## 2 Mathematical Model of a Chaotic System with a Butterfly-Like Curve of Balance Points  
The chaotic dynamical systems with special curves of balance points mentioned in Table 1 are particular cases of a general dynamical model stated as follows:  
\[\left\{\begin{array}{l}\dot{u}=w\\ \dot{v}=w\ \varphi(u,v,w)\\ \dot{w}=\psi(u,v)\end{array}\right. \tag{1}\]  
It is easy to observe that the balance points of the dynamical system (1) lie on the curve \(S\) on the \((u,v)-\)plane, which can be defined as follows:  
\[S=\left\{(u,v,w)\in R^{3}\big{|}\psi(u,v)=0,\ w=0\right\} \tag{2}\]  
In this research work, we define the mappings \(\varphi\) and \(\psi\) as given below:  
\[\left\{\begin{array}{l}\varphi(u,v,w)=-au-b\big{|}v\big{|}-cuw\\ \psi(u,v)=u^{4}+v^{4}-\big{|}u\big{|}\big{|}v\big{|}-u^{2}\end{array}\right. \tag{3}\]  
where \(a,b,c\) are real, constant, parameters. We assume that \(a,b,c\) are all positive.  
Substituting (3) into (1), we obtain a new 3-D mathematical model as given below:  
\[\left\{\begin{array}{l}\dot{u}=w\\ \dot{v}=w\ (-au-b\big{|}v\big{|}-cuw)\\ \dot{w}=u^{4}+v^{4}-\big{|}u\big{|}\big{|}v\big{|}-u^{2}\end{array}\right. \tag{4}\]  
The balance points of the system (4) belong to the following curve \(S\) on the \((u,v)-\)plane.  
\[S=\left\{(u,v,w)\in R^{3}\big{|}u^{4}+v^{4}-\big{|}u\big{|}\big{|}v\big{|}-u^ {2}=0,\ w=0\right\} \tag{5}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Chaotic System** & **The Curve of Balance Points** \\ \hline [PERSON] et al. [25] & Hyperbola \\ [PERSON] et al. [26] & Circle \\ [PERSON] et al. [27] & [PERSON] \\ [PERSON] et al. [28] & [PERSON] \\ [PERSON] et al. [29] & [PERSON] \\ [PERSON] et al. [30] & [PERSON] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Chaotic systems with special curves of balance points.  
The closed curve \(S\) is symmetrical about both \(u\) and \(v\) coordinate cases in \(R^{3}\). In the \((u,v)-\)plane, \(S\) looks like a butterfly as illustrated in Figure 1.  
The new dynamical model (4) exhibits a chaotic attractor if we take \((a,b,c)=(14,0.5,1.4)\) and the initial phase vector \((u(0),v(0),w(0))=(0.05,0.05,0.05)\) as the Lyapunov exponents spectrum can be calculated using [PERSON]'s algorithm [43] as \(\rho_{1}=0.0144\), \(\rho_{2}=0\) and \(\rho_{3}=-0.0155\).  
The presence of \(\rho_{1}>0\) in the Lyapunov exponents spectrum \((\rho_{1},\rho_{2},\rho_{3})\) establishes the existence of a chaotic attractor of the system (4) [44]. Since \(\rho_{1}+\rho_{2}+\rho_{3}=-0.0011<0\), we also conclude that the system (4) is a dissipative chaotic system [44].  
The Kaplan-Yorke dimension [44] of a general n-D dissipative system with its Lyapunov exponents arranged in non-increasing order (\(\rho_{1}\geq\rho_{2}\geq\ldots\geq\rho_{n}\)) is defined as follows:  
\[K_{D}=j+\frac{1}{|\rho_{j+1}|}\sum_{i=1}^{j}\rho_{i} \tag{6}\]  
where \(j\) is the largest index such that \(\sum\limits_{i=1}^{j}\rho_{i}\geq 0\).  
Using the Formula (6), we determine the Kaplan-Yorke dimension of the system (4) as follows:  
\[K_{D}=2+\frac{\rho_{1}+\rho_{2}}{|\rho_{3}|}=2+\frac{0.0144+0}{0.0155}=2.9290 \tag{7}\]  
The large value of \(K_{D}\) near three signifies the high complexity of the model (4).  
The 2-D signal plots of the chaotic model (4) with butterfly-shaped curve of balance points for \((a,b,c)=(14,0.5,1.4)\) and the initial state \((u(0),v(0),w(0))=(0.05,0.05,0.05)\) are represented in Figures 2-4.  
Figure 3: The 2-D signal representation of the chaotic model (4) in the \((v,w)-\)plane for the parameters \((a,b,c)=(14,0.5,1.4)\) and the initial state \((u(0),v(0),w(0))=(0.05,0.05,0.05)\).  
Figure 2: The 2-D signal representation of the chaotic model (4) in the \((u,v)-\)plane for the parameters \((a,b,c)=(14,0.5,1.4)\) and the initial state \((u(0),v(0),w(0))=(0.05,0.05,0.05)\).