\\\ 269&0.229958\\\
270&0.222978\\\ 271&0.198288\\\ 272&0.233587\\\ 273&0.223525\\\
274&0.198087\\\ 275&0.203532\\\ 276&0.253099\\\ 277&0.186744\\\
278&0.217099\\\ 279&0.248554\\\ 280&0.201609\\\ 281&0.213417\\\
282&0.215235\\\ 283&0.199976\\\ 284&0.286658\\\ 285&0.216406\\\
286&0.231279\\\ 287&0.181706\\\ 288&0.219251\\\ 289&0.201476\\\
290&0.292975\\\ 291&0.203687\\\ 292&0.197124\\\ 293&0.209663\\\
294&0.194105\\\ \end{array}\right)$
Table 17:  $\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents
values}\\\ 295&0.174838\\\ 296&0.218176\\\ 297&0.20286\\\ 298&0.229976\\\
299&0.200354\\\ 300&0.232252\\\ 301&0.193725\\\ 302&0.221419\\\
303&0.188222\\\ 304&0.204547\\\ 305&0.200112\\\ 306&0.218766\\\
307&0.237877\\\ 308&0.220502\\\ 309&0.182346\\\ 310&0.210023\\\
311&0.239338\\\ 312&0.188054\\\ 313&0.221709\\\ 314&0.209987\\\
315&0.177275\\\ 316&0.255036\\\ 317&0.224455\\\ 318&0.238405\\\
319&0.185702\\\ 320&0.231805\\\ 321&0.211784\\\ 322&0.210903\\\
323&0.248652\\\ 324&0.274243\\\ 325&0.238526\\\ 326&0.227284\\\
327&0.211009\\\ 328&0.222291\\\ 329&0.211565\\\ 330&0.230572\\\
\end{array}\right)$ Table 18:
$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
331&0.20134\\\ 332&0.219904\\\ 333&0.213297\\\ 334&0.219925\\\ 335&0.209362\\\
336&0.266719\\\ 337&0.215589\\\ 338&0.249748\\\ 339&0.261866\\\
340&0.236032\\\ 341&0.247475\\\ 342&0.215065\\\ 343&0.207613\\\
344&0.229582\\\ 345&0.21395\\\ 346&0.227397\\\ 347&0.227974\\\ 348&0.260846\\\
349&0.194388\\\ 350&0.183481\\\ 351&0.201502\\\ 352&0.211913\\\
353&0.229533\\\ 354&0.228183\\\ 355&0.190445\\\ 356&0.238333\\\
357&0.219304\\\ 358&0.251103\\\ 359&0.241116\\\ 360&0.219417\\\
361&0.202869\\\ 362&0.224552\\\ \end{array}\right)$ Table 19:
$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
363&0.230416\\\ 364&0.240793\\\ 365&0.223057\\\ 366&0.260618\\\
367&0.226144\\\ 368&0.291342\\\ 369&0.230966\\\ 370&0.267265\\\
371&0.290253\\\ 372&0.249541\\\ 373&0.240764\\\ 374&0.272625\\\
375&0.219788\\\ 376&0.196459\\\ 377&0.286189\\\ 378&0.335753\\\
379&0.304186\\\ 380&0.304906\\\ 381&0.300879\\\ 382&0.328899\\\
383&0.313488\\\ 384&0.275138\\\ 385&0.320551\\\ 386&0.356407\\\
387&0.298877\\\ 388&0.112348\\\ 389&0.289324\\\ 390&0.0848054\\\
391&0.302225\\\ 392&0.207087\\\ 393&0.38845\\\ 394&3.251\\\ 395&0.416102\\\
396&0.353294\\\ 397&0.302942\\\ 398&0.315576\\\ 399&0.335486\\\
400&0.331842\\\ 401&0.286329\\\ 402&0.285197\\\ 403&0.34339\\\ 404&0.295304\\\
405&0.299275\\\ 406&0.768472\\\ 407&0.369837\\\ \end{array}\right)$
Table 20:  $\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents
values}\\\ 408&0.349883\\\ 409&0.31186\\\ 410&0.777299\\\ 411&0.312361\\\
412&0.331389\\\ 413&0.284097\\\ 414&0.28867\\\ 415&0.310068\\\ 416&0.304586\\\
417&0.271222\\\ 418&0.221264\\\ 419&0.324382\\\ 420&0.283068\\\
421&0.253938\\\ 422&0.316774\\\ 423&0.294786\\\ 424&0.296038\\\
425&0.262009\\\ 426&0.871476\\\ 427&0.333812\\\ 428&0.221338\\\
429&0.316276\\\ 430&0.299743\\\ 431&0.34594\\\ 432&0.279604\\\ 433&0.271286\\\
434&0.270754\\\ 435&0.258876\\\ 436&0.285789\\\ 437&0.243558\\\
438&0.280651\\\ 439&0.273258\\\ 440&0.365818\\\ 441&0.262362\\\ 442&1.08994\\\
443&0.267152\\\ 444&0.265681\\\ 445&0.272178\\\ \end{array}\right)$ Table 21:
$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
446&0.147717\\\ 447&0.266351\\\ 448&0.310172\\\ 449&0.290583\\\
450&0.183513\\\ 451&0.275272\\\ 452&0.325405\\\ 453&0.276304\\\
454&0.185019\\\ 455&0.361066\\\ 456&0.34653\\\ 457&0.283958\\\ 458&0.275742\\\
459&0.323037\\\ 460&0.326334\\\ 461&0.289564\\\ 462&0.343046\\\
463&0.322932\\\ 464&0.59162\\\ 465&0.26335\\\ 466&0.293619\\\ 467&0.267087\\\
468&0.31462\\\ 469&0.351324\\\ 470&0.411217\\\ 471&0.298389\\\ 472&0.721084\\\
473&0.280495\\\ 474&0.308878\\\ 475&0.297671\\\ 476&0.272683\\\
477&0.295723\\\ 478&0.293\\\ 479&0.334672\\\ 480&0.397376\\\
\end{array}\right)$
Table 22:  $\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents
values}\\\ 481&0.317711\\\ 482&0.238976\\\ 483&0.308637\\\ 484&0.296584\\\
485&0.263766\\\ 486&0.248706\\\ 487&0.254955\\\ 488&0.250237\\\
489&0.348082\\\ 490&0.167979\\\ 491&0.304352\\\ 492&0.448908\\\
493&0.291155\\\ 494&0.313148\\\ 495&0.265916\\\ 496&0.19097\\\ 497&0.287631\\\
498&0.369616\\\ 499&0.233645\\\ 500&0.345756\\\ 501&0.369985\\\ 502&0.30685\\\
503&0.255885\\\ 504&0.569668\\\ 505&0.289266\\\ 506&0.262803\\\ 507&0.31031\\\
508&0.338266\\\ 509&0.351184\\\ 510&0.390376\\\ 511&0.287916\\\ 512&0.27619\\\
513&0.273717\\\ 514&0.551982\\\ 515&0.306474\\\ 516&0.316648\\\
517&0.275735\\\ 518&0.377191\\\ 519&0.301323\\\ 520&0.295325\\\
521&0.329195\\\ \end{array}\right)$ Table 23:
$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
522&0.301517\\\ 523&0.355049\\\ 524&0.45739\\\ 525&0.267099\\\ 526&0.336464\\\
527&0.232015\\\ 528&0.302149\\\ 529&0.261932\\\ 530&0.293415\\\
531&0.365807\\\ 532&0.153578\\\ 533&0.316334\\\ 534&0.340326\\\
535&0.274738\\\ 536&0.368267\\\ 537&0.320209\\\ 538&0.60179\\\ 539&0.264732\\\
540&0.282688\\\ 541&0.220575\\\ 542&0.403453\\\ 543&0.304669\\\
544&0.326846\\\ 545&0.305038\\\ 546&0.390608\\\ 547&0.313079\\\
548&0.270487\\\ 549&0.349348\\\ 550&0.278621\\\ 551&0.29754\\\ 552&0.349668\\\
553&0.292467\\\ 554&0.258608\\\ 555&0.305177\\\ 556&0.153286\\\
557&0.154885\\\ 558&0.50595\\\ 559&0.315583\\\ 560&0.146772\\\ 561&0.299072\\\
\end{array}\right)$ Table 24:
$\left(\begin{array}[]{cc}\text{$\alpha$}&\text{Lyapunov exponents values}\\\
562&0.433062\\\ 563&0.253575\\\ 564&0.197195\\\ 565&0.295641\\\
566&0.348164\\\ 567&0.307269\\\ 568&0.309574\\\ 569&0.364853\\\
570&0.256416\\\ 571&0.314278\\\ 572&0.138215\\\ 573&0.33564\\\ 574&0.301685\\\
575&0.291345\\\ 576&0.26482\\\ 577&0.380644\\\ 578&0.152456\\\ 579&0.317665\\\
580&0.397127\\\ 581&0.327612\\\ 582&0.244467\\\ 583&0.305\\\ 584&0.247417\\\
585&0.301816\\\ 586&0.227344\\\ 587&0.296792\\\ 588&0.721956\\\
589&0.337911\\\ 590&0.257894\\\ 591&0.28919\\\ 592&0.298331\\\ 593&0.268303\\\
594&0.337063\\\ 595&0.319528\\\ 596&0.43679\\\ 597&0.306326\\\ 598&0.477708\\\
599&0.28166\\\ 600&0.348162\\\ \end{array}\right)$ Table 25:

The above listed lyapunov exponents are interpreted in the following plot (see
Figure 6:

Figure 6: Lyponov exponents for the dynamics 1 ,$\alpha\in[0;600]$

Yitang in [6] established a relationship between the existence of the Landau-
Siegel zero of $L(s,\chi)$ and the distribution of zeros of the Dirichlet
$L$-function $L(s,\psi)$, with $\psi$ belonging to a set $\Psi$ of primitive
characters, in a region $\Omega$. He showed that if the Landau-Siegel zero
exists (equivalently, $L(1,\chi)$ is small),Now assume that our dynamics is
reformulated correctly ,our computations showed that dynamics is chaotic ,now
we may consider the analytical solutions of Discrete dynamic 1 (Yitang
Dynamics) are exactly the zeros of $L(1,\chi)$ , The above numerical evidence
showed that the Yitang discrete dynamics has a fixed point $\sigma$ close to
$0$ but not exactly $0$ for $L(1,\chi)$ ($\alpha$ large ), where the system is
non chaotic for $\alpha$ large enough , Assume that $1-\beta<\sigma<1$
,follows from the mean value theorem (since the dynamics is non linear)
,$1-\beta=L(1,\chi)/L^{\prime}(\sigma,\chi)$. Applying the classical bound
$L^{\prime}(\sigma,\chi)=O(\log^{2}q)$ and Zhang’s result (Theorem 4) gives us
the zero-free region that

$1-\beta>C_{2}(\log q)^{-A-2},$

and mayeb this will influence the zeros of other $L$-functions, forcing them
to lie on the critical line (which is expected) and to be very regularly
spaced because for large $\alpha$ the dynamics is non chaotic lead to the
zeros of $L(s,\chi)$ uniforme distributed .

##  6 Conclusion

We have discussed a new discrete dynamical system that affirms the results
obtained by Yitang such that the computation of Lyapunove exponents of the new
dynamics showed us that no transition to chaos for large values of $\alpha$
which means no chaos for small $L(1,\chi)$ and there is transition to chaos
for small $\alpha$ in small range .The behavior of this new dynamics roughly
proves the validity of Yitang latest results on the Landau-Siegel zero .

## Acknowledgments

This research paper is a voluntary work for researchers who do not use the
Math- Overflow website, as it summarized some of the results and showed the
significant positive progress in the case of the validity of the results
obtained by Yitang Zhang about the Landau zero, while this paper highlighted a
new discrete system whose chaotic behavior roughly proves of Yitang latest
results

## References

  * [1] Landau, E. (1918). "Über die Klassenzahl imaginär-quadratischer Zahlkörper". Göttinger Nachrichten (in German): 285–295.
  * [2] Yitang Zhang,Discrete mean estimates and the Landau-Siegel zero, arXiv:2211.02515 , https://arxiv.org/abs/2211.02515
  * [3] Consequences resulting from Yitang Zhang’s latest claimed results on Landau-Siegel zeros, https://mathoverflow.net/q/433949/51189
  * [4] Guégan, D. (1997). Nonparametric Methods for Time Series and Dynamical Systems. In: Babu, G.J., Feigelson, E.D. (eds) Statistical Challenges in Modern Astronomy II. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1968-2_17
  * [5] Paul Voutier,Introduction to L-functions: Dirichlet L-functions,CIMPA-ICTP Research School, Nesin Mathematics Village June 2017
  * [6] Yitang Zhang ,On the Landau-Siegel Zeros Conjecture, arXiv:0705.4306 , https://arxiv.org/abs/0705.4306
  * [7] zeraoulia rafik ,https://mathoverflow.net/q/444189/51189