Multiple Hopf bifurcations, period-doubling reversals and coexisting attractors for a novel chaotic jerk system with Tchebytchev polynomials

Abstract

The study of complex nonlinear phenomena certainly represents a very hot ongoing research topic owing to recently published works. While a large number of contributions deal with systems with symmetry, very few works are concerned with those without any symmetry property. In this contribution, we introduce a novel asymmetric jerk system whose nonlinearity is in the form of a six-order Tchebytchev polynomial of the fundamental variable. The new system distinguishes by the presence of six equilibrium points symmetrically distributed along the x-axis. Three of these points are always unstable (regardless the values of parameter) while each of the three others undergoes Hopf type bifurcation at three different critical values of the control parameter resulting to self-excited coexisting behaviors such as: (i) a limit cycle and a pair of stable fixed points; (ii) a pair of limit cycles and a stable fixed point; (iii) a stable fixed point, a limit cycle and a chaotic attractor; just to cite a few. These features are studied by combining both analytical and numerical methods. More importantly, various parameter ranges are depicted where the new jerk system with Tchebytchev polynomials demonstrates extremely complex and striking nonlinear patterns such as antimonotonicity, hysteresis and coexisting bifurcation branches. These two latter properties give rise to multiple (i.e. two, three or four) coexisting (periodic and chaotic) attractors. Cross sections of the basins of attraction are provided to illustrate the magnetization of the state space caused by the various competing dynamics. The control of multistability in the new system is achieved via linear augmentation scheme. The practical feasibility of the new system is supported by a series of PSPICE simulations utilizing an electronic analogue of the proposed jerk system. The combination of features found in the new jerk system with Tchebytchev polynomials are rarely reported and thus merits dissemination.