A hyperchaotic memristive system with extreme multistability and conservativeness

Abstract

Comparing with dissipative systems, conservative systems have distinguished advantages in information processing and secure communication. It is of practical importance and theoretical significance to design conservative chaotic systems based on memristors due to their special features of complexity and flexibility. In this paper, a novel conservative hyperchaotic memristor system is proposed. The rich dynamics of the system, including extreme multistability and hyperchaos, are analyzed by using phase portraits, time series, bifurcation diagrams and Lyapunov exponents, and confirming the system is conservative. Based on the Hamiltonian theory, a specific energy function of the system is constructed and the generation mechanism of the extreme multistability is revealed and analyzed. Interestingly, a special heart-shaped attractor is found from the system. Finally, the theoretical results are verified and demonstrated through physical circuit implementation, demonstrating its potential for future applications.

Appendices

Appendix

Conservativeness analysis of the hyperchaotic memristive system

Based on the permanent point theory [45], the equilibrium point $({E}_{EP})$ of the system is determined by taking the time derivative of each state variable on the left-hand side of Eq. (2.4) and setting it to be zero.

If the first derivative of the system with respect to time is regarded as velocity, then the second derivative of the system with respect to time is acceleration. The permanent point $({E}_{PP})$ is the point where the acceleration of the system is zero, but the velocity is not zero. The formula for a permanent point is described as follows:

$$ E_{PP} = (E - E \cap E_{EP} ) $$

(A.1)

where $E$ is the point set where the second derivative of the system with respect to time is zero. Therefore, point $E$ satisfies

$$ \sum {{\text{J}} \cdot f_{i} (X)} = 0 $$

(A.2)

where J is the Jacobian matrix of the system and ${f}_{i}(X)$ are the state equations of the system.

The Jacobian matrix corresponding to the equilibrium point ${E}_{EP}=(u, 0, 0, 0, 0)$ of system (2.4) is

$$ {\text{J}} = \left( {\begin{array}{*{20}c} {\frac{{\partial f_{1} }}{\partial x}} & {\frac{{\partial f_{1} }}{\partial y}} & {\frac{{\partial f_{1} }}{\partial z}} & {\frac{{\partial f_{1} }}{\partial w}} & {\frac{{\partial f_{1} }}{\partial h}} \\ {\frac{{\partial f_{2} }}{\partial x}} & {\frac{{\partial f_{2} }}{\partial y}} & {\frac{{\partial f_{2} }}{\partial z}} & {\frac{{\partial f_{2} }}{\partial w}} & {\frac{{\partial f_{2} }}{\partial h}} \\ {\frac{{\partial f_{3} }}{\partial x}} & {\frac{{\partial f_{3} }}{\partial y}} & {\frac{{\partial f_{3} }}{\partial z}} & {\frac{{\partial f_{3} }}{\partial w}} & {\frac{{\partial f_{3} }}{\partial h}} \\ {\frac{{\partial f_{4} }}{\partial x}} & {\frac{{\partial f_{4} }}{\partial y}} & {\frac{{\partial f_{4} }}{\partial z}} & {\frac{{\partial f_{4} }}{\partial w}} & {\frac{{\partial f_{4} }}{\partial h}} \\ {\frac{{\partial f_{5} }}{\partial x}} & {\frac{{\partial f_{5} }}{\partial y}} & {\frac{{\partial f_{5} }}{\partial z}} & {\frac{{\partial f_{5} }}{\partial w}} & {\frac{{\partial f_{5} }}{\partial h}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ {ay({\text{sgn}}(x + 1) - {\text{sgn}}(x - 1))} & {a(|x + 1| - |x - 1|)} & 0 & { - a} & { - a} \\ 0 & 0 & 0 & 0 & b \\ 0 & c & 0 & 0 & 0 \\ 0 & d & { - d} & 0 & 0 \\ \end{array} } \right) $$

(A.3)

Based on (2.4), one obtains

$$ \left( {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ {ay({\text{sgn}}(x + 1) - {\text{sgn}}(x - 1))} & {a(|x + 1| - |x - 1|)} & 0 & { - a} & { - a} \\ 0 & 0 & 0 & 0 & b \\ 0 & c & 0 & 0 & 0 \\ 0 & d & { - d} & 0 & 0 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} y \\ {a(|x + 1| - |x - 1|)y - aw - ah} \\ {bh} \\ {cy} \\ {dy - dz} \\ \end{array} } \right) $$

$$ = \left( {\begin{array}{*{20}c} {a(|x + 1| - |x - 1|)y - aw - ah} \\ {ay^{2} ({\text{sgn}}(x + 1) - {\text{sgn}}(x - 1)) + a(|x + 1| - |x - 1|)(a(|x + 1| - |x - 1|)y - aw - ah) - acy - ady + adz} \\ {bdy - bdz} \\ {c(a(|x + 1| - |x - 1|)y - aw - ah)} \\ {d(a(|x + 1| - |x - 1|)y - aw - ah) - bdh} \\ \end{array} } \right) $$

(A.4)

Obviously, when $y$ is 0, one of the solutions can be obtained as ${E}_{1}=(u, 0, 0, 0, 0)$ from (A.4). However, when $y$ is not 0, the situation becomes complication and needs to be classified and discussed.

It follows from (A.4) that

$$ y({\text{sgn}}(x + 1) - {\text{sgn}}(x - 1)) - c = 0 $$

(A.5)

The classification of $x$ is given as follows:

$$ \left\{ \begin{gathered} (1)x > 1,\;\;\;{\text{Equation(A}}{\text{.5) has no solution}} \hfill \\ (2)x = 1,y = c \hfill \\ (3) - 1 < x < 1,y = \frac{c}{2} \hfill \\ (4)x = - 1,y = c \hfill \\ (5)x < - 1,{\text{Equation(A}}{\text{.5) has no solution}} \hfill \\ \end{gathered} \right. $$

(A.6)

Other equilibrium points are.

$E_{2} = (1,c,c,0,2c),E_{3} = ( - 1,c,c,0, - 2c),E_{4} = (x_{0} ,\frac{c}{2},\frac{c}{2},0,c)( - 1 < x_{0} < 1)$.

According to (A.1), the permanent points exist in the system. In other words, ${E}_{PP}\ne \varnothing$. If the permanent point theory [45] is used to judge whether the system is conservative, it will lead to a wrong conclusion. In fact, the results of numerical simulation show that the system is conservative with some specific parameter values and initial conditions. This reveals some limitation of the permanent point theory.