The Similarity of Dynamic behavior between Different Chaotic Systems

TL;DR

This paper reveals that chaotic systems within the same chaos family exhibit similar dynamic behaviors, and higher-order systems tend to inherit features from lower-order ones, with similarity decreasing as complexity increases.

Contribution

It introduces the novel concept that chaos systems within the same family share dynamic similarities, supported by quantitative analysis across multiple chaotic system types.

Findings

Strong correlation in dynamics within the same chaos family

Higher-order systems inherit behaviors from lower-order systems

Similarity decreases as system complexity increases

Abstract

Chaos is associated with stochasticity, complex, irregular motion, etc. It has some peculiar properties such as ergodicity, highly initial value sensitivity, non-periodicity and long-term unpredictability. These pseudo random features lead chaotic systems to enormous applications such as random number generator, image encryption and secure communication. In general, the concept of chaos is never associated with similarity. However, we found the chaotic systems belonging to one chaos family (OCF) have similar dynamic behavior, which is a novel characteristic of chaos. In this work, three classical chaotic system family are studied, which are Lorenz family, Chua family and hyperbolic sine family. These systems contain different derived chaotic systems (Lorenz system, Chen system and L\"u system), different order chaotic systems (Chua family and hyperbolic sine family), and different kinds of chaotic systems (chaos and hyper-chaos). Their PSPs demonstrate that there exist strong correlation in OCF. Moreover, we found that high order/dimensional chaotic systems will inherit all dynamic behavior of lower ones, and the similarity will decrease as the order/dimensional goes higher, which is analogous to genetic process in biology. All of these features are quantitatively evaluated by PPMCC and SSIM.

Tables

Table 1. Table 1: PPMCC score of N-dimensional Chua family

3D system
$x-y$ plane	3D system $x-z$ plane	3D system $y-z$ plane	4D system $x-y$ plane	4D system $x-z$ plane	4D system $y-z$ plane	5D system $x-y$ plane	5D system $x-z$ plane	5D system $y-z$ plane
3D system $x-y$ plane	1	0.0099	0.3588	0.9531	0.0099	0.3588	0.9531	0.0099	0.3588
3D system $x-z$ plane	0.0099	1	0.0505	0.0106	1	0.0505	0.0106	1	0.0505
3D system $y-z$ plane	0.3588	0.0505	1	0.3588	0.0505	1	0.3588	0.0505	1
4D system $x-y$ plane	0.9531	0.0106	0.3588	1	0.0106	0.3588	1	0.0106	0.3588
4D system $x-z$ plane	0.0099	1	0.0505	0.0106	1	0.0505	0.0106	1	0.0505
4D system $y-z$ plane	0.3588	0.0505	1	0.3588	0.0505	1	0.3588	0.0505	1
5D system $x-y$ plane	0.9531	0.0106	0.3588	1	0.0106	0.3588	1	0.0106	0.3588
5D system $x-z$ plane	0.0099	1	0.0505	0.0106	1	0.0505	0.0106	1	0.0505
5D system $y-z$ plane	0.3588	0.0505	1	0.3588	0.0505	1	0.3588	0.0505	1

Table 2. Table 2: SSIM score of N-dimensional Chua family

3D system
$x-y$ plane	3D system $x-z$ plane	3D system $y-z$ plane	4D system $x-y$ plane	4D system $x-z$ plane	4D system $y-z$ plane	5D system $x-y$ plane	5D system $x-z$ plane	5D system $y-z$ plane
3D system $x-y$ plane	1	0.2417	0.1949	0.8776	0.2417	0.1949	0.8776	0.2417	0.1949
3D system $x-z$ plane	0.2417	1	0.1907	0.2409	1	0.1907	0.2409	1	0.1907
3D system $y-z$ plane	0.1949	0.1907	1	0.1949	0.1907	1	0.1949	0.1907	1
4D system $x-y$ plane	0.8776	0.2409	0.3588	1	0.2409	0.1949	1	0.2409	0.1949
4D system $x-z$ plane	0.2417	1	0.1907	0.0106	1	0.1907	0.2409	1	0.1907
4D system $y-z$ plane	0.1949	0.1907	1	0.1949	0.1907	1	0.1949	0.1907	1
5D system $x-y$ plane	0.8776	0.2409	0.1949	1	0.2409	0.1949	1	0.2409	0.1949
5D system $x-z$ plane	0.2417	1	0.1907	0.2409	1	0.1907	0.2409	1	0.1907
5D system $y-z$ plane	0.1949	0.1907	1	0.1949	0.1907	1	0.1949	0.1907	1

Table 3. Table 3: PPMCC score of 3D ∼ similar-to \sim 10D hyperbolic sine systems

3D system $x_1-x_2$ plane	3D system $x_1-x_3$ plane	3D system $x_2-x_3$ plane
4D system $x_2-x_3$ plane	0.9703	0.3365	0.2488
4D system $x_2-x_4$ plane	0.3635	0.9458	0.2208
4D system $x_3-x_4$ plane	0.2608	0.2294	0.9032
5D system $x_3-x_4$ plane	0.8989	0.3452	0.2616
5D system $x_3-x_5$ plane	0.3542	0.7659	0.2901
5D system $x_4-x_5$ plane	0.2488	0.2979	0.7896
10D system $x_8-x_9$ plane	0.5789	0.4573	0.1673
10D system $x_8-x_{10}$ plane	0.4083	0.5203	0.4454
10D system $x_9-x_{10}$ plane	0.2433	0.3635	0.6765

Table 4. Table 4: SSIM score of 3D ∼ similar-to \sim 10D hyperbolic sine systems

3D system $x_1-x_2$ plane	3D system $x_1-x_3$ plane	3D system $x_2-x_3$ plane
4D system $x_2-x_3$ plane	0.7552	0.6211	0.5985
4D system $x_2-x_4$ plane	0.6023	0.7902	0.5517
4D system $x_3-x_4$ plane	0.5583	0.5198	0.6946
5D system $x_3-x_4$ plane	0.6987	0.6183	0.5993
5D system $x_3-x_5$ plane	0.6284	0.7338	0.5796
5D system $x_4-x_5$ plane	0.5883	0.5805	0.6813
10D system $x_8-x_9$ plane	0.4937	0.4302	0.3957
10D system $x_8-x_{10}$ plane	0.5062	0.5416	0.4508
10D system $x_9-x_{10}$ plane	0.5852	0.6055	0.6201

Table 5. Table 5: PPMCC score of three dimensional Lorenz canonical family

Lorenz system
$x-y$ plane	Lorenz system $x-z$ plane	Lorenz system $y-z$ plane	Chen system $x-y$ plane	Chen system $x-z$ plane	Chen system $y-z$ plane	Lü system $x-y$ plane	Lü system $x-z$ plane	Lü system $y-z$ plane
Lorenz system $x-y$ plane	1	0.2951	0.0521	0.6851	0.2002	0.1232	0.6905	0.1593	0.0825
Lorenz system $x-z$ plane	0.2951	1	0.3905	0.3046	0.6795	0.3957	0.2833	0.6839	0.4295
Lorenz system $y-z$ plane	0.0521	0.3905	1	0.1545	0.6204	0.8260	0.0799	0.5877	0.7532
Chen system $x-y$ plane	0.6851	0.3046	0.1545	1	0.3098	0.2296	0.7188	0.2872	0.2126
Chen system $x-z$ plane	0.2002	0.6795	0.6204	0.3098	1	0.6555	0.2021	0.8126	0.6644
Chen system $y-z$ plane	0.1232	0.3957	0.8260	0.2296	0.6555	1	0.1402	0.5773	0.7709
Lü system $x-y$ plane	0.6905	0.2833	0.0799	0.7188	0.2021	0.1402	1	0.1669	0.0906
Lü system $x-z$ plane	0.1593	0.6839	0.5877	0.2872	0.8126	0.5773	0.1669	1	0.7115
Lü system $y-z$ plane	0.0825	0.4295	0.7532	0.2126	0.6644	0.7709	0.0906	0.7115	1

Table 6. Table 6: SSIM score of three dimensional Lorenz canonical family

Lorenz system
$x-y$ plane	Lorenz system $x-z$ plane	Lorenz system $y-z$ plane	Chen system $x-y$ plane	Chen system $x-z$ plane	Chen system $y-z$ plane	Lü system $x-y$ plane	Lü system $x-z$ plane	Lü system $y-z$ plane
Lorenz system $x-y$ plane	1	0.6789	0.5602	0.7694	0.6074	0.5522	0.7850	0.5507	0.5001
Lorenz system $x-z$ plane	0.6789	1	0.6371	0.6583	0.7064	0.6124	0.6515	0.6730	0.5723
Lorenz system $y-z$ plane	0.5602	0.6371	1	0.5659	0.6721	0.8260	0.5343	0.6074	0.6687
Chen system $x-y$ plane	0.7694	0.3046	0.5659	1	0.6159	0.5604	0.7635	0.5613	0.5088
Chen system $x-z$ plane	0.6074	0.7064	0.6204	0.6159	1	0.6735	0.5776	0.7346	0.6644
Chen system $y-z$ plane	0.5522	0.6124	0.7332	0.5604	0.6555	1	0.1402	0.5917	0.7709
Lü system $x-y$ plane	0.7850	0.6515	0.5343	0.7635	0.5776	0.5208	1	0.5278	0.4761
Lü system $x-z$ plane	0.5507	0.6730	0.6074	0.5613	0.7346	0.5917	0.5278	1	0.5938
Lü system $y-z$ plane	0.5001	0.5723	0.6687	0.5088	0.6317	0.6747	0.4761	0.5938	1

Table 7. Table 7: PPMCC score of N-dimensional Lorenz family

3D system
$x-y$ plane	3D system $x-z$ plane	3D system $y-z$ plane	4D system $x-y$ plane	4D system $x-z$ plane	4D system $y-z$ plane	5D system $x-y$ plane	5D system $x-z$ plane	5D system $y-z$ plane
3D system $x-y$ plane	1	0.2990	0.0709	0.8181	0.1722	0.0326	0.7775	0.3444	0.1281
3D system $x-z$ plane	0.2990	1	0.4305	0.2074	0.7873	0.3701	0.3678	0.8428	0.4016
3D system $y-z$ plane	0.0709	0.4305	1	0.0054	0.4856	0.7968	0.1922	0.4120	0.8072
4D system $x-y$ plane	0.8181	0.2074	0.0054	1	0.0883	0.0731	0.6308	0.2471	0.0557
4D system $x-z$ plane	0.1722	0.7873	0.4856	0.0883	1	0.5132	0.2935	0.7458	0.3927
4D system $y-z$ plane	0.0326	0.3701	0.7968	0.0731	0.5132	1	0.0753	0.3211	0.6965
5D system $x-y$ plane	0.7775	0.3678	0.1922	0.6308	0.2935	0.0753	1	0.4372	0.2277
5D system $x-z$ plane	0.3444	0.8428	0.4120	0.2471	0.7458	0.3211	0.4372	1	0.4308
5D system $y-z$ plane	0.1281	0.4016	0.8072	0.0557	0.3927	0.6965	0.2277	0.4308	1

Table 8. Table 8: SSIM score of N-dimensional Lorenz family

3D system
$x-y$ plane	3D system $x-z$ plane	3D system $y-z$ plane	4D system $x-y$ plane	4D system $x-z$ plane	4D system $y-z$ plane	5D system $x-y$ plane	5D system $x-z$ plane	5D system $y-z$ plane
3D system $x-y$ plane	1	0.6268	0.4856	0.8027	0.5847	0.4665	0.7585	0.6084	0.4848
3D system $x-z$ plane	0.6268	1	0.5494	0.6064	0.7095	0.3701	0.6291	0.7125	0.5287
3D system $y-z$ plane	0.4856	0.5494	1	0.4696	0.5388	0.6244	0.5047	0.5281	0.6120
4D system $x-y$ plane	0.8027	0.2074	0.4696	1	0.5690	0.4562	0.7263	0.5841	0.4651
4D system $x-z$ plane	0.5847	0.7095	0.5388	0.5690	1	0.5406	0.6011	0.6788	0.5060
4D system $y-z$ plane	0.4665	0.5238	0.6244	0.4562	0.5406	1	0.4780	0.5046	0.5826
5D system $x-y$ plane	0.7585	0.6291	0.5047	0.7263	0.6011	0.4780	1	0.6351	0.5080
5D system $x-z$ plane	0.6084	0.7125	0.5281	0.5841	0.6788	0.5046	0.6351	1	0.5365
5D system $y-z$ plane	0.4848	0.5287	0.6120	0.4651	0.5060	0.5826	0.5080	0.5365	1

Equations

$$PPMCC=\frac{{\sum }_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{{\sum }_{i=1}^n(x_i-\bar{x}{)}^2}\sqrt{{\sum }_{i=1}^n(y_i-\bar{y}{)}^2}}$$

$$I(x,y)$$

$$C(x,y)$$

$$S(x,y)$$

$$SSIM=l(x,y{)}^{\alpha }\cdot C(x,y{)}^{\beta }\cdot S(x,y{)}^{\gamma }$$

$$SSIM=\frac{(2{\mu }_x{\mu }_y+C_1)(2{\sigma }_{xy}+C_2)}{({\mu }_x^2+{\mu }_y^2+C_1)({\sigma }_x^2{\sigma }_y^2+C_2)}$$

$$\dot{x}$$

$$\dot{y}$$

$$\dot{z}$$

$$f(x)=m_{1}x+\frac{1}{2}(m_0-m_1)(\left|x+1\right|-\left|x-1\right|)$$

$$\dot{x}$$

$$\dot{y}$$

$$\dot{z}$$

$$\dot{u}$$

$$\dot{x}$$

$$\dot{y}$$

$$\dot{z}$$

$$\dot{u}$$

$$\dot{v}$$

$$\{\begin{array}{rl}& \dot{x_1}=x_2-x_1\\ & \dot{x_2}=x_3-x_2\\ & \cdots \\ & \dot{x_{n-3}}=x_{n-2}-x_{n-3}\\ & \dot{x_{n-2}}=x_{n-1}\\ & \dot{x_{n-1}}=x_n\\ & \dot{x_n}=-x_n-f(x_{n-1})-n{x}_{n-2}-n{x}_{n-3}-\cdots -\frac{1}{2n}{x}_1\end{array}$$

$$\begin{array}{r}\begin{array}{rl}\dot{x_1}& =x_2\\ \dot{x_2}& =x_3\\ \dot{x_3}& =-x_3-\rho sinh(\psi x_2)-3x_1\end{array}\end{array}$$

$$\{\begin{array}{rl}& \dot{x_1}=x_2-x_1\\ & \dot{x_2}=x_3\\ & \dot{x_3}=x_4\\ & \dot{x_4}=-x_4-\rho sinh(\psi x_3)-4x_2-0.125x_1\end{array}$$

$$\{\begin{array}{rl}& \dot{x_1}=x_2-x_1\\ & \dot{x_2}=x_3-x_2\\ & \dot{x_3}=x_4\\ & \dot{x_4}=x_5\\ & \dot{x_5}=-x_5-\rho sinh(\psi x_4)-5x_3-5x_2-0.1x_1\end{array}$$

$$\{\begin{array}{rl}\dot{x_1}=& x_2-x_1\\ \dot{x_2}=& x_3-x_2\\ \dot{x_3}=& x_4-x_3\\ \dot{x_4}=& x_5-x_4\\ \dot{x_5}=& x_6-x_5\\ \dot{x_6}=& x_7-x_6\\ \dot{x_7}=& x_8-x_7\\ \dot{x_8}=& x_9\\ \dot{x_9}=& x_{10}\\ \dot{x_{10}}=& -x_{10}-\rho sinh(\psi x_9)-10x_8-10x_7-10x_6-10x_5-10x_4\\ & -10x_3-10x_2-0.05x_1\end{array}$$

$$\dot{x}$$

$$\dot{y}$$

$$\dot{z}$$

$$\dot{x}$$

$$\dot{y}$$

$$\dot{z}$$

$$\dot{x}$$

$$\dot{y}$$

$$\dot{z}$$

$$\dot{x}$$

$$\dot{y}$$

$$\dot{z}$$

$$\dot{v}$$

$$\dot{x}$$

$$\dot{y}$$

$$\dot{z}$$

$$\dot{u}$$

$$\dot{v}$$

Full text

In this work, three classical chaotic system family are studied, which are Lorenz family, Chua family and hyperbolic sine family. These systems contain different derived chaotic systems (Lorenz system, Chen system and Lü system), different order chaotic systems (Chua family and hyperbolic sine family), and different kinds of chaotic systems (chaos and hyper-chaos). Their PSPs demonstrate that there exist strong correlation in OCF. Moreover, we found that high order/dimensional chaotic systems will inherit all dynamic behavior of lower ones, and the similarity will decrease as the order/dimensional goes higher, which is analogous to genetic process in biology. All of these features are quantitatively evaluated by PPMCC and SSIM.

chaos; dynamic behavior; similarity; genetic process

††preprint: AIP/123-QED

I Introduction

Since Lorenz discovered chaos in a simple system of ordinary differential equations in 1959, a new field of science which has grown ever larger with each passing year has been unleashedSprott (2010). Over decades, chaos has been observed in nature (weather and climateYuan et al. (2018), dynamics of satellites in the solar systemVoosen (2018), time evolution of the magnetic field of celestial bodiesWang, Chen, and Jing (2018), and population growth in ecologyGilpin and Feldman (2017)) and laboratory (electrical circuitsQi (2017), lasersZhao, Yin, and Zhu (2018), chemical reactionsVaidyanathan (2017), fluid dynamicsZhang and Chen (2008), mechanical systemsSong, Sun, and Ling (2017), and magneto-mechanical devicesMalaji and Ali (2018)). Chaotic behavior has also found numerous applications in electrical and engineeringLiu et al. (2018a), information and communication technologiesWang et al. (2018), biology and medicineCoffey (1998). By now, many chaotic models have been developed and studied in great detail, but they continue to present surprises and raise questions. The completely features of chaos is not already known.

There is no universally accepted mathematical definition of chaos. Usually, chaos can often be qualitatively identified with some confidence by observing the strange attractor or chaotic sea in a state space plot or Poincaré section, or quantitatively identified by Lyapunov exponentSprott (1993). This is mainly due to the features and manifestations of chaos. In detail, the main features of a nonlinear dynamical system, exhibiting deterministic chaos for given values of the parameters, are the followingKocarev and Lian (2011):

• •

Sensitive dependence on initial conditions - two nearby initial conditions on the attractor or in the chaotic sea is separated by a distance which grows exponentially in time, and leads to long-term unpredictability.

• •

Ergodicity - the trajectory winds around forever, never repeating, and a sufficient long trajectory will almost fill the square for most of the initial conditions.

• •

Strange attractors - a strange attractor contains an infinite number of points bounded within a definite region of the state space (an attractor with fractal dimension). Moreover, many dynamical systems have multiple coexisting attractors, which can be observed by changing initial conditionsLi, Min, and Li (2018).

These stochastic-like features lead chaotic systems to enormous cryptographic applications. In general, the concept of chaos is never associated with similarity, and there is no correlation among different chaotic systems (DCS). However, we found that the dynamic behaviors of DCS belonging to one chaos family (OCF) are similar, and thereby suggest there exist correlation among DCS of OCF. To illustrate this phenomenon, we have studied Lorenz family, Chua family, and hyperbolic sine family. First, their phase space plots (PSPs) are numerical calculated by 4th-order Runge-Kutta (RK4) method with fixed step size, then the related PSPs among DCS are selected. At last, Pearson correlation coefficient (PPMCC) and structural similarity (SSIM) are utilized to quantitatively evaluate the similarity of PSPs.

The rest of the paper is organized as follows: In Section 2, the similarity evaluation methods utilized in this paper, i.e., the Pearson correlation coecient (PPMCC) and structural similarity (SSIM) are introduced. In Section 3, the PSPs of Lorenz family, Chua family, hyperbolic sine family are calculated, and the similarity among these families are quantitatively evaluated. Conclusions and future work are drawn in Section 5.

II Similarity Calculation method

II.1 Pearson correlation coefficient

The Pearson correlation coefficient (PPMCC) is one of the most commonly used statistical tools to measure the degree of correlation between two data sets $X$ and $Y$Xu and Deng (2018):

[TABLE]

Here $(x_{i},y_{i})$ are individual paired samples from the data sets $X$ and $Y$, and $n$ is the total number of pairs; $\bar{x}$ and $\bar{y}$ are the mean values of the samples in data sets $X$ and $Y$.

The degree of PPMCC always lies in the range of $-1$ to $+1$. PPMCC$>0$ implies the two variable have positive correlation, PPMCC $<0$ means a negative correlation. When PPMCC is higher than $0.5$ (or lower than -0.5) indicate a strong relationship. PPMCC$=0$ implies no relationship.

II.2 Structural Similarity

The structural similarity (SSIM) index is used for measuring the similarity between two imagesWang et al. (2004). Basically, SSIM consists of three local comparison functions between two signal $x$ and $y$, namely luminance comparison, contrast comparison, and structure comparison, which are computed as follows:

[TABLE]

In Eq. (2), $\mu_{x}$ and $\mu_{y}$ are the sample means of $x$ and $y$, $\sigma_{x}$ and $\sigma_{y}$ are the sample standard deviations of $x$ and $y$, and $\sigma_{xy}$ is the sample correlation coefficient between $x$ and $y$. The constants $C_{1}$, $C_{2}$ and $C_{3}$ are used to stabilize the algorithm when the denominators approach to zero.

The general form of SSIM index is given by combining the three comparison functions:

[TABLE]

where $\alpha$, $\beta$ and $\gamma$ are parameters which define the relative importance of the three components.

Usually, $\alpha=\beta=\gamma=1$. Therefore, SSIM can be rewritten as:

[TABLE]

III The similarity of dynamic behavior among different chaotic systems

In order to demonstrate the similarity of dynamic behavior among different chaotic systems, Chua family, Chen family and hyperbolic sine are taken as example in this paper. These is because these three chaotic families have been well studied which involve chaos and hyperchaos, different derived and different order/dimensional chaotic systems.

First, Chua family is studied, the PSPs indicate the dynamic behavior is similar, and thereby suggest there is strong correlation among this OCF. Then the hyperbolic sine family is studied, the resulting PSPs demonstrate that that high order/dimensional chaotic system inherit all dynamic behavior of lower ones, and the similarity will decrease as the order/dimensional goes higher, which is analogous to genetic process in biology. At last, Lorenz family is studied, the PSPs indicate that this phenomenon is widely exist in different derived chaotic systems (Lorenz system, Chen system and Lü system), different order chaotic systems, and different kinds of chaotic systems (chaos and hyper-chaos). All of these features have been quantitatively evaluated by PPMCC and SSIM.

III.1 Chua family case

The dimensionless state equations of canonical Chua’s system are as followsZhang et al. (2014):

[TABLE]

where

[TABLE]

and $\alpha=10$, $\beta=14.87$, $m_{0}=-1.27$ and $m_{1}=-0.68$.

Forth-order Chua’s system is as followsLiu, Wang, and Huang (2007):

[TABLE]

Here, $\alpha=10$, $\beta=-14.87$, $\gamma=-0.0497$, $r=27.3333$, $m_{0}=-1.27$ and $m_{1}=-0.68$.

Fifth-order Chua’s system is described as followsSimin and Jinhu (2007):

[TABLE]

Here, $a=10$, $b=1$, $c=-14.87$, $d=-0.0497$, $e=66.7962$, $r_{1}=27.3333$, $r2=0.05$, $m_{0}=-1.27$ and $m_{1}=-0.68$.

When the step size is set to be $0.001$ and all initial conditions are set to be $0.1$, one can calculate their PSPs as shown Fig. 1. From Fig. 1, the corresponding PSPs are almost the same, which indicate the dynamic behavior are very similar to each other.

In order to quantitatively describe this features, PPMCC and SSIM are utilized to calculate the similarity of PSPs. The resulting scores are shown in Tab. 1 and Tab. 2. From Tab. 1 and Tab. 2, some PPMCC and SSIM scores of corresponding PSPs are $1$, which indicate there exist extreme strong correlation among DCS of Chua family.

III.2 Hyperbolic Sine family case

RefLiu et al. (2018b) introduced a a class of simple chaotic systems with hyperbolic sine nonlinearity. With general nth-order ordinary differential equations (ODEs), any desirable order of hyperbolic sine chaotic systems could be constructed.

The general form is described by

[TABLE]

where $n$ is the order of the system. In this system, the nonlinear function is $f(x_{n-1})$, which is defined by $f(x_{n-1})=\rho sinh(\psi\dot{x})$, where $sinh((\psi\dot{x})=(\frac{e^{\psi\dot{x}}-e^{-(\psi\dot{x})}}{2})$, and $\rho=1.2\times 10^{-6}$, $\psi=\frac{1}{0.026}$. According to this equations, third order hyperbolic sine chaotic system is given by:

[TABLE]

Fourth order hyperbolic sine chaotic system is given by:

[TABLE]

Fifth order chaotic system is given by:

[TABLE]

And tenth order chaotic system is given by:

[TABLE]

When the step size is set to be $0.001$ and all initial conditions are set to be $0.1$, one can calculate their PSPs as shown Fig. 2. From the Fig. 2, the corresponding PSPs are similar, and one can note that the PSPs of lower chaotic system (third order chaotic system) appear in high order chaotic system, which suggest that high order chaotic system inherit all dynamic behavior of lowers ones. However, this similarity is weaken as the orders goes higher.

To quantitatively describe this feature, PPMCC and SSIM are utilized to evaluate the similarity scores of PSPs. From Tab. 3 and 4, the corresponding PSPs got higher scores than non-corresponding PSPs. However, these scores decrease as order goes higher, which suggested that, high order chaotic system inherit all dynamic behavior of lowers ones, and the correlation is weaken as the orders goes higher.

III.3 Lorenz family case

III.3.1 Lorenz canonical family

The equation of canonical Lorenz system is given byXinguo, Yide, and Li (2011):

[TABLE]

where $\sigma=10$, $r=28$, and $b=8/3$.

In 1999, Chen found a similar but nonequivalent chaotic attractor, which is now known to be the dual of Lorenz system. The standard form is as followsChen and Ueta (1999):

[TABLE]

where $(a,b,c)=(35,3,28)$.

In 2002, Lü and Chen reported a new chaotic system which bridging the gap between Lorenz and Chen systems. The equations are given byLü and Chen (2002):

[TABLE]

Here, $a=36$, $b=3$ and $c=20$.

When the step size is set to be $0.001$ and all initial conditions are set to be $0.1$, one can calculate their PSPs as shown Fig. 3. From Fig. 3, the corresponding PSPs are similar, which indicate there exist correlation among the corresponding dynamic behavior.

Tab. 5 and Tab. 6 are the similarity testing results. One can note that the similarity property exist in different derived chaotic system. Compared to testing results of Chua family and hyperbolic sine family, the scores are lower, which suggest that the dynamic behavior of Chen system and Lü system are less similar to the dynamic behavior of Lorenz system.

III.3.2 High dimensional Lorenz family

Stenflo derived a fourth order chaotic ordinary differential equations which is known as Lorenz-Stenflo SystemZhou, Lai, and Yu (1997). The equations is as follows:

[TABLE]

Here, $\sigma=10$, $s=28$, $r=28$ and $b=\frac{8}{3}$.

The 5D hyperchaootic Lorenz system is given by Yang and Chen (2013):

[TABLE]

where $\sigma=10$, $s=1$, $r=28$, $\beta=\frac{8}{3}$, $k_{1}=1.36$ and $k_{2}=64$.

When the step size is set to be $0.001$ and all initial conditions are set to be $0.1$, one can calculate their PSPs as shown Fig. 4. From Fig. 4, the corresponding PSPs are similar, which indicate there exist correlation among the corresponding dynamic behavior.

Tab. 7 and Tab. 8 are the testing results of PPMCC and SSIM. One can note that, the similarity is also exits in different kinds of chaotic systems (chaos and hyper-chaos). Compared to testing results of Chua family and hyperbolic sine family, the scores are lower, which suggest that the dynamic behavior among hyper-chaotic systems are less similar to the dynamic behavior among high-order chaotic systems.

IV Conclusion and future work

Chaos is famous for its stochastic-like properties such as ergodicity, highly initial value sensitivity, non-periodicity and long-term unpredictability. These pseudo random features lead chaotic systems to enormous applications such as random number generator, image encryption and secure communication.

In this paper, we found the dynamic behavior among DCS belonging to OCF is analogous to genetic process in biology, that is: 1. The dynamic behavior is similar in OCF. 2. High order/dimensional chaotic systems will inherit all dynamic behavior of lowers ones, and the similarity will decrease as the orders/dimensional goes higher. 3. The dynamic behavior of different derived chaotic systems and different types of chaotic system are less similar than the dynamic behavior among different orders of chaotic system. This phenomenon is widely exist in derived chaotic systems (Chen system, Lü system and Lorenz system), different order of chaotic systems (Chua system and hyperbolic sine chaotic system), and different dimensional chaotic systems/hyper-chaotic systems (N-dimensional Lorenz family). These novel features has been quantitatively evaluated by PPMCC and SSIM.

There is some work which is worth to study in future.

1. The results need to verify by rigorous mathematical proof.

2. The method to evaluate the similarity among different chaotic systems should be improved.

Acknowledgment

Conflict of Interest: The authors declare that they have no conflict of interest.