A New Hyperchaotic Hyperjerk System with Three Nonlinear Terms, its Synchronization and Circuit Simulation

Abstract

In recent decades, hyperjerk systems have been studied well in the literature because of their simple dynamics structure and complex qualitative properties. In this work, we announce a new hyperchaotic hyperjerk system with three nonlinear terms. Dynamical properties of the hyperjerk system are analyzed through equilibrium analysis, dissipativity, phase portraits and Lyapunov chaos exponents. We show that the new hyperchaotic hyperjerk system has a unique saddle-focus equilibrium at the origin. Thus, the new hyperchaotic hyperjerk system has a self-excited strange attractor. Next, global hyperchaos synchronization of a pair of new hyperchaotic hyperjerk systems is successfully achieved via adaptive backstepping control. Also, an electronic circuit of the hyperchaotic hyperjerk system has been designed via MultiSIM to check the feasibility of the theoretical system.

A Damping-Tunable Snap System: From Dissipative Hyperchaos to Conservative Chaos

“…Many simple hyperchaotic snap systems [7,[19][20][21][22][23][24][25][26] have been presented in a single fourth-order ODE, which can be written in a general form as: ....…”

“…where A is a tuning parameter, B = 2.65 is a constant, f (x) = x 2 , and g( ẍ, ẋ, x) = | ẋ|, as also included in Table 1. It can be noticed from Table 1 that the nonlinear function f (x) in ( 3) is a quadratic function of the form x 2 , whereas f (x) in existing systems [7,[19][20][21][22][23][24][25] is a quartic function of the form x 4 . Although the system in [26] has f (x) = 1, the function g( ẍ, ẋ, x) consists of many trigonometric functions that lead to difficulties in circuit implementation.…”

“…Many simple hyperchaotic snap systems [ 7 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 ] have been presented in a single fourth-order ODE, which can be written in a general form as: where is a damping coefficient [ 5 , 12 ] of the form: are parameters, and and are nonlinear functions. As a result, the damping coefficient in ( 2 ) is a nonlinear damping coefficient.…”

“…As a result, the damping coefficient in ( 2 ) is a nonlinear damping coefficient. Table 1 compares such existing hyperchaotic snap systems [ 7 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 ] and shows that they do not exhibit conservative chaos, where ( to ) are tuning parameters and ( to ), , ( to ), ( , , ), and ( , ) are constants, all of which are not equal to one. Note that is described in Section 2.4 .…”

Variable universe adaptive fuzzy sliding mode projective synchronization of hyperjerk system based on disturbance observer

“…When n = 3 , system (1) degenerates into a jerk system. The main concerns of the researchers are the chaotic or hyperchaotic performance, control and synchronization of jerk system or hyperjerk system with such simple structures [2][3][4][5][6][7][8][9][10][11].…”