Independent Period-2 Motions to Chaos in a van der Pol–Duffing Oscillator

In this paper, an independent bifurcation tree of period-2 motions to chaos coexisting with period-1 motions in a periodically driven van der Pol–Duffing oscillator is presented semi-analytically. Symmetric and asymmetric period-1 motions without period-doubling are obtained first, and a bifurcation tree of independent period-2 to period-8 motions is presented. The bifurcations and stability of the corresponding periodic motions on the bifurcation tree are determined through eigenvalue analysis. The symmetry breaks of symmetric period-1 motions is determined by the saddle-node bifurcations, and the appearance of the independent bifurcation tree of period-2 motions to chaos is also due to the saddle-node bifurcations. Period-doubling cascaded scenario of period-2 to period-8 motions are predicted analytically, and unstable periodic motions are also obtained. Numerical simulations are performed to illustrate motion complexity in such a van der Pol–Duffing oscillator. Such nonlinear systems can be applied in nonlinear circuit design and fluid-induced oscillations.

Periodic Motions and Bifurcations of a Periodically Forced Spring Pendulum Varying With Excitation Amplitude

In this paper, the semi-analytical method of implicit discrete maps is employed to investigate the nonlinear dynamical behavior of a nonlinear spring pendulum. The implicit discrete maps are developed through the midpoint scheme of the corresponding differential equations of a nonlinear spring pendulum system. Using discrete mapping structures, different periodic motions are obtained for the bifurcation trees. With varying excitation amplitude, a bifurcation tree of period-1 motion to chaos is achieved through the bifurcation tree of period-1 to period-2 motions. The corresponding stability and bifurcations are studied through eigenvalue analysis. Finally, numerical illustrations of periodic motions are obtained numerically and analytically.

Period-3 Motions in a Parametrically Exited Inverted Pendulum

In this paper, period-3 motions in a parametrically exited inverted pendulum are analytically investigated through a discrete implicit mapping method. The corresponding stability and bifurcation conditions of the period-3 motions are predicted through eigenvalue analysis. The symmetric and asymmetric period-3 motions are obtained on the bifurcation tree, and the period-doubling bifurcations of the asymmetric period-3 motions are observed. The saddle-node and Neimark bifurcations for symmetric period-3 motions are obtained. The saddle-bifurcations of the symmetric period-3 motions are for symmetric motion appearance (or vanishing) and onsets of asymmetric period-3 motion. Numerical simulations of the period-3 motions in the inverted pendulum are completed from analytical predictions for illustration of motion complexity and characteristics.

On Periodic Motions in a Periodically Driven van der Pol-Duffing Oscillator

In this paper, the symmetric and asymmetric period-1 motions on the bifurcation tree are obtained for a periodically driven van der Pol-Duffing hardening oscillator through a semi-analytical method. Such a semi-analytical method develops an implicit mapping with prescribed accuracy. Based on the implicit mapping, the mapping structures are used to determine periodic motions in the van der Pol-Duffing oscillator. The symmetry breaks of period-1 motion are determined through saddle-node bifurcations, and the corresponding asymmetric period-1 motions are generated. The bifurcation and stability of period-1 motions are determined through eigenvalue analysis. To verify the semi-analytical solutions, numerical simulations are also carried out.

Period-1 to Period-2 Motions in a Discontinuous Oscillator

In this paper, grazing bifurcations on bifurcation trees in a discontinuous dynamical oscillator are discussed. Once the grazing bifurcation occurs, periodic motions switches from the old motion to a new one. Thus, grazing bifurcations on a bifurcation tree of period-1 to period-2 motions varying spring stiffness are presented in a discontinuous oscillator with three domains divided by circular boundaries. The stability and bifurcations of period-1 and period-2 motions are discussed. From analytical predictions, periodic motions are simulated numerically. Stiffness effects on the periodic motions are discussed. Such studies will help one understand parameter effects in discontinuous dynamical systems, which can be applied for system design and control.

Period-1 Motion to Chaos in a Nonlinear Flexible Rotor System

In this paper, a bifurcation tree of period-1 motion to chaos in a flexible nonlinear rotor system is presented through period-1 to period-8 motions. Stable and unstable periodic motions on the bifurcation tree in the flexible rotor system are achieved semi-analytically, and the corresponding stability and bifurcation of the periodic motions are analyzed by eigenvalue analysis. On the bifurcation tree, the appearance and vanishing of jumping phenomena of periodic motions are generated by saddle-node bifurcations, and quasi-periodic motions are induced by Neimark bifurcations. Period-doubling bifurcations of periodic motions are for developing cascaded bifurcation trees, however, the birth of new periodic motions are based on the saddle-node bifurcation. For a better understanding of periodic motions on the bifurcation tree, nonlinear harmonic amplitude characteristics of periodic motions are presented. Numerical simulations of periodic motions are performed for the verification of semi-analytical predictions. From such a study, nonlinear Jeffcott rotor possesses complex periodic motions. Such results can help one detect and control complex motions in rotor systems for industry.

Period-1 to Period-8 Motions in a Nonlinear Jeffcott Rotor System

In this paper, a bifurcation tree of period-1 to period-8 motions in a nonlinear Jeffcott rotor system is obtained through the discrete mapping method. The bifurcations and stability of periodic motions on the bifurcation tree are discussed. The quasi-periodic motions on the bifurcation tree are caused by two (2) Neimark bifurcations of period-1 motions, one (1) Neimark bifurcation of period-2 motions and four (4) Neimark bifurcations of period-4 motions. The specific quasi-periodic motions are mainly based on the skeleton of the corresponding periodic motions. One stable and one unstable period-doubling bifurcations exist for the period-1, period-2 and period-4 motions. The unstable period-doubling bifurcation is from an unstable period-m motion to an unstable period-2m motion, and the unstable period-m motion becomes stable. Such an unstable period-doubling bifurcation is the 3rd source pitchfork bifurcation. Periodic motions on the bifurcation tree are simulated numerically, and the corresponding harmonic amplitudes and phases are presented for harmonic effects on periodic motions in the nonlinear Jeffcott rotor system. Such a study gives a complete picture of periodic and quasi-periodic motions in the nonlinear Jeffcott rotor system in the specific parameter range. One can follow the similar procedure to work out the other bifurcation trees in the nonlinear Jeffcott rotor systems.

On a Global Sequential Scenario of Bifurcation Trees to Chaos in a First-Order, Periodically Excited, Time-Delayed System

In this paper, the global sequential scenario of bifurcation trees of periodic motions to chaos is studied for a first-order, time-delayed, nonlinear dynamical system with periodic excitation. The periodic motions of such a first-order time-delayed system is obtained semi-analytically, and the corresponding stability and bifurcations are determined by eigenvalue analysis. A global sequential scenario of bifurcation trees is given by [Formula: see text] where [Formula: see text] is a global bifurcation tree of an asymmetric period-[Formula: see text] motion to chaos, and [Formula: see text] is a global bifurcation tree of a symmetric period-[Formula: see text] motion to chaos. Each bifurcation tree of a specific periodic motion to chaos is presented in detail. Numerical simulations of periodic motions are performed from analytical predictions. From finite Fourier series, harmonic amplitudes and phases for periodic motions are obtained for frequency analysis. Through this study, the rich dynamics of the first-order, time-delayed, nonlinear dynamical system is presented.