††∗ Corresponding author<EMAIL_ADDRESS>
# Similarity signature curves for forming periodic orbits in the Lorenz system

Jindi Li  Jindi Li
Department of Mathematics, Northeastern University, Shenyang, 110819, P.R.
China<EMAIL_ADDRESS>and  Yun Yang∗ Yun Yang
Department of Mathematics, Northeastern University, Shenyang, 110819, P.R.
China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we systematically investigate the short periodic orbits of the
Lorenz system by the aid of the similarity signature curve, and a novel method
to find the short-period orbits of the Lorenz system is proposed. The
similarity invariants are derived by the equivariant moving frame theory and
then the similarity signature curve occurs along with them. The similarity
signature curve of the Lorenz system presents a more regular behavior than the
original one. By combining the sliding window method, the quasi-periodic
orbits can be detected numerically, all periodic orbits with period
$p\leqslant 8$ in the Lorenz system are found, and their period lengths and
symbol sequences are calculated.

###### Key words and phrases:

Lorenz system; similarity signature curve; periodic orbit; sliding window
method

##  1\. Introduction

The Lorenz system was initially derived from the Oberbeck-Boussinesq
approximation by Lorenz in 1963 [23]. Since then, chaos, as an interesting
phenomenon in nonlinear dynamical systems has become a widely-studied topic of
great interest to specialists and non-specialists alike. Tucker strictly
proved the existence of the Lorenz attractor [33], which laid the foundation
for subsequent research. In fact, there are infinite periodic orbits in
chaotic attractors, and unstable periodic orbits can be used to characterize
the chaotic state of the system. Therefore, exploring the existence and
position of periodic orbits in chaotic systems attracts the researchers’
attention.

In chaos, finding all possible symmetries, i.e., self-equivalences or self-
congruences provides an effective way to understand the mathematical
properties of the whole system. Generally, the efficient solution of an
equivalence problem rests on investigating some related invariants. In fact,
invariants supply us with a moduli space of a certain kind of geometric object
under group transformations. To obtain the invariants of submanifolds for
general transformation groups, Fels and Olver formulated a new, powerful,
constructive approach to the equivariant moving frame theory [10]. Let $M$ be
a smooth, meaning $C^{\infty}$ manifold with a sufficiently high order jet
bundle, and suppose $G$ is a Lie group acting smoothly on $M$ via
prolongation. The practical construction of a moving frame is based on a
choice of cross section $K\subset M$ to the group orbits [10, 30]. For
example, the right-equivariant moving frame map $\rho:M\rightarrow G$ is
locally uniquely obtained by solving the normalization equations $g\cdot z\in
K$ for the group parameters $g=\rho(z)$ in terms of the point $z\in M$. Then
substituting the moving frame formulae $g=\rho(z)$ into the unnormalized
components leads to the fundamental invariants $I_{1}(z),\cdots,I_{m-r}(z)$,
where $r$ and $m$ denote the dimensions of $G$ and $M$, respectively. Then
every invariant $I(z)$ can be locally uniquely expressed as a function of the
fundamental invariants [29].

According to Cartan’s main idea, two regular submanifolds are (locally)
equivalent if and only if they have identical syzygies among all their
differential invariants. More generally, as a consequence of the Fundamental
Basis Theorem [28], one can verify that, for any Lie group action, the entire
algebra of differential invariants can be generated from a finite number of
low order invariants by repeated invariant differentiation. These typically
include the generating differential invariants $I^{1},I^{2},\cdots I^{l}$ as
well as a certain finite collection of their invariant derivatives
$I^{\nu}_{,J}$. These differential invariants serve to define a signature map
$\sigma:S\to\Sigma\subset{\mathbb{R}}^{N}$ whose image is a differential
invariant signature of the original submanifold $S$. Under certain regularity
assumptions, the signature solves the equivalence problem: two $p$-dimensional
submanifolds are locally equivalent under the transformation group if and only
if they have identical signatures [30].

The signature curve $\Sigma\subset{\mathbb{R}}^{2}$ of the Euclidean plane
curve $C\subset{\mathbb{R}}^{2}$ is parametrized by the two lowest order
differential invariants [18]

$\chi:C\rightarrow\Sigma=\left\\{\left(\kappa,\displaystyle{\frac{d\kappa}{ds}}\right)\right\\}\subset{\mathbb{R}}^{2}.$

Similarly, the differential invariant signature for the Euclidean space curve
$C\subset{\mathbb{R}}^{3}$ is defined by

$\Sigma=\left\\{(\kappa,\kappa_{s},\tau)\right\\}\subset{\mathbb{R}}^{3},$

where $\kappa$ and $\tau$ are the curvature and torsion of the curve $C$,
respectively.

The above-mentioned two types of the signature curve are invariant to both the
translation and rotation of the curve. This concept has received considerable
attentions in computer vision, mostly in the issues of object recognition and
symmetry detection. Calabi et al. introduced a new paradigm, the
differentially invariant signature curve or manifold, for the invariant
recognition of visual objects, and discussed various aspects of the numerical
computation of signatures and their applications [6]. Boutin gave the
signature formulas which are invariant under the action of the Euclidean group
and affine group respectively. Another important property of the signature
curve is that it is valid for any fine partition of a given curve [3]. Hoff
and Olver proposed an automated solution to two-dimensional picture puzzles
using the signature curve [17]. Bruckstein and Snaked presented a general
framework for skew-symmetric detection, based on invariant planar curve
descriptions and successfully applied it to mirror-symmetric polygons
distorted by affine and projective viewing transformations [5]. The similarity
signature curve is a generalization of the signature curve under similarity
transformation, which exhibits invariance to translation, rotation, and
scaling. Bruckstein and Netravali generalized the curvature versus arc length
representation that is invariant under Euclidean motions to general geometric
viewing transformations, such as similarity and affine transformations. Such
results can simplify the problem of model-based planar object recognition
under partial occlusion to the matching of locally invariant signature
functions [4]. Ghorbel et al. completed image reconstruction using the
similarity signature curve [15]. As a matter of fact, the Lorenz attractor
exhibits fractal structure and self-similarity, and simpler structures can be
modeled using the similarity signature curve.

Periodic orbit theory is a powerful tool for analyzing the dynamic behavior of
chaotic systems, which can effectively calculate the average value of physical
quantities of dynamic systems [2, 1]. Many methods for finding periodic orbits
have been proposed. The interval method provides a simple computational test
for the uniqueness, existence, and nonexistence of the zeros of a map within a
given interval vector [20] , and supplies research support for the possibility
of complete analysis of piecewise continuous systems by combination with the
Poincaré map [12]. Galias and Zgliczyński introduced the Krawczyk operator to
prove the existence of periodic orbits of infinite-dimensional discrete
dynamical systems and found all periodic orbits of a given period [14]. In
addition, the traditional approach is to find periodic orbits by a general
technique based on the construction of a graph describing the dynamics of the
system. Galias and Tucker successfully studied the existence of periodic
orbits for a particular type of system by using symbolic dynamics combined
with the interval method [13]. Moreover, a variational principle was utilized
to determine the periodic orbits of a class of continuous systems and unstable
spatiotemporally periodic solutions of extended systems, such as the Rössler
attractor [11].

It has also been suggested that it is difficult to obtain an efficient set of
Poincaré sections when finding the periodic orbits of turbulent or high-
dimensional flows. In response to this situation, some people proposed the
variational method. Lan replaced the Poincaré section with the variational
method and verified the existence of periodic orbits in the numerical
calculation [21].

In this paper, we present a method that can be employed to model periodic
orbits of the Lorenz system, which relies on the similarity signature curve of
the system, the interval method, and the sliding window method. The similarity
signature curve of the system is calculated, and then the trajectory of the
Lorenz system is segmented by the sliding window method. Each quasi-periodic
orbit is verified by using the Krawczyk operator. This makes it possible to
find periodic orbits in the trajectories between the segmentation points.

The structure of this paper is as follows. In Sec.2, we introduce the Lorenz
equation which sketches the chaotic behavior, together with the typical
parameters. On the other hand, the similarity invariants are derived by using
the equivariant moving frame theory, and the similarity signature curve of the
Lorenz attractor is described. In Sec. 3, the sliding window method is
improved, and the algorithm based on the similarity signature curve to segment
trajectories is proposed, which can obtain the sequence of segmentation
points. The trajectories between these points form quasi-periodic orbits. In
Sec. 4, all periodic orbits with period $p\leqslant 8$ in the Lorenz system
are shown, and the period number, period length, and motion trajectory of each
periodic orbit are listed. The results of this paper are summarized in Sec. 5.

### Acknowledgements

This work was supported by the Fundamental Research Funds for the Central
Universities under grant-N2104007, and the second author would also like to
express his deep gratitude to Professor Peter J. Olver for his encouragement
and help during his stay in School of Mathematics, University of Minnesota as
a Visiting Professor, while part of this work was completed.

##  2\. Lorenz system and its similarity signature curve

###  2.1. Lorenz system dynamics

The Lorenz system, as the benchmark system for chaotic dynamics, is the first
chaotic dissipative system discovered in numerical experiments. In this
section, we discuss the relationship between the similarity signature curve
and the periodic orbit. In details, the Lorenz equations can be described as a
system of three differential equations, namely,

$\left\\{\begin{array}[]{l}\dot{x}\left(t\right)=\sigma\big{(}{y\left(t\right)-x\left(t\right)}\big{)},\\\
\dot{y}\left(t\right)=rx\left(t\right)-y\left(t\right)-x\left(t\right)z\left(t\right),\\\
\dot{z}\left(t\right)=x\left(t\right)y\left(t\right)-\eta
z\left(t\right),\end{array}\right.$

where $\dot{x}$ denotes the derivative of $x$ with respect to time $t$. The
positive parameters $\sigma$, $\eta$, and $r$ raise up from the physical
context and Lorenz chose in his original work

$\sigma=10$, $\eta=\displaystyle{\frac{{\text{8}}}{{\text{3}}}}$, $r=28$.

In this paper, we consider the numerical simulation of the Lorenz system with
above parameter values. Fig. 1 shows a numerical approximation with the help
of the computer software Matlab.

Figure 1. A trajectory of the Lorenz system.

First of all, the Lorenz equations contain symmetry. Applying the coordinate
transformation

$\left({x,y,z}\right)\to\left({-x,-y,z}\right),$

the equations remain the same.

For $r<1$ (and $\sigma$, $\eta$ arbitrarily), the point $\left({0,0,0}\right)$
is the only fixed point, which is automatically asymptotic stable. It holds
that all solutions converge to the origin as $t\to\infty$. At $r=1$, it occurs
a supercritical Pitchfork–bifurcation and two new fixed points appear. They
fulfill

$x=y=\sqrt{b\left({r-1}\right)},\qquad z=r-1.$

When $r>{r_{H}}\approx 24.737$, the Lorenz system, which exhibits chaotic
behavior, is the object of our study.

In addition, its stability matrix can be obtained from the Lorenz equations

$M=\left[{\begin{array}[]{*{20}{c}}{-\sigma}&{\sigma}&0\\\
{r-z_{0}}&{-1}&{-x_{0}}\\\ y_{0}&x_{0}&{-\eta}\end{array}}\right],$

where $x_{0}$, $y_{0}$, and $z_{0}$ are the coordinates of the stable points.
The eigenvalues at stable points indirectly describe the degree to which
trajectories in its vicinity are attracted or repelled. The eigenvalue at the
origin is
$\left({{\lambda^{\left(1\right)}},{\lambda^{\left(2\right)}},{\lambda^{\left(3\right)}}}\right)=\left({11.83,-2.67,-22.83}\right)$,
and the eigenvalues at the other two symmetrical stable points are
$\left({{\mu^{\left(1\right)}}\pm{\omega^{\left(1\right)}}i,{\lambda^{\left(3\right)}}}\right)=\left({0.094\pm
10.19i,-13.85}\right)$. The unstable eigenplane of the origin is spanned by Re
${e^{\left(1\right)}}$ and Im ${e^{\left(1\right)}}$. The Lorenz flow has an
unstable eigenvector ${e^{\left(1\right)}}$ and two stable eigenvectors
${e^{\left(2\right)}}$, ${e^{\left(3\right)}}$ near the origin.

In [8], the motion of the Lorenz attractor is given in numerical terms. The
periodic time scale in the neighborhood of the equilibrium
$\left(x,y,z\right)=\left(0,0,0\right)$ is of order $\approx T$. The
contraction/expansion radially by the multiplier ${\Lambda_{radial}}$, and by
the multiplier ${\Lambda_{j}}$ along the ${e^{\left(j\right)}}$ eigen-
direction per a turn of the spiral:

$T=2\pi/\omega,\quad{\Lambda_{radial}}={e^{T\mu}},\quad{\Lambda_{j}}={e^{T\mu\left(j\right)}}.$

The trajectory period around the critical point is about $T=2\pi/10.19\approx
0.62s$. Near the two stable points the unstable manifold trajectories spiral
out, with very small radial per-turn expansion multiplier
${\Lambda^{1}}=\exp\left({0.094T}\right)\approx 1.06$, and very strong
contraction multiplier ${\Lambda^{3}}=\exp\left({-13.85T}\right)\approx
1.957\times{10^{-4}}$ onto the unstable manifold. This contraction confines
the Lorenz attractor to a two-dimensional surface. In the neighborhood of the
origin, the trajectories have an extremely strong contraction along the
${e^{\left(3\right)}}$ direction and the slowest contraction along the
${e^{\left(2\right)}}$ direction. In the plane, the expansion of the attractor
in the ${e^{\left(1\right)}}$ direction takes precedence over the contraction
in the ${e^{\left(2\right)}}$ direction, which makes it difficult for a few
trajectories can approach the origin.

###  2.2. Similarity signature curve

The goal of this part is to establish a basic functional relation or syzygy
among the differential invariants under the action of the similarity group
$\mathrm{Sim}(3)=\\{\lambda Ax+b\\}$, where $A$ is a real orthogonal $3\times
3$ matrix, $b\in{\mathbb{R}}^{3}$ is a real vector and $\lambda>0$ is a real
number. In fact, the similarity invariants have been obtained in [7, 9]. Here
we mainly use the equivariant moving frame theory to build the syzygy and the
similarity signature curve.

We denote points in ${\mathbb{R}}^{3}$ by
$z=\left(x,u,v\right)\in{\mathbb{R}}^{3}$, with
$z(t)=\left(x(t),u(t),v(t)\right)$ for $t\in I\subset{\mathbb{R}}$ being a
smoothly parametrized curve.

Let us consider the action

$\left(X,U,V\right)=Z=g\cdot z=g\cdot\left(x,u,v\right)$

of the similarity group $\mathrm{Sim}(3)$ in ${\mathbb{R}}^{3}$, so that

(2.1) $X=k\alpha\cdot z+a,\quad U=k\beta\cdot z+b,\quad V=k\gamma\cdot z+c,$

where $k>0,a,b,c$ are real numbers, $\alpha,\beta,\gamma$ are the rows of the
real orthogonal $3\times 3$ matrix.

To construct a (right) equivariant moving frame, we prolong the
$\mathrm{Sim}(3)$ action to the curve jet spaces $\mathrm{J}^{n}$, which has
local coordinates

$z^{(n)}=\left(x,u,v,u_{x},v_{x},u_{xx},v_{xx},\cdots,u_{n},v_{n}\right).$

The formulae for the prolonged action on $\mathrm{J}^{n}$ are provided by
implicit differentiation, based on

$dX=k\alpha\cdot z_{t}dt,\qquad D_{X}=\displaystyle{\frac{1}{k\alpha\cdot
z_{t}}\frac{d}{dt}}.$

Then we have

$\displaystyle U_{X}=\frac{\beta\cdot z_{t}}{\alpha\cdot z_{t}},\qquad
V_{X}=\frac{\gamma\cdot z_{t}}{\alpha\cdot z_{t}},$ $\displaystyle
U_{XX}=\frac{\left(\alpha\cdot z_{t}\right)\left(\beta\cdot
z_{tt}\right)-\left(\alpha\cdot z_{tt}\right)\left(\beta\cdot
z_{t}\right)}{k\left(\alpha\cdot z_{t}\right)^{3}},$ $\displaystyle
V_{XX}=\frac{\left(\alpha\cdot z_{t}\right)\left(\gamma\cdot
z_{tt}\right)-\left(\alpha\cdot z_{tt}\right)\left(\gamma\cdot
z_{t}\right)}{k\left(\alpha\cdot z_{t}\right)^{3}}.$

To compute the equivariant moving frame, we must normalize the
$\mathrm{dim\\{Sim(3)\\}}=7$ independent group parameters by setting 7 of the
transformed jet variables equal to conveniently chosen constants. This
corresponds to the choice of a cross section to the prolonged group orbits,
or, equivalently, to placing the curve in normal form, [28]. The standard
cross section that produces the classical moving frame is given by

$x=u=v=u_{x}=v_{x}=v_{xx}=0,\qquad u_{xx}=1.$

The determination of the moving frame associated with the above cross-section
relies on solving the corresponding normalization equations

$X=U=V=U_{X}=V_{X}=V_{XX}=0,\qquad U_{XX}=1,$

for the group parameters $g\in\mathrm{Sim}(3)$, using the explicit formulas
for the prolonged Euclidean transformations that were obtained by implicit
differentiation.

The order zero normalizations $X=U=V=0$ prescribe the translation parameters
of the group element $g\in\mathrm{Sim}(3)$. Since they play no further role in
the prolonged action or other moving frame formulae, we can effectively ignore
them from here on. After some obvious simplification, we find

$\beta\cdot z_{t}=0,\qquad\gamma\cdot z_{t}=0,\qquad\gamma\cdot
z_{tt}=0,\qquad\displaystyle{\frac{\beta\cdot z_{tt}}{k\left(\alpha\cdot
z_{t}\right)^{2}}=1}.$

We impose the nondegeneracy condition $|z_{t}\wedge z_{tt}|\neq 0$. Therefore,
the right-equivariant moving frame induced by the cross-section is given by

(2.2) $\displaystyle\begin{split}&\gamma=\frac{z_{t}\wedge
z_{tt}}{\left|z_{t}\wedge
z_{tt}\right|},\quad\beta=\frac{|z_{t}|^{2}z_{tt}-(z_{t}\cdot
z_{tt})z_{t}}{\left||z_{t}|^{2}z_{tt}-(z_{t}\cdot
z_{tt})z_{t}\right|}=\frac{|z_{t}|^{2}z_{tt}-(z_{t}\cdot
z_{tt})z_{t}}{|z_{t}||z_{t}\wedge z_{tt}|},\\\
&\alpha=\beta\wedge\gamma=\frac{z_{t}}{|z_{t}|},\quad k=\frac{\beta\cdot
z_{tt}}{\left(\alpha\cdot z_{t}\right)^{2}}=\frac{|z_{t}\wedge
z_{tt}|}{|z_{t}|^{3}}.\end{split}$

As always, a complete system of functionally independent differential
invariants is obtained by invariantization, that is, substituting the moving
frame formulae into the unnormalized transformation rules. All other
differential invariants are obtained by differentiating the curvature and
torsion with respect to the similarity length element,

$d\tilde{s}=\displaystyle{\frac{|z_{t}\wedge z_{tt}|}{|z_{t}|^{2}}dt},$

which comes from substituting the moving frame formula for $\alpha$ and $k$
into $dX$.

$\displaystyle U_{XXX}=$ $\displaystyle\frac{1}{k^{2}\left(\alpha\cdot
z_{t}\right)^{5}}\times\Big{(}(\beta\cdot z_{ttt})(\alpha\cdot z_{t})^{2}$
$\displaystyle\quad-(\beta\cdot z_{t})(\alpha\cdot z_{t})(\alpha\cdot
z_{ttt})-3(\alpha\cdot z_{tt})(\beta\cdot z_{tt})(\alpha\cdot
z_{t})+3(\alpha\cdot z_{tt})^{2}(\beta\cdot z_{t})\Big{)},$ $\displaystyle
V_{XXX}=$ $\displaystyle\frac{1}{k^{2}\left(\alpha\cdot
z_{t}\right)^{5}}\times\Big{(}(\gamma\cdot z_{ttt})(\alpha\cdot z_{t})^{2}$
$\displaystyle\quad-(\gamma\cdot z_{t})(\alpha\cdot z_{t})(\alpha\cdot
z_{ttt})-3(\alpha\cdot z_{tt})(\gamma\cdot z_{tt})(\alpha\cdot
z_{t})+3(\alpha\cdot z_{tt})^{2}(\gamma\cdot z_{t})\Big{)}.$

The generating differential invariants are the similarity curvature
$\tilde{\kappa}$, obtained from $u_{xxx}$, and the similarity torsion
$\tilde{\tau}$ , obtained from $v_{xxx}$. By using (LABEL:mfv), a short
computation produces the required expressions for the curvature and torsion
invariants

$\displaystyle\tilde{\kappa}$ $\displaystyle=\iota(u_{xxx})=\frac{(z_{t}\wedge
z_{ttt})\cdot(z_{t}\wedge z_{tt})|z_{t}|^{2}-3(z_{t}\cdot z_{tt})|z_{t}\wedge
z_{tt}|^{2}}{|z_{t}\wedge z_{tt}|^{3}},$ $\displaystyle\tilde{\tau}$
$\displaystyle=\iota(v_{xxx})=\frac{|z_{t}|^{3}[z_{t},z_{tt},z_{ttt}]}{|z_{t}\wedge
z_{tt}|^{3}}.$

######  Remark 2.1.

In fact, taking $k=1$ in (2.1) and using cross section
$x=u=v=u_{x}=v_{x}=v_{xx}=0$, a similar process yields the Euclidean arc
element $ds=|z_{t}|dt$, the Euclidean curvature
$\kappa=\iota(u_{xx})=\displaystyle{\frac{|z_{t}\wedge z_{tt}|}{|z_{t}|^{3}}}$
and the Euclidean torsion
$\tau=\iota(v_{xxx})/\kappa=\displaystyle{\frac{[z_{t},z_{tt},z_{ttt}]}{|z_{t}\wedge
z_{tt}|^{2}}}$. At the same time, $\iota(u_{xxx})=\kappa_{s}$.

By using the inductive construction method of moving frames introduced by
Kogan [19], we can see $\mathrm{Sim}(3)=B\cdot\mathrm{SE}(3)$, where
$B=\left(\begin{array}[]{ccc}k&0&0\\\ 0&k&0\\\ 0&0&k\\\ \end{array}\right)$
and $B\cap\mathrm{SE}(3)=\\{I,-I\\}$ is finite. Now we prolong the action of
$B$ up to third order

$\displaystyle x\to kx,\quad u\to ku,\quad v\to kv,\quad u_{x}\to u_{x},\quad
v_{x}\to v_{x},$ $\displaystyle u_{xx}\to\displaystyle{\frac{u_{xx}}{k}},\quad
v_{xx}\to\displaystyle{\frac{v_{xx}}{k}},\quad
u_{xxx}\to\displaystyle{\frac{u_{xxx}}{k^{2}}},\quad
v_{xxx}\to\displaystyle{\frac{v_{xxx}}{k^{2}}}.$

Restricting these transformations to the Euclidean cross section
$K_{E}=\\{x=u=v=u_{x}=v_{x}=v_{xx}=0\\}$, we obtain

$\displaystyle u_{xx}\to\frac{u_{xx}}{k},\quad
u_{xxx}\to\frac{u_{xxx}}{k^{2}},\quad v_{xxx}\to\frac{v_{xxx}}{k^{2}}.$

The above action is free on the open subset $\\{z\in K_{E}|u_{xx}\neq 0\\}$,
and we choose the cross section $K=\\{z\in K_{E}|u_{xx}=1\\}$ to the orbits of
$B$ on $K_{E}$. This produces a moving frame $\rho_{B}:K_{E}\to B$, that is,

$k=u_{xx}.$

Thus we obtain the expression of the similarity invariants $\tilde{\kappa}$
and $\tilde{\tau}$ in terms of Euclidean invariants

$\tilde{\kappa}=\displaystyle{\frac{\kappa_{s}}{\kappa^{2}}},\qquad\tilde{\tau}=\displaystyle{\frac{\tau}{\kappa}}$.

Let us introduce the following basis for the infinitesimal generators in the
Lie algebra of $\mathrm{Sim}(3)$

$\displaystyle\mathbf{v}_{1}=\partial_{x},\quad\mathbf{v}_{2}=\partial_{u},\quad\mathbf{v}_{3}=\partial_{v},\quad\mathbf{v}_{4}=v\partial_{u}-u\partial_{v},$
$\displaystyle\mathbf{v}_{5}=-u\partial_{x}+x\partial_{u},\quad\mathbf{v}_{6}=-v\partial_{x}+x\partial_{v},\quad\mathbf{v}_{7}=x\partial_{x}+u\partial_{u}+v\partial_{v}.$

The prolonged infinitesimal generators of the group $\mathrm{Sim}(3)$ action
on curve jets are

$\displaystyle\centering\mathrm{pr}\mathbf{v}_{1}\@add@centering$
$\displaystyle=\;\partial_{x},\qquad\mathrm{pr}\mathbf{v}_{2}\;=\;\partial_{u},\qquad\mathrm{pr}\mathbf{v}_{3}\;=\;\partial_{v},$
$\displaystyle\mathrm{pr}\mathbf{v}_{4}$
$\displaystyle=\;v\partial_{u}-u\partial_{v}+v_{x}\partial_{u_{x}}-u_{x}\partial_{v_{x}}+v_{xx}\partial_{u_{xx}}-u_{xx}\partial_{v_{xx}}+\cdots,$
$\displaystyle\mathrm{pr}\mathbf{v}_{5}$
$\displaystyle=\;-u\partial_{x}+x\partial_{u}+(1+u^{2}_{x})\partial_{u_{x}}+u_{x}v_{x}\partial_{v_{x}}+3u_{x}u_{xx}\partial_{u_{xx}}$
$\displaystyle\qquad+\left(u_{xx}v_{x}+2u_{x}v_{xx}\right)\partial_{v_{xx}}+\left(3u^{2}_{xx}+4u_{x}u_{xxx}\right)\partial_{u_{xxx}}$
$\displaystyle\qquad+\left(u_{xxx}v_{x}+3u_{xx}v_{xx}+3u_{x}v_{xxx}\right)\partial_{v_{xxx}}+\cdots,$
$\displaystyle\mathrm{pr}\mathbf{v}_{6}$
$\displaystyle=\;-v\partial_{x}+x\partial_{v}+v_{x}u_{x}\partial_{u_{x}}+\left(1+v^{2}_{x}\right)\partial_{v_{x}}+\left(v_{xx}u_{x}+2v_{x}u_{xx}\right)\partial_{u_{xx}}$
$\displaystyle\qquad+3v_{x}v_{xx}\partial_{v_{xx}}+\left(v_{xxx}u_{x}+3v_{xx}u_{xx}+3v_{x}u_{xxx}\right)\partial_{u_{xxx}}$
$\displaystyle\qquad+\left(3v^{2}_{xx}+4v_{x}v_{xxx}\right)\partial_{v_{xxx}}+\cdots,$
$\displaystyle\mathrm{pr}\mathbf{v}_{7}$
$\displaystyle=\;x\partial_{x}+u\partial_{u}+v\partial_{v}-u_{xx}\partial_{u_{xx}}-v_{xx}\partial_{v_{xx}}-2u_{xxx}\partial_{u_{xxx}}-2v_{xxx}\partial_{v_{xxx}}+\cdots.$

The invariant arc length derivative $\mathcal{D}=\iota(D_{x})$ of any
differential invariant $I=\iota(F)$ obtained by invariantizing a differential
function $F$ is specified by the recurrence relation

(2.3)
$\mathcal{D}I=\mathcal{D}\iota(F)=\iota(D_{x}F)+\sum_{i=1}^{7}R^{i}\iota(\mathrm{pr}\mathbf{v}_{i}(F)),$

where $R^{1},R^{2},\cdots,R^{7}$ are the Maurer-Cartan invariants. To
determine their formulas, we write out (2.3) for the seven phantom invariants
which come from the cross section variables $x$, $u$, $v$, $u_{x}$, $v_{x}$,
$u_{xx}$, $v_{xx}$

$\displaystyle 0=\mathcal{D}\iota(x)=\iota(1)+R^{1},\quad
0=\mathcal{D}\iota(u)=R^{2},\quad 0=\mathcal{D}\iota(v)=R^{3},\quad
0=\mathcal{D}\iota(u_{x})=1+R^{5},$ $\displaystyle
0=\mathcal{D}\iota(v_{x})=R^{6},\quad
0=\mathcal{D}\iota(u_{xx})=\tilde{\kappa}-R^{7},\quad
0=\mathcal{D}\iota(v_{xx})=\tilde{\tau}-R^{4}.$

Thus, the general recurrence relation (2.3) becomes

$\mathcal{D}\iota(F)=\iota(D_{x}F)-\iota(\mathrm{pr}\mathbf{v}_{1}(F))+\tilde{\tau}\iota(\mathrm{pr}\mathbf{v}_{4}(F))-\iota(\mathrm{pr}\mathbf{v}_{5}(F))+\tilde{\kappa}\iota(\mathrm{pr}\mathbf{v}_{7}(F)).$

Then it is obvious that

$\iota(u_{xxxx})=\tilde{\kappa}_{\tilde{s}}+3+2\tilde{\kappa}^{2}-\tilde{\tau}^{2},\quad\iota(v_{xxxx})=\tilde{\tau}_{\tilde{s}}+3\tilde{\kappa}\tilde{\tau}.$

######  Definition 2.2.

The differential invariant signature for the similarity space curve
$C\subset{\mathbb{R}}^{3}$ is is parametrized by the three lowest order
differential invariants

$\Sigma=\left\\{(\tilde{\kappa},\tilde{\kappa}_{\tilde{s}},\tilde{\tau})\right\\}\subset{\mathbb{R}}^{3}$,

where $\tilde{s}$ is the similarity arc length parameter, $\tilde{\kappa}$ and
$\tilde{\tau}$ are the similarity curvature and torsion of the curve $C$,
respectively.

In the actual calculation, the curve is composed of discrete points
distributed in space. In [6], the authors provided an approach to the
numerical approximation of differential invariants, based on a suitable
combination of invariants of the group action to compute differential
invariant signatures numerically in a fully group-invariant manner. In Fig. 2,
suppose ${P_{i-1}}$, ${P_{i}}$, ${P_{i+1}}$, and ${P_{i+2}}$ are four
successive points on a three-dimensional curve $C$. The Euclidean distance
between points ${P_{i}}$ and ${P_{j}}$ is
$\mathrm{dist}\left({{P_{i}},{P_{j}}}\right)$. Let

$\displaystyle a=\mathrm{dist}\left({{P_{i-1}},{P_{i}}}\right),\quad
b=\mathrm{dist}\left({{P_{i}},{P_{i+1}}}\right),\quad
c=\mathrm{dist}\left({{P_{i-1}},{P_{i+1}}}\right),$ $\displaystyle
d=\mathrm{dist}\left({{P_{i+1}},{P_{i+2}}}\right),\quad
e=\mathrm{dist}\left({{P_{i}},{P_{i+2}}}\right),\quad
f=\mathrm{dist}\left({{P_{i-1}},{P_{i+2}}}\right).$

Figure 2. Consecutive points for signature approximations.

The equations used to approximate the signature are as follows:

$\kappa\big{(}{{P_{i}}}\big{)}=\displaystyle{\frac{{4\Delta}}{{abc}}},\quad{\kappa_{s}}\big{(}{{P_{i}}}\big{)}=\displaystyle{\frac{{3\big{(}\kappa\left({{P_{i+1}}}\right)-\kappa\left({{P_{i}}}\right)\big{)}}}{{a+b+c}}},\quad\tau\big{(}{{P_{i}}}\big{)}=\displaystyle{\frac{{6H}}{{def\kappa\left({{P_{i}}}\right)}}},$

where $\Delta$ is the area of the triangle with sides $a,b,c$, and $H$ is the
height of the tetrahedron with sides $a,b,c,d,e$, and $f$ [18].

###  2.3. Analysis of the similarity signature curve for the Lorenz attractor

Compared with the Euclidean signature curve, the similarity signature curve is
not only invariant to translation and rotation but also invariant to scale
invariance. Fig. 3 and Fig. 4 are the projections of the signature curve and
the similarity signature curve of the Lorenz attractor on three coordinate
planes. The similarity signature curve of the Lorenz attractor has a stronger
self-similarity, which indirectly describes the geometric structure of the
Lorenz attractor and is shown as in Fig. 5.

(a) $x-y$ plane (b) $y-z$ plane (c) $x-z$ plane

Figure 3. Projection of signature curve of Lorenz attractor on coordinate
plane.

(a) $x-y$ plane (b) $y-z$ plane (c) $x-z$ plane

Figure 4. Projection of similarity signature curve of Lorenz attractor on
coordinate plane.

It is worth noting that multiple curves with a given signature can be
constructed. Hickman et al. constructed an example of a non-congruence curve
with the same Euclidean signature curve [16, 25]. Therefore, the same
consequence may occur in the similarity signature curve of the system. Later,
we will use the set method to further explain that for different periodic
symbol sequences, their similarity signature curves will not completely
coincide.

Due to constraints in each direction, the intrinsic fractal structure and
self-similarity of the Lorenz attractor are formed. As the decompositions of
all the motion states of the Lorenz attractor, Fig. 6 and Fig. 7 depict two
completely different states of the similarity signature curve. One is the
stable state, and the other is the mutation state. The motion behavior of the
Lorenz attractor can be roughly summarized as the random switching of its
trajectory between two states. When the similarity signature curve is in a
stable state, its corresponding trajectory will continuously rotate around a
non-zero stable point. When the state changes, its corresponding trajectory
will transition to another non-zero stable point.

Figure 5. The Lorenz attractor with self similarity and its similarity
signature curve

##  3\. Method of finding periodic orbits

The existence of periodic orbits in the Lorenz attractor has been strictly
proved [33]. On the basis that the trajectories of the Lorenz system never
intersect, we propose a theorem and an algorithm to illustrate the feasibility
of using the similarity signature curve to find periodic orbits. The algorithm
combines the sliding window method with the similarity signature curve and
uses the Krawczy operator to verify whether there is a real periodic orbit
near a quasi-periodic orbit.

###  3.1. Interval methods

Interval methods provide simple computational tests for the uniqueness,
existence, and nonexistence of zeros of a map within a given interval vector
[13]. Simplification of continuous-time systems to discrete systems benefits
from the Poincaré map. The Poincaré map reflects the dependence of the
successor point on the predecessor point when the trajectory repeatedly
crosses the same plane which can simplify the quasi-periodic orbit into a
series of discrete points. The Poincaré map for the autonomous system is as
follows.

######  Definition 3.1 ([31]).

Let ${x^{*}}$ be a point on a limit cycle and let $\Pi$ be an
$\left({n-1}\right)$-dimensional hyperplane transversal to $\Gamma$ at
${x^{*}}$. The trajectory emanating from ${x^{*}}$ will hit $\Pi$ at ${x^{*}}$
in $T$ seconds, where $T$ is the minimum period of the limit cycle. Due to the
continuity of ${\phi_{t}}$ with respect to the initial condition, trajectories
starting on $\Pi$ in a sufficiently small neighborhood of ${x^{*}}$ will, in
approximately $T$ seconds, intersect $\Pi$ in the vicinity of ${x^{*}}$.
Hence, ${\phi_{t}}$ and $\Pi$ define a mapping ${P_{A}}$ of some neighborhood
$U\subset\Pi$ of ${x^{*}}$ onto another neighborhood $V\subset\Pi$ of
${x^{*}}$. ${P_{A}}$ is a Poincaré map of the autonomous system.

Figure 6. Similarity signature curve in stable state and the trajectory of its
corresponding the Lorenz attractor. Figure 7. Similarity signature curve in
mutation state and the trajectory of its corresponding the Lorenz attractor.

The Poincaré map can be simply called as $\Omega:\Pi\to\Pi$ and it is defined
as $\Omega\left(x\right)=\varphi\left({t\left(x\right),x}\right)$ where
$\Pi={\Pi_{1}}\cup\cdots\cup{\Pi_{m}}$ is the union of hyperplanes, and
$t\left(x\right)$ is the return time after which the trajectory
$\varphi\left({t,x}\right)$ returns to $\Pi$. Periodic points of $\Omega$
correspond to periodic orbits of the continuous system [13].

A map is constructed to study the existence of period-$p$ orbits of $\Omega$:

${\left[{F\left(z\right)}\right]_{k}}={x^{\left({\left({k+1}\right)\bmod
p}\right)}}-\Omega\left({{x^{k}}}\right),$

where $0\leq k\leq p$ and
$z=\left({{x^{\left(0\right)}},{x^{\left(1\right)}},\cdots,{x^{\left({p-1}\right)}}}\right)$.
Zeros of $F$ correspond to period-$p$ points of $\Omega$, i.e.,
$F\left(z\right)=0$ if and only if
${\Omega^{p}}\left({{x^{\left(0\right)}}}\right)={x^{\left(0\right)}}$.

The interval method can test the uniqueness, existence, and non-existence of
the zeros of the Poincaré map, so it can be used to find the periodic solution
of the Lorenz system on the Poincaré section. The Krawczyk operator [20, 26]
is:

$K\left(\mathbf{z}\right)=\hat{z}-MF\left({\hat{z}}\right)-\left({MF^{\prime}\left(\mathbf{z}\right)-I}\right)\left({\mathbf{z}-\hat{z}}\right),$

where $\hat{z}\in\mathbf{z}$, $F\left({\hat{z}}\right)$ is the interval matrix
containing the Jacobian matrices $F\left({\hat{z}}\right)$ for all
$z\in\mathbf{z}$, and $M$ is an invertible matrix. Using the method given in
[11], the map $\Omega\left(x\right)$ and its Jacobian matrix in the Krawczyk
operator can be obtained. Generally, we take $\hat{z}$ as the center of
$\mathbf{z}$ and ${M}$ as the inverse of $F^{\prime}\left(z\right)$. If
$\mathbf{z}\cap K\left(\mathbf{z}\right)=\phi$, then there are no period-$p$
orbits in $\mathbf{z}$. If $K\left(\mathbf{z}\right)\subset{\mathop{\rm
int}}\mathbf{z}$, there exists precisely one period-$p$ orbit in $\mathbf{z}$
[21].

###  3.2. Periodic orbits by the symbolic dynamics based approach

The following theorem guarantees the feasibility of the algorithm proposed in
this paper.

######  Theorem 3.2.

For two different periodic orbits $P_{1}$ and $P_{2}$, their periodic symbol
sequences are
${S_{1}}=\left({{s_{11}},{s_{12}},\ldots,{s_{1{n_{1}}}}}\right)$,
${S_{2}}=\left({{s_{21}},{s_{22}},\ldots,{s_{2{n_{2}}}}}\right)$,
$1<{n_{2}}<{n_{1}}$, and $S_{2}$ is a ordered subset of $S_{1}$. The
similarity signature curve of $P_{2}$ will not completely coincide with the
similarity signature curve of $P_{1}$.

###### Proof.

For a given periodic symbol sequence
$S_{0}=\left({{s_{0}},{s_{1}},\ldots,{s_{p-1}}}\right)$, there exists at most
one point $x\in N$ with the symbol sequence $S$ such that
${R^{p}}\left(x\right)=x$ [32]. Then periodic orbits with the same sequence of
periodic symbols will not be strictly similar. If the long periodic orbits do
not contain short periodic orbits, it means that there will be no periodic
orbit with period symbol $S_{2}$ in the long period orbit, so this theorem is
correct. Otherwise, we will face the following two cases.

(1) $P_{1}$ starts with $P_{2}$ or ends with $P_{2}$, that is,
${S_{1}}=\left({{S_{2}},{s_{{n_{2}}+1}},\cdots,{s_{{n_{1}}}}}\right)$ or
${S_{1}}=\left({{s_{11}},\cdots,{s_{{n_{1}}-{n_{2}}}},{S_{2}}}\right)$. For
the given $S_{2}$, there exists at most one point $x\in N$ with the symbol
sequence $S_{2}$ such that ${R^{{n_{2}}}}\left(x\right)=x$. The point $x$
exists in the trajectory satisfying the periodic symbol sequence and belongs
to the intersection of attractor and trapping region $N$, then $x$ also
satisfies ${R^{{n_{1}}}}\left(x\right)=x$. The part of ${S_{1}}$ excluding
${S_{2}}$ can be regarded as a whole, which is called ${S_{3}}$, that is,
${S_{1}}=\left({{S_{2}},{S_{3}}}\right)$ or
${S_{1}}=\left({{S_{3}},{S_{2}}}\right)$. ${S_{3}}$ is also a sequence of
periodic symbols, where
${S_{3}}=\left({{s_{{n_{2}}+1}},\cdots,{s_{{n_{1}}}}}\right)$ or
${S_{3}}=\left({{s_{11}},\cdots,{s_{{n_{1}}-{n_{2}}}}}\right)$.

(2) $P_{2}$ is included in $P_{1}$, that is,
${S_{1}}=\left({{s_{11}},\cdots,{s_{1k}},{S_{2}},{k+{n_{2}}+1},\cdots,{s_{{n_{1}}}}}\right)$,
and then the proof process is similar as the above case. ∎

In [32], it was established that there existed a finite set of rectangles
$N={N^{-}}\cup{N^{+}}\subset\Pi$ such that $N$ is mapped into itself by the
return map $R:\Pi\to\Pi$. They formed the forward invariant set
$\mathscr{A}=\bigcap\nolimits_{k=0}^{\infty}{{P^{k}}\left(N\right)}$. In fact,
$\mathscr{A}$ is the intersection of the Lorenz attractor with $N$. For a
trajectory $\left\\{{{x_{n}}}\right\\}_{n=0}^{\infty}\subset N$ where
${x_{n+1}}=R\left({{x_{n}}}\right)$. Suppose that $x^{\prime}$ is the
similarity signature of $x$, and we need at least 4 points to approximate
$x^{\prime}$. The points in the Lorenz attractor will not intersect. According
to the continuity of the Lorenz system, we know that the similarity signature
curve of the Lorenz attractor is also continuous. When point $x$ satisfies the
condition $R{{}^{p}}\left(x\right)=x$, the similarity signature of $x$ is also
invariant, so there is an invariant set in the similarity signature of the
Lorenz attractor. The similarity signature at this point also exists and is
unique, otherwise, there will be conflicts.

###  3.3. Sliding window method based on similarity signature curve

The sliding window method can solve the problems of finding the properties,
such as the length, of a continuous interval that satisfies certain
conditions. All trajectories of the Lorenz attractor are processed
successively by its similarity signature curve and sliding window method, and
the segmented fragments have obvious self-similarity, which is a manifestation
of the fractal structure of the Lorenz attractor. It is wise to construct a
real periodic orbit based on a quasi-periodic orbit but the initial value of
the system limits its accuracy. Experiments show that in the process of
finding quasi-periodic orbits, the choice of initial point will affect the
final result. A suitable initial point can ensure that the quasi-periodic
orbits are similarly positioned in the Lorenz system. In addition to the
initial point, the appropriate window size can more intuitively reflect the
relationship between the properties of the data in each window, which is
helpful for the division of periodic orbits. The algorithm for finding the
periodic orbits of the Lorenz system is implemented according to the rules
proposed above.

The algorithm simplifies the problem of finding the periodic orbits of the
Lorenz system into the search and combination of two different states of the
trajectory. For purposes of clarity, we generate the following pseudo-code and
algorithm structure.

Algorithm 1 Sliding window method

0: The Lorenz system dataset $D$; Window size $W$; Initial point ${P_{1,1}}$

0: Periodic orbit segmentation points $z$

1: Initialize $z=\left[{{P_{1,1}}}\right]$, $poi=0$, $i=1$;

2: Compute $\tilde{\kappa}$, ${\tilde{\kappa}_{\tilde{s}}}$ and $\tilde{\tau}$
of the Lorenz system;

3: while $poi+W\leqslant length\left(D\right)$ do

4: for $j=1:W$ do

5: Compute $d\left(j\right)=dist\left({{P_{i,1}},{P_{i,j}}}\right)$;

6: Compute $\left[{val,index}\right]=\min d\left(j\right)$;

7: Compute ${P_{i+1,1}}={P_{i,index}}$;

8: $z=\left[{z,{P_{i+1,1}}}\right]$;

9: end for

10: $poi=poi+index$;

11: $i=i+1$;

12: end while

13: return $z$;

The similarity signature curve for the Lorenz systems is acquired through the
numerical calculation by using the formulas in Sec. 2. Actually, a proper
selections of the initial point and window size for the algorithm can ensure
the feasibility of the algorithm. The initial point should satisfy two
conditions. (1) The closest point to the initial value of the system. (2) The
extreme point of the similarity signature curve in the $z$-axis direction.
These ensure that the segmentation points on the trajectory of the Lorenz
attractor are more stable. When the trajectory transitions from rotating
around one stable point to another stable point, the period of its similarity
signature curve is the longest, so the maximum periodic orbit length of the
system should be taken into account in the selection window size. If the
window size is too small, a complete period may not exist in the window.
Suppose the window size is $W$ which satisfies the condition slightly smaller
than the length of two periods of the similarity signature curve. Let
$W=2500$, equates to about $1.25s$. If the length of the last remaining data
is less than the size of the window, it will not be considered. Suppose that
the initial point of each window is ${p_{i1}}\left({1\leqslant i\leqslant
n}\right)$, and the other points in a window period are
${p_{ij}}\left({2\leqslant j\leqslant 2500}\right)$, where ${n}$ is equal to
the number of windows required in the end. By calculating the distances
between all points in the longer trajectory and the fixed initial point, we
observe the existence of cycles in their distance image, as shown in Fig. 8.
After completing the closest point search for the initial points of the
similarity signature curve, the trajectory between these two points is always
in the form of two circle-like nests. As the degree of the mutation of the
similarity signature curve gradually increases, the amplitude of the distance
map also increases, which means that the Lorenz attractor will transition from
a stable state to a mutation state.

Figure 8. Distance between all points in a window and the initial point.

Second, as the trajectory continues to rotate around a non-zero stable point,
the point closest to the initial point on each approximate circle in its
similarity signature curve gradually moves away from the initial point. As
shown in Fig. 9, the distance between them gradually increases. When the point
${p_{ik}}$ closest to the initial point ${p_{i1}}$ is found, the initial point
needs to be updated. The point ${p_{ik}}$ will be called the new initial
point, and then we recalculate the distance image and update the initial point
continuously until the subsequent trajectory ends. Then point ${p_{ik}}$ is
equal to point ${p_{i+1,1}}$. This part is a detailed explanation of the loop
part in the pseudocode.

Figure 9. The points in the graph represent the distance between the points
closest to the starting point in each cycle.

Third, a sequence of points
$p=\left\\{{{p_{11}},{p_{21}},\cdots,{p_{n1}}}\right\\}$ is obtained by
looping, where $n$ is the number of segmentation points and windows required.
Filtering segmentation points leaves only points along the same direction of
the $z$-axis. The filtered points form a new point sequence, which is assumed
to be $P=\left\\{{{P_{11}},{P_{21}},...,{P_{k1}}}\right\\}$, where point
${P_{i1}}$ is equivalent to point ${p_{k1}}\in p$. The trajectory between
every two points in the point sequence $P$ may be a quasi-periodic orbit.
Except for the first and last points in the sequence, the other points are the
end of one orbit and the beginning of the next. The endpoints of the quasi-
periodic orbits obtained by applying the similarity signature curve can be
kept within the neighborhood of 0.0005.

Finally, the periodic orbits are closed. This method is based on the existence
of a real periodic orbit in a neighborhood of the quasi-periodic orbit [22].
Table 1 gives the data for quasi-periodic orbits with periods not greater than
5.

In short, there are three advantages of using the similarity signature curve
to find periodic orbits: (1) It can reflect the self-similarity of the Lorenz
system from the perspective of invariants; (2) High precision is achieved by
common methods, which is helpful for quasi-periodic orbits positioning; (3)
The order of the periodic orbits is guaranteed and no omission occurs.

Table 1. Data of periodic orbits with period $p\leqslant 5$, including the coordinates of the segmentation points and the distance between the endpoints. $s$ | $x$ | $y$ | $z$ | dist
---|---|---|---|---
LR | -9.727902750 | -16.398960521 | 17.292212679 | 0.002599810
LLR | -9.836367146 | -16.255530452 | 18.121784817 | 0.002386087
LLLR | -9.835075562 | -16.131007675 | 18.411435859 | 0.003081481
LLRR | -9.917578308 | -15.670162223 | 19.764832024 | 0.003614604
LLLLR | -9.918073021 | -15.155786811 | 20.761179930 | 0.001839419
LLLRR | -9.849285414 | -15.553696219 | 19.723034119 | 0.002946679
LLRLR | -9.803205079 | -16.451041061 | 17.539781868 | 0.003687509

###  3.4. Method summary

In this method, quasi-periodic orbits are obtained by dividing and recombining
the Lorenz attractor by its similarity signature curve. A better approximation
of the position of the periodic orbit near the quasi-periodic orbit is found
by standard (non-interval) Newton iteration. Finally, the existence of real
periodic orbits can be proved by using the Krawczyk operator.

##  4\. Periodic orbits for the Lorenz system

According to the algorithm proposed in the previous section, a sequence
$P=\left({{P_{11}},{P_{21}},\cdots,{P_{k1}}}\right)$ is obtained, where
$P_{i1}$ is the point in the Lorenz attractor and is also the starting and
ending points of orbits. The standard Newton method contributes to sharpening
the approximation and thus obtaining quasi-periodic trajectories. Finally, the
Krawczyk operator is used to prove the existence of real periodic orbits in
the neighborhood of their quasi-periodic orbits and all periodic orbits with
period $p\leqslant 8$ are found. The orange part in Fig. 10 is composed of the
segmentation points of each periodic orbit. The positions of the segmentation
points exhibit similarity due to the invariance to scaling of the similarity
signature curve. Each trajectory is labeled in the following ways. Let $q$ be
the first intersection of the two-dimensional flow and the return plane $\Pi$.
When the trajectory intersects the plane to the left of $q$, it is recorded as
L, otherwise, recorded as R. Period-$p$ orbits can be better represented using
the sign sequence $S=\left({{s_{1}},{s_{2}},\cdots,{s_{p}}}\right)$, for
${s_{i}}=L$ or ${s_{i}}=R$, for $i=1,2,\cdots,p$.

Figure 10. Divided similarity signature curve and the positions of the
segmentation points on the Lorenz attractor.

There are infinitely long periodic orbits in the Lorenz system, and these
long-period orbits can be composed of multiple short-period orbits. Only the
coprime periodic orbits are considered when recording periodic orbits. If a
sequence is a periodic orbit, only its non-repetitive serial number is
recorded as the equivalent form of the periodic orbit.

The one-dimensional symbolic dynamics of the Lorenz system are established to
show all the found periodic orbits with periods no greater than 8. We give the
period $p$, the period length $T$, and the corresponding symbol sequence $s$
for each periodic orbit discovered. The results are reported in Table 2. Fig.
11 illustrates the periodic orbits with period $p\leqslant 6$. For each pair
of symmetric orbits, only periodic orbits with a greater number of L than R
are plotted.

Table 2. Short periodic orbits, where $p$ is the period of the orbit, $T$ is the flow-time, $s$ is the corresponding symbol sequence. $p$ | ${T}$ | $s$ | $p$ | ${T}$ | $s$
---|---|---|---|---|---
2 | 1.55865 | LR | 7 | 5.39421 | LLRLLRR
3 | 2.30591 | LLR | 7 | 5.42912 | LLRLRLR
4 | 3.02358 | LLLR | 8 | 5.78341 | LLLLLLLR
4 | 3.08428 | LLRR | 8 | 5.92499 | LLLLLLRR
5 | 3.72564 | LLLLR | 8 | 5.99044 | LLLLLRRR
5 | 3.82025 | LLLRR | 8 | 5.99732 | LLLLLRLR
5 | 3.86953 | LLRLR | 8 | 6.01003 | LLLLRRRR
6 | 4.41776 | LLLLLR | 8 | 6.03523 | LLLLRLLR
6 | 4.53410 | LLLLRR | 8 | 6.08235 | LLLLRLRR
6 | 4.56631 | LLLRRR | 8 | 6.08382 | LLLLRRLR
6 | 4.59381 | LLLRLR | 8 | 6.10805 | LLLRLRRR
6 | 4.63714 | LLRLRR | 8 | 6.12145 | LLLRLLRR
7 | 5.10303 | LLLLLLR | 8 | 6.12233 | LLLRRLLR
7 | 5.23419 | LLLLLRR | 8 | 6.13512 | LLLRRLRR
7 | 5.28634 | LLLLRRR | 8 | 6.15472 | LLLRLRLR
7 | 5.30120 | LLLLRLR | 8 | 6.17587 | LLRLLRLR
7 | 5.33091 | LLLRLLR | 8 | 6.18751 | LLRLRRLR
7 | 5.36988 | LLLRLRR | 8 | 6.19460 | LLRLRLRR
7 | 5.37052 | LLLRRLR |  |  |

(a) p=2, LR (b) p=3, LLR (c) p=4, LLRR (d) p=4, LLLR (e) p=5, LLLRR (f) p=5,
LLLLR (g) p=5, LLRLR (h) p=6, LLLRRR (i) p=6, LLLLLR (j) p=6, LLLLRR (k) p=6,
LLLRLR (l) p=6, LLRLRR

Figure 11. Periodic orbits of the Lorenz system with $p\leq 6$, where $p$ is
the period, and the order in which L and R appear is the trajectory of the
periodic orbit.

##  5\. Conclusion

In this paper, a novel method based on the similarity signature curve is
proposed for forming the periodic orbits of the Lorenz system, and some
experimental results are exhibited to show the efficacy of this method.

Firstly, through the similarity invariants of the Lorenz system and a series
of experiments, we can demonstrate the chaotic patterns of the Lorenz system
evolution, and ensure the feasibility of the method of applying the similarity
signature curve to calculate the periodic orbits.

Secondly, in order to continuously obtain the required quasi-periodic orbits,
the conditions for the window moving in the sliding window method are
improved. The window is updated by repeatedly selecting the points in the
subsequent data which are closest to the initial point on the similarity
signature curve.

In this way, all periodic orbits with period $p\leqslant 8$ are found. Also,
it is a common fact to predict long-period orbits by using some short-period
orbits. However, the longer the period, the larger the error and the more
difficult to locate the corresponding position. Searching exact periodic
orbits with similarity signature curves offers certain advantages comparing
with other methods.

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