On Existence and Bifurcations of Periodic Motions in Discontinuous Dynamical Systems

In this paper, the existence and bifurcations of periodic motions in a discontinuous dynamical system is studied through a discontinuous mechanical model. One can follow the study presented herein to investigate other discontinuous dynamical systems. Such a sampled discontinuous system consists of two subsystems on boundaries and three subsystems in subdomains. From the theory of discontinuous dynamical systems, switchability conditions of a flow at and on the boundaries are developed. From such switchability conditions, grazing motions of a flow at boundaries are discussed, and sliding motions of a flow on boundaries are presented. Based on the motions in each domain and on each boundary, generic mappings are introduced. Using the generic mappings, mapping structures for specific periodic motions are developed. Based on the grazing conditions and appearance and vanishing conditions of sliding motions, parametric dynamics of the existences of the specific periodic motions are presented. In addition, the traditional saddle-node bifurcation, Neimark bifurcations and period-doubling bifurcation are used for parametric dynamics of periodic motions. Bifurcation trees of periodic motions varying with a system parameter are presented first, and phase trajectories of periodic motions are illustrated. The [Formula: see text]-functions are presented for the illustration of the motion switchability at the boundaries and sliding motions on the boundaries. Codimension-2 parametric dynamics of periodic motions are studied and how to develop the 2D parametric maps for specific periodic motions are presented. In the end, periodic motions with grazing are illustrated.

Periodic Motions in an Autonomous System With a Discontinuous Vector Field

In this paper, periodic motions in an autonomous system with a discontinuous vector field are discussed. The periodic motions are obtained by constructing a set of algebraic equations based on motion mapping structures. The stability of periodic motions is investigated through eigenvalue analysis. The grazing bifurcations are presented by varying the spring stiffness. Once the grazing bifurcation occurs, periodic motions switches from the old motion to a new one. Numerical simulations are conducted for motion illustrations. The parameter study helps one understand autonomous discontinuous dynamical systems.

On Symmetric Periodic Motions With Different Excitation Periods in a Discontinuous System With a Hyperbolic Boundary

In this paper, symmetric periodic motions with different excitation periods in a discontinuous dynamic system with a hyperbolic boundary are presented analytically. The switchability conditions of flows at the hyperbolic boundaries are developed. Periodic motions with specific mapping structures are predicted analytically, and numerical simulations of periodic motions are carried out. The corresponding G-functions are presented for illustration of motion switchability at the hyperbolic boundaries.

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.

The Coexisting Behaviors on the Boundary of a Duffing-like Oscillator with Signum Nonlinearity and Its FPGA-Based Implementation

In this paper, the nonlinear behaviors of a Duffing-like system with signum function are investigated through the theory of discontinuous dynamical systems. The necessary and sufficient conditions for motion switchability on the boundary between two domains are analyzed to understand the switching mechanism. The switching velocity with varying different system parameters and the parameter mappings are carried out to illustrate the dynamical motions. The attraction basins are depicted to express the coexistence of the Duffing-like oscillator with different initial values, and the coexisting trajectories in phase space with various initial conditions are also exhibited. The effectiveness of the analysis conditions of periodic and chaotic motions with different mapping structures are verified through numerical simulations. Moreover, a hardware circuit of the Duffing-like system is established via Field Programmable Gate Array for the validation of the numerical analysis.

Period-1 Motions to Chaos With Varying Excitation Frequency in a Parametrically Driven Pendulum

In this paper, with varying excitation frequency, period-1 motions to chaos in a parametrically driven pendulum are presented through period-1 to period-4 motions. Using the implicit discrete maps of the corresponding differential equations, discrete mapping structures are developed for different periodic motions, and the corresponding nonlinear algebraic equations of such mapping structures are solved for analytical predictions of bifurcation trees of periodic motions. Both period-1 static points to period-2 motions and period-1 motions to period-4 motions are illustrated. The corresponding stability and bifurcations are studied. Finally, numerical illustrations of various periodic motions on the bifurcation trees are presented in verification of the analytical prediction.

On Period-1 and Period-2 Motions of a Jeffcott Rotor System

In this paper, the semi-analytical solutions of period-1 and period-2 motions in a nonlinear Jeffcott rotor system are presented through the discrete mapping method. The periodic motions in the nonlinear Jeffcott rotor system are obtained through specific mapping structures with a certain accuracy. A bifurcation tree of period-1 to period-2 motion is achieved, and the corresponding stability and bifurcations of periodic motions are analyzed. For verification of semi-analytical solutions, numerical simulations are carried out by the mid-point scheme.

Periodic Motions in a Discontinuous Dynamical System With Two Circular Boundaries

In this paper, periodic motions in a discontinuous dynamical systems are studied. The discontinuous dynamical system has three domains partitioned through two circular boundaries. On the three domains, there are three distinct dynamical systems. From the G-functions, the switchability conditions of a flow from one domain to anther domain at the boundary are developed. The flow mappings from a boundary to a bounbary are developed for each domain and boundary. From the mapping structures, periodic motions in the discontinuous dynamical system are predicted. Numerical simulations of periodic motions and motion switchability at boundaries are presented in the discontinuous dynamical system.