A theory for non-smooth dynamic systems on the connectable domains

Abstract

In this paper, a local theory of non-smooth dynamical systems on connectable and accessible sub-domains is developed. The properties for separation boundaries based on the characteristics of flows are determined, and the sliding dynamics on a specified separation boundary is introduced. The local singularity and transversality of a flow on the separation boundary from a domain into its adjacent domains are investigated, and the bouncing and tangency of the flows to the separation boundary for non-smooth dynamical systems are discussed as well. The sufficient and necessary conditions for the local singularity, transversality and bouncing of the flows are developed. These conditions are applicable for determining complicated dynamical behaviors of non-smooth dynamical systems. Ó 2004 Elsevier B.V. All rights reserved. PACS: 05.45.)a; 45.20.)d

Study of a piecewise linear dynamic system with negative and positive stiffness

“…Experimental and numerical results were arranged into bifurcation diagrams and Lissajous curves, compared, and analyzed. Luo proposed a local theory for non-smooth dynamic system to analyze the local singularity and transversality on the connecting boundary between adjacent domains [3]. Luo developed a methodology based on a perturbation method to investigate the local singularity of a periodically forced, piecewise linear system [4,5].…”

The Nonsmooth Vibration of a Relative Rotation System with Backlash and Dry Friction

“…In [16], the idea of mapping structure was used to investigate a periodic forced piecewise linear system. In addition, the investigations by Nordmark [17], Błazejczyk et al [18], Blazejczyk-Okolewska et al [19], Luo et al [20], di Bernardo et al [21,22], and Luo et al [23][24][25][26] can also provide us with a plenty of meaningful conclusions.…”

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

“…When a flow in a domain reaches the sliding or stick boundary at time and then the sliding or stick motion occurs, such time may not be the onset time of the sliding or stick motion on this boundary. When conditions (36), (37), (40), (41) or (38), (39), (42), and (43) cannot be satisfied, the passable flow vanishes and the sliding or stick motion will begin at time . From the flow switchability theory on the discontinuous dynamical systems, time is the switching time from semipassable flow to nonpassable flow.…”

“…At time 1 = 0.1888 s, the blue filled circle is the vanishing point of the sliding motion on the elliptic boundary Ω 12 , because the motion of the mass enters domain Ω 2 and is the free-fight motion after such a point. To describe the analytical conditions of sliding motion on the elliptic boundary Ω 12 for the mass , the time histories offunctions are depicted in Figure 11 = (3) < 0, which satisfies condition (38) in Theorem 8(ii); therefore the flow of the periodic motion is to pass through the velocity boundary Ω 23 and enters into domain Ω 3 from domain Ω 2 , as shown in Figure 12(a). In the meantime, the motion of the mass in domain Ω 2 goes back to the starting point.…”

“…In 2005, Luo [38,39] developed a theory for nonsmooth dynamical systems on connectable domains and the mapping dynamics of periodic motions for a piecewise linear system under a periodic excitation. In 2005, 2006 and 2008, Luo [40][41][42] introduced the concepts of imaginary, sink, and source flows in nonsmooth dynamical systems and developed a general theory for the local singularity of a flow to the discontinuous boundary and a theory for flow switchability in discontinuous dynamical systems.…”