A plethora of coexisting strange attractors in a simple jerk system with hyperbolic tangent nonlinearity

Symmetry Breaking, Coexisting Bubbles, Multistability, and Its Control for a Simple Jerk System with Hyperbolic Tangent Nonlinearity

“…Here, k is the symmetry control parameter of the model. Specifically, for k � 0, system (1) exhibits a perfect symmetry and reduces to the case previously studied by Kengne and coworkers [18]. e case k ≠ 0 corresponds to an asymmetric model for which more complex nonlinear phenomena arise (that cannot be explained by using the symmetry arguments) including, for instance, the presence of multiple coexisting asymmetric attractors, coexisting bifurcation branches, and crisis events (see Section 4).…”

“…On this line, Kengne and colleagues reported the coexistence of four self-excited mutually symmetric attractors in a jerk system possessing a cubic nonlinearity [23] based on both numerical and experimental methods. is striking feature of multiple attractors is mainly due to the system's symmetry and thus is also obtained with a hyperbolic sine [29], a hyperbolic tangent [18], a composite tanh-cubic nonlinearity [21], or a voltage controlled memristor [30], whose intrinsic current-voltage characteristics has the form of a pinched hysteresis loop. Despite the pertinence and the importance of the abovementioned results, we would like to stress that all cases of multistability discussed so far is restricted to symmetric jerk systems; also, multistability in jerk dynamic systems in case of a broken symmetry is very little studied.…”

“…us, the novel chaotic flow is smoothly tuned to behave either symmetrically or to develop no symmetry property using a single parameter. Importantly, the investigations clearly reveal that the modified system can experience coexisting bubbles of bifurcation, coexisting multiple (symmetric or asymmetric) attractors, and crises phenomena not found in the original symmetric system [18]. Despite the fact that the addition of a constant term may be viewed as a purely mathematical technique to induce new nonlinear phenomena, one of the key motivations is that symmetries are rarely exact in real physical systems, and some symmetrybreaking imperfections are always present [31][32][33].…”

“…In this work, we consider a simple jerk system with hyperbolic tangent nonlinearity [18] whose symmetry is broken by the introduction of an additive constant k. We address the chaos generation mechanism, the formation of bubbles of bifurcation, and the coexistence of multiple attractors in both the symmetric (k � 0) and the asymmetric (k ≠ 0) regimes of operation. For convenience, recall that jerk dynamic systems [19][20][21][22][23] refer to 3D ordinary differential equations (ODEs) in the form x ... � J(x, _ x, € x) where the nonlinear vector function J(·) indicates the "jerk" (i.e., the time derivative of the acceleration).…”

“…Despite the pertinence and the importance of the abovementioned results, we would like to stress that all cases of multistability discussed so far is restricted to symmetric jerk systems; also, multistability in jerk dynamic systems in case of a broken symmetry is very little studied. Motivated by previous results on jerk dynamical systems, this paper focuses on the effects engendered by symmetry break in a simple autonomous jerk system with hyperbolic tangent nonlinearity previously analyzed in [18]. us, the novel chaotic flow is smoothly tuned to behave either symmetrically or to develop no symmetry property using a single parameter.…”