Complex dynamics in two-dimensional coupling of quadratic maps

Anca Rǎdulescu∗,111Assistant Professor, Department of Mathematics, State
University of New York at New Paltz; New York, USA; Phone: (845) 257-3532;
Email<EMAIL_ADDRESS>Ashelee Collier1

1 Department of Mathematics, SUNY New Paltz

###### Abstract

In the context of complex quadratic networks (CQNs) introduced previously, we
study escape radius and synchronization properties in two dimensional
networks. This establishes the first step towards more general results in
higher-dimensional networks.

##  1 Introduction

In our previous work we have introduced complex quadratic networks (CQNs) as
networks of coupled nodes, the dynamics of which is governed by discrete
quadratic iterations in the complex plane. As mathematical objects, they have
rich and complex behavior, which can exhibit a wide range of phenomena, far
transcending those described for single iterated maps. These behaviors are
governed by the interplay between the properties of individual nodes, the
strength of the coupling between them and the topology of the network, making
them a fascinating subject of study in the emerging field of network science.
Understanding the effect of edge configuration and weights becomes of crucial
interest in the context of understanding and classifying the long-term
behavior of the system. In particular, the study of these networks has
revealed that even simple changes to the edge weights can lead to large-scale
changes in the system’s dynamics. This has significant implications not only
for network science, but also for other fields such as physics, biology, and
engineering, where understanding the dynamics of complex systems is a critical
component.

In order to efficiently quantify the asymptotic dynamics of the system in
response to network changes, we defined, for any $n$-dimensional network, an
extension of the traditional prisoner set (as the set of bounded initial
conditions that have bounded orbits in $\mathbb{C}^{n}$) [4]. We also defined
the _equi-Mandelbrot set_ , an extension of the traditional Mandelbrot set, as
the set of parameters $c\in\mathbb{C}$ for which the given network with
identical nodes $z\to z^{2}+c$ is postcritically bounded in $\mathbb{C}^{n}$).
Since the properties of the critical orbit are complicated in this case by the
fact that is has multiple components, this definition is not equivalent to the
connectedness locus in $\mathbb{C}$ of the network prisoner set, but we
conjectured that some relationship remains between the two objects in the
parameter plane [2].

In the context of CQNs, one can study a similar concept with that of
traditional “synchronization” of nodes’ activity described in the context of
networks of continuous-time oscillators. Instead of requiring for two nodes to
eventually converge to the same attractor, synchronization of two nodes in
CQNs required that the two nodes are simultaneously bounded [1]. To fix ideas,
We further defined node-wise equi-Mandelbrot sets as the parameter regions for
which individual nodes remain bounded (as the critical orbit evolves in
$\mathbb{C}^{n}$). Then we defined two nodes $z_{i}$ and $z_{j}$ to be
M-synchronized is they have the same node-wise equi-Mandelbrot sets, i.e.,
${\cal M}(z_{i})={\cal M}(z_{j})$. With this relationship, the nodes of a
network can be classified into _clusters_ of nodes with identical equi-M sets.

A substantial focus of our work has been on understanding the contribution of
the coupling to the topological properties of the equi-M sets, and to the
grouping and shapes of the synchronization clusters. In our previous work, we
have approached CQNs from a network science standpoint [1, 3], and have
focused primarily on the contribution of the network architectural aspect to
the emerging dynamics. However, in order to have nontrivial architecture, the
network has to be reasonably large. This leap to access higher-dimensional,
interesting “connectivity patterns” omitted a specific analyses of two-
dimensional networks, which can only have virtually trivial architecture
schemes between two nodes.

This case is, nontheless, extremely important in understanding analytically
how coupling between two nodes affects the equi-M sets and their
synchronization. In this paper we focus precisely on this aspect, and embrace
a bottom-up, constructive approach. We show that, by building upon results on
interaction between two nodes taken in isolation, one can determine how this
type of mutual coupling acts more generally when embedded into higher-
dimensional networks.

##  2 Dynamics of two coupled maps

We consider the system of two coupled complex variables $(z_{1},z_{2})$,
evolving discretely according to quadratic complex dynamics $f(z)=z^{2}+c$,
$c\in\mathbb{C}$, and with linear coupling specified by the matrix
$A=\left(\begin{array}[]{cc}a&b\\\ d&f\end{array}\right)$. More precisely, the
2-dimensional map describing the coupled dynamics is given by:

$\displaystyle z_{1}(n+1)$ $\displaystyle=$
$\displaystyle(az_{1}+bz_{2})^{2}+c$ $\displaystyle z_{2}(n+1)$
$\displaystyle=$ $\displaystyle(dz_{1}+fz_{2})^{2}+c$ (1)

For consistency with our more general prior work in complex quadratic networks
(CQNs), we will refer to our two-map coupled systems as 2D-CQNs throughout
this paper.

###  2.1 Escape radius

In this section, we will show that the system of two coupled maps described
above has an escape radius, for most connectivity matrices $A$. To do this, we
will prove a series of lemma, considering separately different cases for the
balance of the connectivity parameters $(a,b,d,f)$.

######  Lemma 2.1.

Suppose the connectivity matrix $A$ is such that $\Delta=\lvert
af\rvert-\lvert bd\rvert\neq 0$. Then there exists a large enough $M>0$ such
that, for an iteration step $n\geq 0$ we have $\lvert z_{1}(n)\rvert>M$ or
$\lvert z_{2}(n)\rvert>M$, then the next iterates $\lvert z_{1}(n+1)\rvert>2M$
or $\lvert z_{2}(n+1)\rvert>2M$.

###### Proof.

Since $\Delta=\lvert af\rvert-\lvert bd\rvert\neq 0$, we can consider
$\displaystyle\sqrt{K_{1}}=\frac{\lvert b\rvert+\lvert
f\rvert}{\lvert\Delta\rvert}$, and $\displaystyle\sqrt{K_{2}}=\frac{\lvert
a\rvert+\lvert d\rvert}{\lvert\Delta\rvert}$, and further:

$M_{1}=K_{1}+\sqrt{K_{1}(K_{1}+\lvert c\rvert)}\text{ and
}M_{2}=K_{2}+\sqrt{K_{2}(K_{2}+\lvert c\rvert)}$

Take $M>\max\\{M_{1},M_{2}\\}$. Suppose that, for some $n\geq 0$, we have
$z_{1}(n)>M$ or $z_{2}(n)>M$. We will show that $z_{1}(n+1)>2M$ or
$z_{2}(n+1)>2M$. We will argue by contradiction; throughout the argument, we
will abbreviate $z_{1}(n)$ and $z_{2}(n)$ to $z_{1}$ and $z_{2}$, when there
is no danger of confusion. Suppose both $z_{1}(n+1)\leq 2M$ and
$z_{2}(n+1)\leq 2M$. Then

$2M\geq\lvert z_{1}(n+1)\rvert=\lvert(az_{1}+bz_{2})^{2}+c\rvert\geq\lvert
az_{1}+bz_{2}\rvert^{2}-\lvert c\rvert\geq(\lvert az_{1}\rvert-\lvert
bz_{2}\rvert)^{2}-\lvert c\rvert$

It follows that

$\lvert\lvert az_{1}\rvert-\lvert bz_{2}\rvert\rvert\leq\sqrt{2M+\lvert
c\rvert}$

Similarly, starting from $\lvert z_{2}(n+1)\rvert\leq 2M$, one can show that:

$\lvert\lvert dz_{1}\rvert-\lvert fz_{2}\rvert\rvert\leq\sqrt{2M+\lvert
c\rvert}$

Call now $\xi_{1}=\lvert f\rvert(\lvert az_{1}\rvert-\lvert bz_{2}\rvert)$ and
$\xi_{2}=\lvert b\rvert(\lvert dz_{1}\rvert-\lvert fz_{2}\rvert)$. Then

$\lvert\xi_{1}-\xi_{2}\rvert=\lvert(\lvert af\rvert-\lvert
bd\rvert)z_{1}\rvert\leq\lvert\xi_{1}\rvert+\lvert\xi_{2}\rvert\leq(\lvert
b\rvert+\lvert f\rvert)\sqrt{2M+\lvert c\rvert}$

hence

$\lvert z_{1}\rvert\leq\frac{\lvert b\rvert+\lvert
f\rvert}{\lvert\Delta\rvert}\sqrt{2M+\lvert c\rvert}$

From our assumption for $M$ large it follows that the right side of (2.1) is
no larger than $M$. Squaring both sides, this is equivalent to
$h(M)=M^{2}-K_{1}(2M+\lvert c\rvert)>0$, which is guaranteed by taking
$M>M_{1}$, the larger of the quadratic roots of $h$. In conclusion, we obtain
that $\lvert z_{1}(n)\rvert\leq M$.

Similarly, if we call $\psi_{1}=\lvert f\rvert(\lvert az_{1}\rvert-\lvert
bz_{2}\rvert)$ and $\psi_{2}=\lvert b\rvert(\lvert dz_{1}\rvert-\lvert
z_{2}\rvert)$, we get that

$\lvert\psi_{1}-\psi_{2}\rvert=\lvert(\lvert af\rvert-\lvert
bd\rvert)z_{2}\rvert\leq(\lvert a\rvert+\lvert d\rvert)\sqrt{2M+\lvert
c\rvert}$

Hence

$z_{2}(n)\leq\sqrt{K_{2}}\sqrt{2M+\lvert c\rvert}\leq M$

(the second part of the double inequality follows from the condition that
$M>M_{2}$). Since $z_{1}(n)$ and $z_{2}(n)$ cannot be simultaneously smaller
than $M$, the contradiction follows.

∎

We then have the following

######  Theorem 2.2.

In the case of $\lvert af\rvert\neq\lvert bd\rvert$, $M=\max\\{M_{1},M_{2}\\}$
acts as an escape radius for the iteration, with $M_{1}$ and $M_{2}$ described
in the proof of Lemma 2.1.

Hence, when $\lvert af\rvert\neq\lvert bd\rvert$, $M=\max\\{M_{1},M_{2}\\}$
acts as an escape radius for the iteration, with $M_{1}$ and $M_{2}$ described
in the proof of Lemma 2.1. Suppose now $\lvert af\rvert\neq\lvert bd\rvert$.
To fix our ideas, suppose that $af=bd$ (the opposite sign case is similar). We
want to see if, in this case, there still exists a large enough $M>0$ such
that, for an iteration step $n\geq 0$ we have $\lvert z_{1}(n)\rvert>M$ or
$\lvert z_{2}(n)\rvert>M$, then the next iterates $\lvert z_{1}(n+1)\rvert>2M$
or $\lvert z_{2}(n+1)\rvert>2M$. We need to analyze a few distinct cases, as
follows:

Case 1: $a,b\neq 0$. Since $af=bd$, we can call $\displaystyle
k=\frac{d}{a}=\frac{f}{b}$. Say that, for some fixed iteration $n$, $\lvert
z_{1}(n)\rvert>M$ and $\lvert z_{2}\rvert>M$.

Consider $g=\min\\{\lvert a\rvert,\lvert b\rvert\\}$. It follows that $\lvert
az_{1}+bz_{2}\rvert>gM$ or $\lvert az_{1}-bz_{2}\rvert>gM$. Indeed, suppose by
contradiction that both $\lvert az_{1}+bz_{2}\rvert\leq gM$ and $\lvert
az_{1}-bz_{2}\rvert\leq gM$. Then

$2\lvert az_{1}\rvert\leq\lvert az_{1}+bz_{2}\rvert+\lvert
az_{1}-bz_{2}\rvert\leq 2gM<2\lvert a\rvert M$

implying that $z_{1}\leq M$. Similarly,

$2\lvert bz_{1}\rvert\leq\lvert az_{1}+bz_{2}\rvert+\lvert
bz_{2}-az_{1}\rvert\leq 2gM<2\lvert b\rvert M$

implies that $z_{2}\leq M$. Since we cannot have simultaneously $\lvert
z_{1}\rvert\leq M$ and $\lvert z_{2}\rvert\leq M$, we get our contradiction.

Suppose $\lvert az_{1}+bz_{2}\rvert>gM$; the case of $\lvert
az_{1}-bz_{2}\rvert>gM$ works similarly and we will skip the proof. We
consider:

$az_{1}(n+1)+bz_{2}(n+1)=(a+bk^{2})(az_{1}+bz_{2})^{2}+c(a+b)$

Hence

$\lvert az_{1}(n+1)+bz_{2}(n+1)\rvert\geq g^{2}M^{2}\lvert
a+bk^{2}\rvert-\lvert c(a+b)\rvert$

Case (i): If $a+bk^{2}\neq 0$, we can consider the quadratic function
$f(M)=g^{2}M^{2}\lvert a+bk^{2}\rvert-2(\lvert a\rvert+\lvert b\rvert)M-\lvert
c(a+b)\rvert$. The larger root $M_{1}>0$ of this function is defined
explicitly as:

$M_{1}=\frac{\lvert a\rvert+\lvert b\rvert+\sqrt{(\lvert a\rvert+\lvert
b\rvert)^{2}+g^{2}\lvert c\rvert(\lvert a\rvert+\lvert b\rvert)\lvert
a+bk^{2}\rvert}}{g^{2}\lvert a+bk^{2}\rvert}$

If $M>M_{1}$, then $f(M)>0$, and $\lvert
az_{1}(n+1)+bz_{2}(n+1)\rvert>2(\lvert a\rvert+\lvert b\rvert)M$. It follows
easily that either $\lvert z_{1}(n+1)\rvert>2M$ or $\lvert
z_{2}(n+1)\rvert>2M$.

Case (ii): If $a+bk^{2}=0$, then $a=-bk^{2}$, $d=-bk^{3}$ and $f=bk$. Hence
the system becomes: $z_{1}(n+1)=b^{2}(-k^{2}z_{1}+z_{2})^{2}+c$ and
$z_{2}(n+1)=b^{2}k^{2}(-k^{2}z_{1}+z_{2})^{2}+c$. Call
$\xi(n)=-k^{2}z_{1}(n)+z_{2}(n)$. Then one can easily see that
$\xi(n)=c(1-k^{2})$ is constant for $n\geq 1$. It follows that
$z_{1}(n+1)=b^{2}\xi^{2}(n)+c$ and $z_{2}=b^{2}k^{2}\xi^{2}(n)+c$ are also
constant for $n\geq 1$.

Case 2: $a=0$, $b\neq 0$. Then automatically $d=0$, and the system becomes:
$z_{1}(n+1)=(bz_{2})^{2}+c$, $z_{2}(n+1)=(fz_{2})^{2}+c$. The second equation
is decoupled. If $f\neq 0$, one can make the change of variables
$\xi=f^{2}z_{2}$, and rewrite the equation as $\xi(n+1)=\xi^{2}+c_{f}$, where
$c_{f}=f^{2}c$. It follows from the traditional theory on escape radius for
single quadratic maps that $M=\max\\{2/f^{2},\lvert c\rvert\\}$ is an escape
radius for $z_{2}$. Hence the whole system has an escape radius, since $z_{1}$
only depends on $z_{2}$. It $f=0$, the system is trivial (both $z_{2}$ and
$z_{1}$ are constant).

Case 3: $b=0$, $a\neq 0$. Then automatically $f=0$, and the system becomes
$z_{1}(n+1)=(az_{1})^{2}+c$, $z_{2}(n+1)=(dz_{1})^{2}+c$. The proof follows
similarly with Case 2.

Case 4: $a=b=0$. Then $z_{1}(n)=c$ for all $n\geq 0$ (constant sequence) and
$z_{2}(n+1)=(dc+fz_{2})^{2}+c$. Withe the change of variable
$\xi=f^{2}z_{2}+dc$, the second equation becomes:
$\xi(n+1)=\xi^{2}+c(f^{2}+d)$, which has an escape radius.

Altogether, these cases imply that the escape radius property remains valid
when $\lvert af\rvert=\lvert bd\rvert$, as long as $a+bk^{2}\neq 0$. One can
easily find singular matrices $A$ (i.e., $af-bd=0$) with $a+bk^{2}=0$, for
which the corresponding system does not satisfy the escape radius property.
Take for example $a=1$, $b=-1$, $k=1$, in which case
$z_{1}(n+1)=z_{2}(n+1)=(z_{2}-z_{1})^{2}+c$. Initial conditions with
arbitrarily large $z_{1}(0)=z_{2}(0)$ immediately collapse to
$z_{1}(n)=z_{2}(n)=c$, for all $n\geq 1$, showing that there is no radius
$M>0$ past which orbits automatically escape to infinity. A different and more
difficult question is whether there are networks such that, given any large
$M$, there are $c$ values for which the critical orbit can grow larger than
$N$, but remain asymptotically bounded. This can inform us if we can have a
viable computational test for checking whether a $c$ can be unquestionably
excluded from the equi-M set of a given network.

###  2.2 Synchronization in systems of two coupled maps

In previous work on CQNs, we showed that nodes can independently have
different behaviors, with some nodes remaining bounded, while other nodes in
the same coupled network escaping asymptotically to infinity.

######  Lemma 2.3.

If $z_{j}$ has a nontrivial input to $z_{k}$, and if $z_{j}(n)$ is bounded in
$\mathbb{C}$ as $n\to\infty$, then $z_{k}(n)$ is also bounded.

###### Proof.

We can assume without loss of generality that $z_{1}$ is bounded, and that the
input $b$ from $z_{2}$ to $z_{1}$ is nonzero, and prove that $z_{2}$ is also
bounded. Since $z_{1}$ is bounded, there exists $M>0$ such that $\lvert
z_{1}(n)\rvert\leq M$, for all $n\geq 0$. Since $\lvert
z_{1}(n+1)\rvert\geq\lvert az_{1}+bz_{2}\rvert^{2}-\lvert c\rvert$, it follows
that $\lvert bz_{2}\rvert-\lvert az_{1}\rvert\leq\sqrt{\lvert
z_{1}(n+1)\rvert+\lvert c\rvert}$. Then:

$\lvert bz_{2}\rvert\leq\sqrt{\lvert z_{1}(n+1)\rvert+\lvert c\rvert}+\lvert
az_{1}\lvert\leq\sqrt{M+\lvert c\rvert}+\lvert a\rvert M$

Then, if $b\neq 0$, we have, for all $n\geq 0$:

$\lvert z_{1}(n)\rvert\leq K=\frac{\leq\sqrt{M+\lvert c\rvert}+\lvert f\rvert
M}{\lvert d\rvert}$

insuring that $z_{2}(n)$ is bounded.

∎

######  Proposition 2.4.

If $z_{1}$ and $z_{2}$ are nontrivially interconnected (i.e., both $b,d\neq
0$), then they are simultaneously bounded.

In previous work, we have looked in particular at the possible behaviors of
the components of the critical orbit of the network, and their
“synchronization” into different clusters with identical bounded versus
unbounded behavior. More precisely:

######  Definition 2.5.

For each node $z_{k}$, $1\leq k\leq n$, we define the node-wise equi-M set
${\cal M}_{k}$ as the parameter locus $c\in\mathbb{C}$ for which the component
corresponding to the node $z_{k}$ of the critical multi-orbit is bounded. We
say that two network nodes $z_{i}$ and $z_{j}$ are M-synchronized, if their
equi-M sets ${\cal M}(z_{i})$ and ${\cal M}(z_{j})$ are identical. We also say
that the nodes belong to the same M-synchronization cluster (or simply
M-cluster) of the network.

In this context, we can reinterpret the results in this sections as

######  Corollary 2.6.

If in the network described by (1) the nodes $z_{1}$ and $z_{2}$ are
nontrivially interconnected (i.e., $b,d\neq 0$), then the nodes are
simultaneously bounded. In particular, ${\cal M}(z_{1})={\cal M}(z_{2})$.

Figure 1: _Evolution of the equi M set as the value of $d$ increases. The
other entries of the matrix $A$ were fixed to $a=0.6$, $b=0.8$, $f=0.4$. The
panel illustrates the equi-M contour (identical between the two nodes), using
different colors for different values of $d$, as follows: $d=-0.2$ (grey);
$d=0.01$ (cyan); $d=0.1$ (orange); $d=0.3$ (blue, scaling of the traditional
Mandelbrot set, since $d$ is the critical values for which $\det(A)=0$); $d=1$
(pink); $d=2$ (green); $d=3$ (brown)._

The proof follows directly from Proposition 2.4. Notice that if in addition
the matrix $A$ is singular, then $k=\frac{d}{a}=\frac{f}{b}$ and, with the
notation $\xi=(a+bk^{2})(az_{1}+bz_{2})$, we have
$\xi(n+1)=\xi^{2}+(a+bk^{2})(a+b)c$. The values of $c$ for which $z_{1}$ and
$z_{2}$ are (simultaneously) bounded are the same as the values of $c$ for
which $(a+bk^{2})(a+b)c$ is in the traditional Mandelbrot set. Hence the equi
M set (identical between the two nodes) is in this case a rescaling of the
traditional set. Figure 1 illustrates how the shape of the equi M set of the
two synchronized nodes evolves as the value of $d$ increases through the
negative and positive range as the other three entries of $A$ are fixed,
crossing through the critical value which makes $A$ singular. Notice that, as
$d$ is increasing from negative values, the shape slowly approaches that of
the traditional Mandelbrot set (achieved for the critical $d=0.3$), and then
slowly degrades away from this shape as $d$ continues to increase.

This sheds some light on the effects of the coupling when the nodes are
interconnected ($b,d\neq 0$). At the other end of the spectrum, we have the
case when the nodes both act independently. This case is trivial, as described
in the following lemma:

######  Lemma 2.7.

If $b=d=0$ and $a,f\neq 0$, then the two node-wise equi M sets are both scaled
versions of the traditional Mandelbrot set.

###### Proof.

We have $z_{1}(n+1)=(az_{1})^{2}+c$, and $z_{2}(n+1)=(fz_{2})^{2}+c$. Then
$c\in{\cal M}(z_{1})\Longleftrightarrow ac\in{\cal M}\Longleftrightarrow{\cal
M}(z_{1})=\frac{1}{a^{2}}{\cal M}$. Similarly, ${\cal
M}(z_{2})=\frac{1}{f^{2}}{\cal M}$. ∎

A natural question to ask next is whether the nodes need to necessarily be
nontrivially interconnected in order to be M-synchronized. The answer to that
is negative. The following lemma gives a weaker condition that insures
M-synchronization of the two nodes (although it does not deliver the stronger
result in Corollary 2.4).

######  Lemma 2.8.

Suppose $z_{1}(n+1)=(az_{1})^{2}+c$ (decoupled), and
$z_{2}(n+1)=(dz_{1}+fz_{2})^{2}+c$. Then, for $\lvert f\rvert$ sufficiently
small, if the $z_{1}$ component of the critical orbit is bounded, then the
$z_{2}$ component is also bounded.

###### Proof.

Since $z_{1}$ is bounded, there exists a large enough $M>0$ such that
$z_{1}(n)\leq M$, for all $n\geq 0$. (Note: the value of $M$ depends on $a$
and $\lvert c\rvert$.) Assume $\lvert f\rvert$ is small enough so that

$4M\lvert fd\rvert+4\lvert f\rvert^{2}\lvert c\rvert<1$ (2)

Then the discriminant of the quadratic function in $X$

$h(X)=\lvert f\rvert^{2}X^{2}+(2M\lvert df\rvert-1)X+M^{2}\lvert
d\rvert^{2}+\lvert c\rvert$

is $\Delta=1-4M\lvert fd\rvert-4\lvert f\rvert^{2}\lvert c\rvert>0$. If we
call $X_{1}<X_{2}$ the two distinct roots of $h(X)=0$, then the larger root
$X_{2}$is positive. Consider $\max\\{X_{1},0\\}<K<X_{2}$. Clearly, we have
that $h(K)<0$. We will show inductively that $K$ is an upper bound for
$z_{2}(n)$, for all $n\geq 0$. Indeed, $z_{2}(0)=0<K$. Now suppose $\lvert
z_{2}\rvert=\lvert z_{2}(n)\rvert<K$, and consider

$\displaystyle\lvert z_{2}(n+1)\rvert$ $\displaystyle=$
$\displaystyle\lvert(dz_{1}+fz_{2})^{2}+c\rvert\leq\lvert
dz_{1}+fz_{2}\rvert^{2}+\lvert c\rvert\leq(\lvert d\rvert M+\lvert
fz_{2}\rvert)^{2}+\lvert c\rvert$ (3) $\displaystyle=$ $\displaystyle\lvert
d\rvert^{2}M^{2}+2\lvert fd\rvert M\lvert z_{2}\rvert+\lvert f\rvert^{2}\lvert
z_{2}\rvert^{2}+\lvert c\rvert\leq\lvert d\rvert^{2}M^{2}+2\lvert fd\rvert
MK+\lvert f\rvert^{2}K^{2}+\lvert c\rvert$ $\displaystyle=$ $\displaystyle
h(K)+K<K$

Hence $\lvert z_{2}(n+1)\rvert<K$, and the induction is complete. In
conclusion, $K$ is an upper bound for the critical component $z_{2}$.
∎

######  Corollary 2.9.

If $z_{1}(n+1)=(az_{1})^{2}+c$, and $z_{2}(n+1)=(dz_{1}+fz_{2})^{2}+c$, with
$a,d\neq 0$ and $\lvert f\rvert$ sufficiently small, then ${\cal
M}(z_{1})={\cal M}(z_{2})$.

###### Proof.

Follows easily from Lemmas 2.3 and 2.8. ∎

Remark. Corollary 2.9 states that, if node $z_{2}$ depends on $z_{1}$, but
$z_{1}$ does not depend on $z_{2}$, then a sufficiently weak self-dependence
of $z_{2}$ will still insure M-synchronization. What weak means in this
context, and how this bound depends on the other system parameters will become
obvious in the proof of the lemma.

Figure 2 illustrates the behavior of the equi M contours for positive values
of $f$, as $\lvert f\rvert$ decreases, when the first node is decoupled
($b=0$), for different fixed values of $a\neq 0$ and $d\neq 0$. Notice that
the nodes eventually synchronize in each case for small enough values of
$\lvert f\rvert$. In addition, as expected from the proof of Corollary 2.9,
the threshold depends on the values of $a$ and $d$.

Figure 2: _Equi-M Evolution of M synchrinization in a 2D CQN when changing
$f$. Each panel shows the two contours of the node-wise equi M sets, in green
(for $z_{1}$) and in blue (for $z_{2}$). Top-down, the panels illustrate,
respectively:._

In future work, we are interested to investigate necessary and sufficient
conditions under which this result can be extended to larger size networks.

## References

  * [1] Anca Radulescu, Danae Evans, Amani-Daisa Augustin, Anthony Cooper, Johan Nakuci, and Sarah Muldoon.  Synchronization and clustering in complex quadratic networks.  arXiv:2205.02390, 2022.
  * [2] Anca Rǎdulescu and Simone Evans.  Asymptotic sets in networks of coupled quadratic nodes.  Journal of Complex Networks, 7(3):315–345, 2019.
  * [3] Anca Radulescu, Johan Nakuci, Simone Evans, and Sarah Muldoon.  Computing brain networks with complex dynamics.  arXiv:2209.05268, 2022.
  * [4] Anca Rǎdulescu and Ariel Pignatelli.  Real and complex behavior for networks of coupled logistic maps.  Nonlinear Dynamics, 87(2):1295–1313, 2017.


Bulk-boundary and RPS Thermodynamics
from Topology perspective
Jafar Sadeghi ⋆111Email<EMAIL_ADDRESS>Mohammad Reza Alipour ⋆222Email:
<EMAIL_ADDRESS>Saeed Noori Gashti†,⋆333Email:
<EMAIL_ADDRESS>Mohammad Ali S. Afshar ⋆444Email:
<EMAIL_ADDRESS>
⋆Department of Physics, Faculty of Basic Sciences,
University of Mazandaran P. O. Box 47416-95447, Babolsar, Iran
†School of Physics, Damghan University, P. O. Box 3671641167, Damghan, Iran

###### Abstract

In this article, we investigate the bulk-boundary and restricted phase space
(RPS) thermodynamics of Rissner-Nordström (R-N) AdS and 6-dimensional charged
Gauss-Bonnet AdS black holes. Also, we examine the topological characteristics
of the considered black holes and compare them with the extended
thermodynamics results. In fact, we have found that the topological behavior
of the bulk-boundary thermodynamics is the same as that of the extended
thermodynamics, whereas the RPS thermodynamics exhibits a distinct behavior.
We also demonstrate that within the RPS formalism, there is only one critical
point with a topological charge of +1 $(Q_{t}=+1)$. Additionally, for RPS
formalism, the inclusion of higher derivative curvature terms in the form of
Gauss-Bonnet gravity does not alter the topological classification of critical
points in charged AdS black holes.

Keywords: Bulk-Boundary, Restricted Phase Space, Topological Charge,
Thermodynamics

###### Contents

  1. 1 Introduction
  2. 2 Thermodynamic topology
  3. 3 Bulk-Boundary thermodynamics
    1. 3.1 Reissner-Nordström AdS black hole
    2. 3.2 Charged Gauss-Bonnet AdS black holes
  4. 4 Restricted Phase Space
    1. 4.1 Reissner-Nordström AdS black hole
    2. 4.2 Charged Gauss-Bonnet AdS black holes
  5. 5 Conclusions
  6. 6 Acknowledgments

##  1 Introduction

Thermodynamics of black holes is the study of how black holes behave as
thermodynamic systems, such as temperature, entropy, heat capacity, etc. One
of the main challenges is to understand the phase transitions and critical
points of black holes in different contexts, such as anti-de Sitter (AdS) or
de Sitter (dS) space, with or without electric charge, nonlinear
electromagnetic fields, etc. some important points in thermodynamics concepts
are critical points and phase transitions. Phase transition is a change of
state of matter from one phase to another, such as from liquid to gas or from
solid to liquid [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. In
thermodynamics, phase transitions are characterized by discontinuities or
singularities in some thermodynamic quantities, such as pressure, volume,
temperature, etc. Phase transitions can be classified into different orders
depending on the behavior of these quantities near the transition point. A
recent approach to studying the thermodynamics and phase transitions of black
holes is to use Duan’s topological current-mapping theory, which introduces
topology to the study of black hole thermodynamics by assigning a topological
charge to each critical point in the phase diagram. This method can reveal
some new features and classifications of black hole thermodynamics that are
not captured by conventional methods. For example, it can distinguish between
conventional and novel critical points, which have different implications for
the first-order phase transition. It can also classify different types of
black holes into different topological classes based on their topological
charges. Many important works have been done so far on investigating
thermodynamics from the point of view of topology, which have explored
different aspects of this topic. For further study and research, you can see
the Ref.s[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. In
this article, we aim to conduct thermodynamic studies by comparing two
different spaces, namely Bulk-Boundary and Restricted Phase Space, for
different types of black holes, which have not been investigated so far. We
will compare the results with each other and with other works in the
literature.

Restricted phase space thermodynamics (RPST) is a new formalism for studying
the thermodynamics of AdS black holes. The construction is based on Visser’s
holographic thermodynamics but with the AdS radius must be fixed. This means
that the pressure and the conjugate volume are excluded from the thermodynamic
variables, and only the central charge $C$ of the dual CFT and the chemical
potential $\mu$ are included. The RPST formalism has some advantages over the
conventional extended phase space (EPS) formalism, which treats the
cosmological constant as pressure and introduces an extra pair of state
variables (P, V). For example, the RPST formalism can avoid some ambiguities
and inconsistencies in the EPS formalism, such as the definition of mass, the
Smarr relation, the first law of thermodynamics, etc. The RPST formalism can
also make the Euler relation hold automatically, and the first-order
homogeneity of the mass and the zeroth-order homogeneity of the intensive
variables are made explicit. The RPST formalism can be applied to various
types of AdS black holes, such as Kerr-AdS, RN-AdS [37, 38], and reveal some
interesting thermodynamic behaviors and phase transitions. For example, it can
show that there is a first-order supercritical phase equilibrium in the $T-S$
processes at a fixed nonvanishing charge or angular momentum, while at
vanishing charge or angular momentum or at fixed potentials, there is always a
non-equilibrium transition from a small unstable black hole state to a large
stable black hole state. Moreover, there is a Hawking-Page phase transition in
the $\mu-C$ processes. The RPST formalism can also classify different types of
black holes into different topological classes based on their topological
charges of critical points [32, 33, 34, 35, 36, 37, 38].

Bulk-boundary thermodynamic equivalence is a concept that states that the
thermodynamics and phase transitions of AdS black holes in the bulk gravity
theory are equivalent to those of the dual CFT on the boundary, from the point
of view of topology. This means that the bulk and boundary thermodynamics have
the same topological charges of critical points, which can be calculated by
the residue method. This concept was proposed by Zhang and Jiang in their
paper [39] Bulk-boundary thermodynamic equivalence: a topology viewpoint,
where they set both the cosmological constant and Newton’s constant to be
dynamical, and developed mass/energy formulas in terms of thermodynamic state
functions for three different scenarios: the extended thermodynamics, which
includes both pressure and volume; the mixed thermodynamics, which includes
only pressure; and the boundary CFT thermodynamics, which includes neither
pressure nor volume. They applied these formulas to study the thermodynamics
and phase transitions of charged AdS black holes in various dimensions and
with various charges or angular momenta, and compared them with each other and
with other works in the literature. They used the residue method to study the
topological properties of the phase transitions. They defined a vector field
$\varphi$ on a complex plane $z$, which is related to the temperature T and
entropy S. They also defined a complex function $\Omega$ on $z$, which is
related to different thermodynamic potentials for extended thermodynamics, ,
mixed thermodynamics, and boundary CFT thermodynamics. They used these
definitions to calculate the residues of $\varphi/\Omega$ at different
critical points on the z-plane and used them as topological charges to
classify different types of phase transitions. They found that the bulk and
boundary thermodynamics are topologically equivalent for both criticalities
and first-order phase transitions in the canonical ensembles, as well as for
the Hawking-Page (-like) phase transitions in the grand canonical ensembles.
They also found that these three kinds of phase transitions are distinguished
by their unique topological charges. They claimed that their results exemplify
the gravity-gauge duality in terms of topology [39, 40, 41].

Therefore, we choose different black holes and study their topological
behavior in two spaces bulk-boundary, restricted phase space, and compare the
results.
_In[39], they showed that the total topological charge remains constant in
extended, mixed, and CFT thermodynamics. On the other hand, we show that in
RPS thermodynamics, the total topological charge changes in comparison with
other thermodynamics. This effect may arise from the fixed cosmological
constant._
Based on the above concepts, we can organize the article as follows.
In section 2, we will briefly explain the thermodynamic topology. In section
3, we will perform calculations related to thermodynamic topology from the
point of view of bulk-boundary for two black holes, namely the R-N AdS black
hole and the charged Gauss-Bonnet AdS black hole. In section 4, we will do a
similar process for restricted Phase Space and compare the results with each
other and with the latest results in the literature. Finally, we will
summarize the results in section 5.

##  2 Thermodynamic topology

The critical points of the phase diagram of thermodynamic systems can be
classified into three categories: conventional, novel, and neutral, based on
the topological charge given by the winding number of Duan’s $\phi$-mapping
theory. This theory uses a scalar thermodynamic function $\Phi$ and a vector
field $\phi$ to study the topological features and stability conditions of
black hole solutions in various gravity theories. The zero points of $\phi$
correspond to the critical points of $\Phi$, and the topological charge is
determined by the deflection angle of $\phi$ along a closed contour around the
zero point. The conventional class has a negative charge, the novel class has
a positive charge, and the neutral class has a zero charge. Different gravity
theories and parameters may affect the topological class of black hole
solutions [16, 42, 21]. Researchers have recently discovered that black holes
can be classified into two types based on their thermodynamic behavior. These
types are called conventional and novel, and they are determined by the
topological charge of the critical points on the phase diagram. The phase
diagram shows how the temperature of the black hole depends on other
properties like entropy, pressure, and some extra parameters ($x^{i}$) that
vary depending on the gravity theory and the black hole solution. These
properties are considered as variables in the extended thermodynamic systems.
The temperature function for these systems is given by [16],

$T=T(S,P,x^{i}).$ (1)

An important fact is that the critical points on the phase diagram of a
thermodynamic system, i.e. where the system changes abruptly from one phase to
another, can be found by solving the conditions for a stationary point of
inflection,

$(\frac{\partial T}{\partial S})=0.$ (2)

We have a scalar thermodynamic function in the following form

$\Phi=\frac{1}{\sin\theta}T(S,x^{i}).$ (3)

The above equation is obtained by eliminating the variable $P$ using the
equation of (2) and adding the factor $\frac{1}{\sin\theta}$ to simplify the
analysis. They also introduce a new vector field to develop the framework of
Duan’s $\phi$-mapping theory.

$\phi=(\phi^{S},\phi^{\theta}),$ (4)

where

$\phi^{S}=(\partial_{S}\Phi),\hskip
28.45274pt\phi^{\theta}=(\partial_{\theta}\Phi).$ (5)

Also, the charge can be considered as,

$Q_{t}=\int_{\Sigma}\Sigma_{i=1}^{N}\beta_{i}\eta_{i}\delta^{2}(\overrightarrow{x}-\overrightarrow{z}_{i})d^{2}x=\Sigma_{i=1}^{N}\beta_{i}\eta_{i}=\Sigma_{i=1}^{N}\omega_{i},$
(6)

where $\eta_{i}$ is the sign of the zero component of the topological current
$J^{0}(\phi/x)_{z_{i}}$ at the i-th zero point of $\phi$, and it can be either
$+1$ or $-1$. $\beta_{i}$ and $\omega_{i}$ are positive integers (Hopf index)
that count the number of loops and the winding number around the i-th zero
point of $\phi$. The equation above shows that the topological charge $Q_{t}$
is non-zero only at the zero points of $\phi$. Thus, each critical point has a
topological charge given by the winding number, and it can be positive or
negative. This leads to two different topological classes (conventional and
novel), where the negative charge corresponds to conventional and the positive
charge corresponds to novel. To visualize these topological classes, one can
use a plot of $(\theta-r)$ or $(\theta-S)$ with some smooth curves $C$ that go
around the zero points in a positive direction. As a simplification, chooses
$C$ to be an ellipse centered at $(r_{0},\frac{\pi}{2})$ with a parameter
$\vartheta$ that ranges from $0$ to $2\pi$ [42],

$r=a\cos\vartheta+r_{0},\hskip 28.45274pt\theta=b\sin\vartheta+\frac{\pi}{2}.$
(7)

Therefore, the topological charge that given by the winding number can be
obtained by calculating the deflection $\Omega(\vartheta)$ of the vector field
$\phi$. So we will have,

$Q_{t}=\frac{1}{2\pi}\Omega(2\pi),$ (8)

where

$\Omega(\vartheta)=\int_{0}^{\vartheta}\epsilon_{ab}n^{a}\partial_{\vartheta}n^{b}d\vartheta.$
(9)

Now, we apply the equations mentioned in this section to two types of black
holes: R-N AdS black hole and charged Gauss-Bonnet AdS black hole. We use the
bulk-boundary and restricted phase space respectively, and describe the
results of our work in detail in the following sections.

##  3 Bulk-Boundary thermodynamics

Bulk-boundary thermodynamics is a formalism that relates the thermodynamic
quantities of AdS black holes in the bulk to those of the dual CFT on the
boundary via the AdS/CFT correspondence. The bulk quantities include the mass,
temperature, entropy, angular momentum, and electric potential of the black
hole, while the boundary quantities include the energy, temperature, entropy,
angular velocity, and chemical potential of the CFT. The bulk-boundary
thermodynamics shows that the first law of thermodynamics and the Smarr
relation are satisfied by both the bulk and the boundary quantities. The bulk-
boundary thermodynamics also reveals some interesting phase transitions and
critical behaviors of the AdS black holes and their dual CFTs.

###  3.1 Reissner-Nordström AdS black hole

A R-N AdS black hole is a type of charged black hole that has a negative
cosmological constant. It is a solution to the Einstein-Maxwell equations that
describe the gravitational field of a charged, non-rotating, spherically
symmetric body. It has two event horizons: an inner and an outer one. The R-N
AdS black hole has some interesting properties, such as casting a shadow,
exhibiting a phase transition similar to the van der Waals system, and having
quantum gravitational corrections to its metric. Now we apply the equations
mentioned in the section 2 in R-N AdS black hole. The metric of the R-N AdS
black hole is given by [37]:

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2}_{2}$

where $d\Omega^{2}_{2}$ is the metric of a unit two-sphere, and

$f(r)=1-\frac{2GM}{r}+\frac{GQ^{2}}{4r^{2}}+\frac{r^{2}}{\ell^{2}}$

where $M$ and $Q$ are related to the mass and charge of the black hole
respectively, and $\ell$ is the radius of the AdS space. So, the temperature
of this black hole is expressed in the following form

$\begin{split}T=\frac{-G\ell^{2}Q^{2}+\ell^{2}r_{h}^{2}+3r_{h}^{4}}{4\pi\ell^{2}r_{h}^{3}}\end{split}$
(10)

We remove the $\ell$ parameter by setting $\frac{\partial T}{\partial
r_{h}}=0$ and consider the $\Phi$ function as follows,

$\begin{split}\Phi=\frac{r_{h}^{2}-2GQ^{2}}{2\pi
r_{h}^{3}\sin(\theta)}\end{split}$ (11)

Then, using relation (11), we obtain vector field
$\phi^{r}=\frac{\partial\Phi}{\partial r_{h}}$ and
$\phi^{\theta}=\frac{\partial\Phi}{\partial\theta}$,

$\begin{split}\phi^{r}=\frac{\csc(\theta)\left(6GQ^{2}-r_{h}^{2}\right)}{2\pi
r_{h}^{4}},\qquad\qquad\phi^{\theta}=\frac{\cot(\theta)\csc(\theta)\left(2GQ^{2}-r_{h}^{2}\right)}{2\pi
r_{h}^{3}}\end{split}$ (12)

The normalized vector field is therefore
$n=\bigg{(}\frac{\phi^{r}}{\|\phi\|},\frac{\phi^{\theta}}{\|\phi\|}\bigg{)}$.
So, we have,

$\begin{split}&n^{r}=\frac{\left(6GQ^{2}-r_{h}^{2}\right)}{\sqrt{r_{h}^{2}\cot^{2}(\theta)\left(r_{h}^{2}-2GQ^{2}\right)^{2}+\left(r_{h}^{2}-6GQ^{2}\right)^{2}}}\\\
&n^{\theta}=\frac{r_{h}\cot(\theta)\left(2GQ^{2}-r_{h}^{2}\right)}{\sqrt{r_{h}^{2}\cot^{2}(\theta)\left(r_{h}^{2}-2GQ^{2}\right)^{2}+\left(r_{h}^{2}-6GQ^{2}\right)^{2}}}\end{split}$
(13)

To determine the topological charge of a critical point (where
$\phi^{r}=\phi^{\theta}=0$ ), it is necessary to find its winding number
$w_{i}$. For this purpose, we seek assistance from the orthogonal $\theta-
r_{h}$ plane. In the following discussion, we set $P=1$ and
$\ell=\sqrt[4]{\frac{3C}{8\pi}}$.

Figure 1: The vector field $n$ is represented by the arrows on the
$r_{h}-\theta$ plane. With the $Q=1,C=1,P=1$

As shown in Figure (1), the topological charge for the R-N AdS black hole in
bulk-boundary thermodynamics is $Q_{t}=-1$. In fact, this finding aligns with
the topological charge obtained in the extended black hole thermodynamics
outlined in [16].

###  3.2 Charged Gauss-Bonnet AdS black holes

A charged Gauss-Bonnet AdS black hole is a type of charged black hole that has
a negative cosmological constant and a Gauss-Bonnet term in the action. The
charged Gauss-Bonnet AdS black hole has some interesting properties, such as
having a thermodynamic geometry, exhibiting a phase transition similar to the
R-N AdS black hole, and having effects on the shadow, energy emission rate and
quasinormal modes of the black hole. The metric of the 6-dimensional charged
Gauss-Bonnet AdS black hole is given by [43, 17],

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2}_{2}$

where $d\Omega^{2}_{2}$ is the metric of a unit two-sphere, and

$f(r)=1+\frac{r^{2}}{2\tilde{\alpha}}\left(1-\sqrt{1+\frac{16\pi\tilde{\alpha}GM}{\Sigma_{k}r^{5}}-\frac{2\tilde{\alpha}GQ^{2}}{3r^{8}}-\frac{4\tilde{\alpha}}{\ell^{2}}}\right)$

where $\tilde{\alpha}=6\alpha$ $M$, $Q$ are related to the mass and charge of
the black hole respectively, $\alpha$ is the Gauss-Bonnet coupling parameter,
and $\ell$ is the radius of the AdS space. To simplify, the area of a
6-dimensional unit sphere is denoted as $\Sigma_{k}=1$. We use the articles
[43, 17] and have the temperature of the charged Gauss-Bonnet AdS black holes
in 6 dimensions,

$\begin{split}T=\frac{\tilde{\alpha}-\frac{GQ^{2}}{2r_{h}^{4}}+\frac{5r_{h}^{4}}{\ell^{2}}+3r_{h}^{2}}{4\pi
r_{h}\left(2\tilde{\alpha}+r_{h}^{2}\right)}\end{split}$ (14)

By setting $\frac{\partial T}{\partial r_{h}}=0$, we eliminate the $\ell$
parameter and focus on the $\Phi$ function,

$\begin{split}\Phi=\frac{1}{\sin(\theta)}\bigg{(}\frac{-2GQ^{2}+3r_{h}^{6}+2\tilde{\alpha}r_{h}^{4}}{2\pi
r_{h}^{7}+12\pi\tilde{\alpha}r_{h}^{5}}\bigg{)}\end{split}$ (15)

and

$\begin{split}&\phi^{r}=\frac{\partial\Phi}{\partial
r_{h}}=\frac{\csc(\theta)\left(2GQ^{2}\left(30\tilde{\alpha}+7r_{h}^{2}\right)-3r_{h}^{4}\left(r_{h}^{2}-2\tilde{\alpha}\right)^{2}\right)}{2\pi
r_{h}^{6}\left(6\tilde{\alpha}+r_{h}^{2}\right)^{2}},\\\
&\phi^{\theta}=\frac{\partial\Phi}{\partial\theta}=\frac{\cot(\theta)\csc(\theta)\left(2GQ^{2}-3r_{h}^{6}-2\tilde{\alpha}r_{h}^{4}\right)}{2\pi
r_{h}^{7}+12\pi\tilde{\alpha}r^{5}}\end{split}$ (16)

We also obtain the normalized vector field as follows,

$\begin{split}&n^{r}=\frac{2GQ^{2}\left(30\alpha+7r^{2}\right)-3r^{4}\left(r^{2}-2\alpha\right)^{2}}{\sqrt{\left(3r^{4}\left(r^{2}-2\alpha\right)^{2}-2GQ^{2}\left(30\alpha+7r^{2}\right)\right)^{2}+r^{2}\cot^{2}(\theta)\left(6\alpha+r^{2}\right)^{2}\left(-2GQ^{2}+3r^{6}+2\alpha
r^{4}\right)^{2}}}\\\
&n^{\theta}=\frac{r\cot(\theta)\left(6\alpha+r^{2}\right)\left(2GQ^{2}-3r^{6}-2\alpha
r^{4}\right)}{\sqrt{\left(3r^{4}\left(r^{2}-2\alpha\right)^{2}-2GQ^{2}\left(30\alpha+7r^{2}\right)\right)^{2}+r^{2}\cot^{2}(\theta)\left(6\alpha+r^{2}\right)^{2}\left(-2GQ^{2}+3r^{6}+2\alpha
r^{4}\right)^{2}}}\end{split}$ (17)

Then, to determine the topological charge of the critical points, we draw the
$r_{h}-\theta$ diagram using equation (17).

Figure 2: The vector field $n$ is represented by the arrows on the
$r_{h}-\theta$ plane. With the $C=30,P=0.015,\tilde{\alpha}=0.5$ and (a)
$Q=0.04$, (b) $Q=1$.

As shown in [43], for the 6D charged Gauss-Bonnet AdS black hole in bulk-
boundary thermodynamics, when $Q<Q_{3}$, we have three critical points, and
when $Q>Q_{3}$, we have only one critical point. According to Figure (2), we
find that in all cases in bulk-boundary thermodynamics, the total topological
charge is -1 $(Q_{t}=-1)$. This result is consistent with [17] that the total
topological charge of the 6D charged Gauss-Bonnet AdS black hole in the
extended thermodynamics is -1 $(Q_{t}=-1)$ in all cases.

##  4 Restricted Phase Space

Restricted phase space (RPS) is a new formalism for thermodynamics of AdS
black holes that excludes the pressure and the conjugate volume as
thermodynamic variables, but includes the central charge of the dual CFT and
the chemical potential as intensive variables. The RPS formalism is based on
Visser’s holographic thermodynamics, but with the AdS radius must be fixed.
The RPS formalism has some advantages over the extended Phase Space (EPS)
formalism, such as avoiding the ambiguity of the Smarr relation and the Euler
relation, and making explicit the scaling properties of the equations of
state. The RPS formalism has been applied to various AdS black hole solutions,
such as R-N AdS, Kerr-AdS, and revealed some interesting thermodynamic
behaviors, such as phase transitions, criticalities, and Hawking-Page
transitions. Now, we will perform the same process as in the previous sections
for the mentioned black hole and this time in restricted phase space. We will
compare the results with other works in the literature.

###  4.1 Reissner-Nordström AdS black hole

We use [37] and for temperature, in the RPS formalism, we have

$\begin{split}T=\frac{\pi CS-\pi^{2}\hat{Q}^{2}+3S^{2}}{4\pi^{3/2}\ell
S\sqrt{CS}}\end{split}$ (18)

First, we remove the $\hat{Q}$ parameter when the condition $\frac{\partial
T}{\partial S}=0$ is met and form the $\phi$ function as follows,

$\begin{split}\Phi=\frac{1}{\sin(\theta)}T(S,C)=\frac{\pi
C+6S}{\sin(\theta)\left(6\pi^{3/2}\ell\sqrt{CS}\right)}\end{split}$ (19)

Then, utilizing the relation of (19), we construct the vector field,

$\begin{split}&\phi^{S}=-\frac{C(\pi
C-6S)\csc(\theta)}{12\pi^{3/2}\ell(CS)^{3/2}},\\\ &\phi^{\theta}=-\frac{(\pi
C+6S)\cot(\theta)\csc(\theta)}{6\pi^{3/2}\ell\sqrt{CS}}\end{split}$ (20)

and

$\begin{split}&n^{S}=\frac{\partial\phi^{S}}{\partial S}=\frac{6S-\pi
C}{\sqrt{\pi^{2}C^{2}\left(4S^{2}+1\right)+12\pi
CS\left(4S^{2}-1\right)+36\left(4S^{4}+S^{2}\right)}},\\\
&n^{\theta}=\frac{\partial\phi^{\theta}}{\partial\theta}=-\frac{2S(\pi
C+6S)\cot(\theta)}{\sqrt{\pi^{2}C^{2}\left(4S^{2}+1\right)+12\pi
CS\left(4S^{2}-1\right)+36\left(4S^{4}+S^{2}\right)}}\end{split}$ (21)

According to equation (21), we determine that the critical point is
$(S,\theta)=(\frac{C}{6},\frac{\pi}{2})$. The obtained result is consistent
with [37].

Figure 3: The vector field $n$ is represented by the arrows on the $S-\theta$
plane. With the $C=1.$

As shown in Figure (3), this critical point possesses a topological charge of
+1 $Q_{t}=+1$. This finding contradicts [16], implying that in the RPS
formalism, it results in a modification of the topological charge.

###  4.2 Charged Gauss-Bonnet AdS black holes

We use the article [43] and with the definition $\hat{Q}=Q\sqrt{C}$ [37] and
we have,

$\begin{split}T=\frac{\tilde{\alpha}-\frac{G\hat{Q}^{2}}{2Cr_{h}^{4}}+\frac{5r_{h}^{4}}{\ell^{2}}+3r_{h}^{2}}{4\pi
r_{h}\left(2\tilde{\alpha}+r_{h}^{2}\right)}\end{split}$ (22)

Then, by utilizing relation (2), we eliminate the $\hat{Q}$ parameter.
Furthermore, in accordance with relation (2), we obtain the function,

$\begin{split}\Phi=\frac{1}{\sin(\theta)}\bigg{(}\frac{2\tilde{\alpha}\ell^{2}+9\ell^{2}r_{h}^{2}+20r_{h}^{4}}{14\pi\ell^{2}r_{h}^{3}+20\pi\tilde{\alpha}\ell^{2}r_{h}}\bigg{)}\end{split}$
(23)

We also have normalized vector fields,

$\begin{split}&n^{r}=-\frac{\csc(\theta)\left(l^{2}\left(20\alpha^{2}+63r^{4}-48\alpha
r^{2}\right)-20\left(7r^{6}+30\alpha r^{4}\right)\right)}{2\pi
l^{2}r^{2}\left(10\alpha+7r^{2}\right)^{2}},\\\
&n^{\theta}=-\frac{\cot(\theta)\csc(\theta)\left(2\alpha
l^{2}+9l^{2}r^{2}+20r^{4}\right)}{14\pi l^{2}r^{3}+20\pi\alpha
l^{2}r}\end{split}$ (24)

Figure 4:  The vector field $n$ is represented by the arrows on the
$r_{h}-\theta$ plane. With the $\ell=1$ and $\tilde{\alpha}=0.5$.

According to Figure (4), we find that there is only one critical point with a
topological charge of +1 $(Q_{t}=+1)$.
In the following, for further investigation between these two formalisms, we
will draw the $T-S$ or $T-r_{h}$ diagram and discuss them.

Figure 5: (a)Spinodal curve for the R-N AdS black hole in bulk-boundary
thermodynamics. With the $Q=C=1$ . (b) Spinodal curve for the R-N AdS black
hole in RPS thermodynamics. With the $C=\ell=1$.

From Figure (5), we can observe that R-N AdS exhibits a maximum point in bulk-
boundary thermodynamics with a topological charge of -1. According to proposal
[16], a first-order phase transition occurs within it. However, for the RPS
formalism in Figure (5), the critical point is located at the minimum of
$T-S$. Since the cosmological constant is considered fix in the RPS formalism,
we do not associate the black hole with the concept of volume. Therefore, the
processes considered in RPS are not related to any volume work. Therefore, in
this formalism, there is a behavior similar to that of van der Waals. Also,
this critical point can show us the phase transition, which is stated in [37].
Such phase transitions are considered as non-equilibrium transitions.

Figure 6: Spinodal curve for the Charged Gauss-Bonnet AdS black hole in bulk-
boundary thermodynamics. With the $C=30,\tilde{\alpha}=0.5$, (a) $Q=0.04$, and
(b) $Q=1$ . (c) Spinodal curve for the Charged Gauss-Bonnet AdS black hole in
RPS thermodynamics. With the $\ell=1,\tilde{\alpha}=0.5$.

From Figure (LABEL:fig6a,fig6b), it is evident that the number of critical
points and the total topological charge for charged Gauss-Bonnet AdS black
holes in bulk-boundary thermodynamics is identical to that in extended
thermodynamics. Furthermore, their phase transitions exhibit similar behavior.
However, as shown in Figure (6), the behavior of the critical point and its
topological charge is entirely distinct between boundary and extended
thermodynamics.
In fact, when comparing two black holes in two different thermodynamic
systems, we observe that the behavior of the number of critical points and the
total topological charge of the black hole is the same in both the boundary
and extended thermodynamics. However, in the RPS formalism, their behavior
differs between the boundary and extended thermodynamics. We also find that in
the RPS formalism, there is only one critical point with a topological charge
of +1 $(Q_{t}=+1)$.

Case |  Thermodynamics |  Total Topological Charge
---|---|---
|  $EPST$ |  $Q_{t}=-1$
R-N AdS black hole |  $BBT$ |  $Q_{t}=-1$
|  $RPST$ |  $Q_{t}=+1$
|  $EPST$ |  $Q_{t}=-1$
Charged Gauss-Bonnet AdS black hole |  $BBT$ |  $Q_{t}=-1$
|  $RPST$ |  $Q_{t}=+1$
Table 1: Summary of the results. Extended phase space thermodynamics (EPST),
bulk-boundary thermodynamics (BBT) and restricted phase space thermodynamics
(RPST).

##  5 Conclusions

We investigated R-N AdS and charged Gauss-Bonnet AdS black holes in bulk-
boundary thermodynamics and RPS thermodynamics. We find that the number of
critical points and the total topological charge of the two black holes
considered in bulk-boundary thermodynamics are the same as the extended
thermodynamics and this result is in agreement with [39]. However, the
topological charge of these two black holes in the RPS formalism differs from
that of the bulk-boundary and extended thermodynamics. Also, in the RPS
formalism, we only find one critical point with a topological charge of +1 for
the charged AdS black hole.
These three thermodynamics are considered to be the same in one respect:
additional terms and parameters to charged AdS black holes do not affect the
topological class of the black hole. However, there is a difference between
bulk-boundary and extended thermodynamics. In both cases, the total
topological charge for the black hole is -1 $(Q_{t}=-1)$ in all modes.
Conversely, in RPS thermodynamics, the total topological charge is +1
$(Q_{t}=+1)$.
The cosmological constant plays a significant role in the topological charge.
Since the cosmological constant is variable in both bulk-boundary and extended
thermodynamics, their total topological charges are the same, but in RPS
thermodynamics, the cosmological constant is fixed, and hence its total
topological charge is different from the other two thermodynamics.

##  6 Acknowledgments

We are grateful to Prof Shao-Wen Wei for providing some valuable points.

## References

  * [1] J. D. Bekenstein, Lett. Nuovo Cim 4, 737 (1972).
  * [2] J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973).
  * [3] A. G. Tzikas, Physics Letters B, 788, 219 (2019).
  * [4] B. K. Singh, R. P. Singh, and D. V. Singh, The European Physical Journal Plus 135, 862 (2020).
  * [5] J. Sadeghi, S. N. Gashti, and E. N. Mezerji, Physics of the Dark Universe 30, 100626 (2020).
  * [6] A. Kumar, R.K. Walia, and S. G. Ghosh, Universe, 8, 232 (2022).
  * [7] D. V. Singh, S. G. Ghosh, and S. D. Maharaj, Annals Phys. 412, 168025 (2020).
  * [8] J. Sadeghi, M. Shokri, S. N. Gashti, and M. R. Alipour, General Relativity and Gravitation 54, 129 (2022).
  * [9] A. Merriam, and M. Z. Sarwar, Int.J.Mod.Phys.D 31, 2150128 (2022).
  * [10] J. Sadeghi, M. Shokri, M. R. Alipour, and S. Noori Gashti, Chinese Physics C 47, 015103 (2023).
  * [11] A. Kumar, D. V. Singh, and Sushant G. Ghosh, The European Physical Journal C 79, 275 (2019).
  * [12] M. Amir, M. S. Ali, and S. D. Maharaj, Class.Quant.Grav. 37, 145014 (2020).
  * [13] J. Sadeghi, S. N Gashti, I. Sakalli, and B. Pourhassan arXiv:2011.05109 (2020).
  * [14] J. Man, and Hongbo Cheng, General Relativity and Gravitation 46, 1660 (2014).
  * [15] J. Sadeghi, B. Pourhassan, S. Noori Gashti, S. Upadhyay, and E. Naghd Mezerji, Physica Scripta 98, 025305 (2023).
  * [16] S. W. Wei, and Y. X. Liu, Phys. Rev. D. 105, 104003 (2022).
  * [17] P. K. Yerra, and C. Bhamidipati, Phys. Rev. D 105, 104053 (2022).
  * [18] N. C. Bai, L. Li, and J. Tao, Phys. Rev. D 107.064015 (2023).
  * [19] P. K. Yerra, and C. Bhamidipati, Phys. Lett. B 835, 137591 (2022).
  * [20] E. Witten, Adv. Theor. Math. Phys. 2, 253-291 (1998).
  * [21] S. W. Wei, Y. X. Liu, and R. B. Mann, Phys. Rev. Lett. 129, 19, 191101 (2022).
  * [22] Y. S. Duan, and M. L. Ge, Sci. Sin. 9, 11, 1072 (1979).
  * [23] Y. Duan, SLAC-PUB-3301.
  * [24] C. H. Liu, and J. Wang, Phys. Rev. D 107, 064023 (2023).
  * [25] Y. Du, and X. Zhang, arxiv:2302.11189 (2023).
  * [26] D. Wu, Eur. Phys. J. C 83, 365 (2023).
  * [27] D. Wu, arxiv:2306.02324 (2023).
  * [28] D. Wu, Phys. Rev. D 107, 024024 (2023).
  * [29] D. Wu, and S. Q. Wu, Phys. Rev. D 107, 084002 (2023).
  * [30] J. Sadeghi, S. N. Gashti, M. R. Alipour, M. A. S. Afshar, Annals of Physics, 169391 (2023).
  * [31] M. R. Alipour, M. A. S. Afshar, S. N. Gashti, J. Sadeghi, arXiv:2305.05595 (2023).
  * [32] J. Sadeghi, M. Shokri, S. N. Gashti, M. R. Alipour, General Relativity and Gravitation 54 (129) (2022).
  * [33] Y. Ladghami, B. Asfour, A. Bouali, A. Errahmani, and T. Ouali, Physics of the Dark Universe41, 101261 (2023).
  * [34] M. R. Alipour, J. Sadeghi, and M. Shokri, Nuclear Physics B 990, 116184 (2023).
  * [35] T. Wang, and L. Zhao, Physics Letters B 827, 136935 (2022).
  * [36] X. Kong, Zh. Zhang, and L. Zhao, arXiv: 2211.00963 (2022).
  * [37] Z. Gao and L. Zhao, Class. Quantum Grav. 39, 075019 (2022).
  * [38] Z. Gao, X. Kong, and L. Zhao, The European Physical Journal C 82, 112 (2022).
  * [39] M. Zhang, and J. Jiang, J. High Energ. Phys. 2023, 115 (2023).
  * [40] Y. Hasegawa, arXiv:2203.12421, (2022).
  * [41] P. K. Yerra, Ch. Bhamidipati, Su. Mukherji, arXiv:2304.14988 (2023).
  * [42] S. W. Wei, Phys. Rev. D 102, 064039 (2020).
  * [43] Y. Qu, J. Tao, H. Yang, Nuclear PhysicsB 992, 116234 (2023).