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.

On the Dynamics of a Quadratic-Oscillator-Based, Infinite-Equilibrium System

In this paper, the local and global dynamics of a periodically forced, quadratic-oscillator-based, infinite-equilibrium system is discussed. The local analysis of regular equilibriums and infinite-equilibriums is completed, and the global responses of the periodically forced infinite-equilibrium system are presented through numerical simulations. Near the infinite-equilibrium surface, the periodically forced infinite-equilibrium system can be reduced to a one-dimensional system and new contraction regions can be formed. The infinite-equilibrium surface can be artificially designed to control the motions of the original quadratic nonlinear oscillator. Such a property is like a discontinuous dynamical system, which can be used for controller design in nonlinear systems.

Periodic Motions With Grazing in a Discontinuous Dynamical System With Two Circular Boundaries

In this paper, studied are periodic motions with grazing in a discontinuous dynamical system with two circular boundaries. The grazing motion is for a periodic motion switching to another periodic motions. Thus, the sufficient and necessary conditions of motion switching, grazing and sliding on the boundaries are discussed first. Periodic motions with grazing in the discontinuous system are presented for illustration of motions switching.

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.

Analysis of Discontinuous Dynamical Behaviors of a Friction-Induced Oscillator with an Elliptic Control Law

This paper develops the passability conditions of flow to the discontinuous boundary and the sticking or sliding and grazing conditions to the separation boundary in the discontinuous dynamical system of a friction-induced oscillator with an elliptic control law and the friction force acting on the mass M through the analysis of the corresponding vector fields and G-functions. The periodic motions of such a discontinuous system are predicted analytically through the mapping structure. Finally, the numerical simulations are given to illustrate the analytical results of motion for a better understanding of physics of motion in the mass-spring-damper oscillator.

Complex Dynamics of Bouncing Motions at Boundaries and Corners in a Discontinuous Dynamical System

In this paper, from the local theory of flow at the corner in discontinuous dynamical systems, obtained are analytical conditions for switching impact-alike chatter at corners. The objective of this investigation is to find the dynamics mechanism of border-collision bifurcations in discontinuous dynamical systems. Multivalued linear vector fields are employed, and generic mappings are defined among boundaries and corners. From mapping structures, periodic motions switching at the boundaries and corners are determined, and the corresponding stability and bifurcations of periodic motions are investigated by eigenvalue analysis. However, the grazing and sliding bifurcations are determined by the local singularity theory of discontinuous dynamical systems. From such analytical conditions, the corresponding parameter map is developed for periodic motions in such a multivalued dynamical system in the single domain with corners. Numerical simulations of periodic motions are presented for illustrations of motions complexity and catastrophe in such a discontinuous dynamical system.