Effects of Symmetric and Asymmetric Nonlinearity on the Dynamics of a Third-Order Autonomous Duffing–Holmes Oscillator

Abstract

A generalized third-order autonomous Duffing–Holmes system is proposed and deeply investigated. The proposed system is obtained by adding a parametric quadratic term m x 2 to the cubic nonlinear term − x 3 of an existing third-order autonomous Duffing–Holmes system. This modification allows the system to feature smoothly adjustable nonlinearity, symmetry, and nontrivial equilibria. A particular attention is given to the effects of symmetric and asymmetric nonlinearity on the dynamics of the system. For the specific case of m = 0 , the system is symmetric and interesting phenomena are observed, namely, coexistence of symmetric bifurcations, presence of parallel branches, and the coexistence of four (periodic-chaotic) and six (periodic) symmetric attractors. For m ≠ 0 , the system loses its symmetry. This favors the emergence of other behaviors, such as the coexistence of asymmetric bifurcations, involving the coexistence of several asymmetric attractors (periodic-periodic or periodic-chaotic). All these phenomena have been numerically highlighted using nonlinear dynamic tools (bifurcation diagrams, Lyapunov exponents, phase portraits, time series, frequency spectra, Poincaré section, cross sections of the attraction basins, etc.) and an analog computer of the system. In fact, PSpice simulations of the latter confirm numerical results. Moreover, amplitude control and synchronization strategies are also provided in order to promote the exploitation of the proposed system in engineering.

Dynamical Analysis and Offset Boosting in a 4-Dimensional Quintic Chaotic Oscillator with Circulant Connection of Space Variables

“…Basing on the numerical results obtained previously, it can be seen that the quintic cyclic system can experience relevant dynamical phenomena when the appropriate parameters are used. Terefore, it is pertinent to verify the feasibility of such diverse oscillations types [16,19,49]. Here, we focus on the PSpice simulations of the dynamics of the system represented in Figure 1 and particularly on the double-band chaotic attractor.…”

“…Some based their analysis on inversion symmetry because a system that provides it brings in the notion of bistability [13,14]; therefore, increasing the number of coexisting attractors in the concerned systems. Te complexity of such chaotic oscillators lets a good number of researchers to break the inversion symmetry which leads to new phenomena like the coexistence of asymmetric type of oscillations [15][16][17][18][19]. In fact, it can be found in any of these references the simultaneous existence of nonsymmetrical attractors for the same rank of parameters.…”

“…Additionally, the authors of reference [18] show the coexistence of two diferent periodic states and two diferent chaotic attractors. Te authors of reference [19] emphasized the appearance of fve asymmetric attractors for the same parameter's rank using diferent initial states. Moreover, reference [20] revealed the coexistence of a periodic state with a chaotic state.…”

A circulant inertia three Hopfield neuron system: dynamics, offset boosting, multistability and simple microcontroller- based practical implementation

“…Toward the end of the same year, Bao et al [19] enriched the literature with the memristive synape-based neural network with simplest cyclic connection able to exhibit coexisting attractors made up of stable points and unstable periodic or chaotic orbits. Concerning multistability, many other chaotic models have been published with such states of functioning [20][21][22][23]. These dynamical systems are divided into two groups: self-exited model and hidden ones [24,25].…”