Complex dynamics in a novel jerk system with septic nonlinearity: analysis, control, and circuit realization

Abstract

This work proposes a new chaotic jerk system with septic nonlinearity. The new system presents odd symmetry and undergoes typical behaviors including period doubling, merging crisis, spontaneous symmetry breaking, coexisting attractors and coexisting bubbles of bifurcations as well. The most gratifying feature discovered in this article, is the occurrence of up to eight coexisting attractors for appropriate sets of parameters. This latter feature is uncommon for a chaotic system as simple as the model proposed in this work (e.g. not reported in cubic, quintic or hyperbolic sine models). Multistability control is achieved by following the linear augmentation approach. We numerically prove that the multistable septic chaotic system can be adjusted to develop a monostable behavior when smoothly monitoring the coupling strength. More interestingly, it is found that the coupling breaks the symmetry of the chaotic jerk system and thus induces new patterns including asymmetric Hopf bifurcations; coexisting non-symmetric bubbles, critical phenomena, coexisting multiple asymmetric attractors, just to name a few. On this line, the linear augmentation scheme can be regarded as a simple means for inducing new features in odd symmetric chaotic systems. PSPICE simulation results captured from an electronic analog of the proposed septic jerk system are consistent with the theoretical investigations.

A compact multi-output chaotic system with amplitude/frequency control

“…x m y z y xz z axy bxz x y 0.9 , , . The amplitude-frequency controller m is explored by employing Lyapunov exponents and average value, as shown in figure 6, the maximum Lyapunov exponents of the system (9), ( 10), (11), and (12) trend positively and significantly larger when m is varied in [1,4]. The increasing frequency can also be further demonstrated by the rise of the maximum Lyapunov exponents.…”

“…The comparison of the rate of the slope of the curve for different systems is shown in table 3. The average value of the absolute value of the state variable x under different parameter m was shown in figure 6(b), the value becomes positive and significantly increases when the parameter m is varied in [1,4], and the ACS4I system has the largest rate of the average value of the absolute value of the variable x. Similarly, the mean values of the absolute values of the state variables y, z of the different systems become positively and significantly larger when the parameter m is varied in [1,4], as shown in figures 6(c)-(d).…”

“…The average value of the absolute value of the state variable x under different parameter m was shown in figure 6(b), the value becomes positive and significantly increases when the parameter m is varied in [1,4], and the ACS4I system has the largest rate of the average value of the absolute value of the variable x. Similarly, the mean values of the absolute values of the state variables y, z of the different systems become positively and significantly larger when the parameter m is varied in [1,4], as shown in figures 6(c)-(d). The rate of change of the mean value of the absolute value of the state variable y is the same for the ACS4 system and the ACS4II system, but the rate of change of the mean value of the absolute value of the state variable y is the largest for the ACS4III system.…”

“…Chaos is a widespread phenomenon in physics, engineering, and other scientific disciplines [1][2][3]. Chaos has essential applications in information and communication and has been explored for decades of years [4][5][6][7]. In some chaotic systems, the change of initial condition could trigger different behaviors from coexisting attractors [8][9][10][11].…”

Bifurcation Analysis, Synchronization and FPGA Implementation of a New 3-D Jerk System with a Stable Equilibrium

“…Jerk systems have several applications in science [15][16][17][18][19]. Li and Zheng [15] proposed a 3-D jerk system with a sinusoidal term and noted that the system has an infinite number of equilibrium points.…”