Numerical Computation And Bifurcation Analysis Of Connecting Orbits In Planar Piecewise Smooth Dynamical Systems.

In recent years, there is growing interest in piecewise smooth dynamical systems for their wide applications in applied science and engineering. Physical systems may have different modes, and the transition from one mode to another can sometimes be idealized as an instantaneous transition, since the transition time from one mode to another is often much smaller than the time scale of the dynamics of each individual mode. Thus the mathematical models of these physical systems lead to piecewise smooth dynamical systems, which switch between different modes, where the dynamics in each mode is described by an ordinary differential equation. Piecewise smooth models often appear in many different disciplines, such as power electronic converters, mechanical systems with clearances or elastic constraints, earthquake engineering, stick-slip models with frictions and many other physical processes. The bifurcation analysis and numerical computation of piecewise-smooth systems have become a pressing open problem.The theory of bifurcations in smooth dynamical systems is well understood, but much less is known about the bifurcations in piecewise smooth systems although recently a lot of research work has been devoted towards this topic. All kinds of bifurcations in smooth systems occur to the piecewise smooth systems, but some of them could behave quite differently from those in smooth systems due to the influence of the discontinuity of the system. The local bifurcations such as saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation present special features in piecewise smooth systems comparing to the counterparts in smooth systems. Especially the geometrical property of the discontinuity is involved in the detection of the generalized Hopf bifurcation which yields the piecewise smooth bifurcating periodic orbits. Some bifurcations related to discontinuity only occur to piecewise smooth systems, such as border-collision, corner-collision, grazing bifurcation, sliding bifurcation, discontinuous bifurcation and non-smooth bifurcation and so on.The connecting orbit in a piecewise smooth dynamical system plays a very important role in determining the global stability and structural stability of the dynamical system. Therefore it is very worthwhile to study the properties of the piece-wise smooth connecting orbit and its numerical computational methods in piecewise smooth dynamical systems. The numerical methods for computing connecting orbits in smooth dynamical systems are well studied. But these well posed methods are unable to be directly applied to piecewise smooth dynamical systems due to the influence of the discontinuity of the systems. Kuznetsov et. have constructed defining equations for several codimension 1 sliding homoclinic orbits and sliding periodic orbits. But there hasn't been a systematic result about the numerical methods for computing piecewise smooth connecting orbits.In this paper, we focus ourselves on studying the bifurcating properties and numerical analysis of the piecewise smooth connecting orbits in a piecewise smooth dynamical system. Our main work consists of·Set up a numerical method for approximating a connecting orbit which transversally intersects the line of discontinuity. We define a nondegenerate condition for the piecewise smooth connecting orbit, which ensures the regularity of an extending equation for computing this piecewise smooth connecting orbit together with its bifurcation parameter. We carry out numerical computations for the piecewise smooth connecting orbit by truncating the extending equation to a finite interval using the projection boundary conditions, then we estimate the truncation errors. Afterwards, we apply this method to computing piecewise smooth homo-clinic orbits and heteroclinic orbits in a special piecewise smooth lienard system and a free rocking block model, respectively, to illustrate the validity of our method.·Apply the method for computing a piecewise smooth connecting orbits to an impacting-oscillation system, and develop a numerical method to approximate the corresponding Melnikov function. We also analyze the errors in computing the Melnikov function caused by the truncation of the time interval of the homoclinic orbits. Numerical experiments are carried on to illustrate the validity of our method.·Study the stability of a piecewise smooth homoclinic orbit via the saddle quantity. We find that the nonzero saddle quantity is still the first quantity for determining the stability of a piecewise smooth homoclinic orbit. That is, a piecewise smooth homoclinic orbit is asymptotically stable if the saddle quantity is minus and is asymptotically unstable if the saddle quantity is plus. We also study the properties of piecewise smooth periodic orbits bifurcating from a piecewise smooth homoclinic orbit. We find that the stability and the transversality ensure the appearance of a family of piecewise smooth periodic orbits. We illustrate our theoretical result by numerical continuations of a family of piecewise smooth periodic orbits bifurcating from a piecewise smooth homoclinic orbit in a special piecewise smooth lienard system.·Analyze the homoclinic bifurcations in a special piecewise smooth lienard system by numerical methods. Theoretical analysis and numerical computation show that this piecewise smooth system possesses not only four bifurcation curves of smooth homoclinic orbits but also a bifurcation curve of piecewise smooth homoclinic orbits. The curve of the piecewise smooth homoclinic orbits terminates one end at a piecewise smooth saddle-node homoclinic orbit and terminates the other end at a degenerate piecewise smooth homoclinic orbit. The degenerate piecewise smooth homoclinic orbit has a tangent intersection with the line of discontinuity as it crosses the line of discontinuity. We also analyze the local bifurcations of this system, including equilibria bifurcation, saddle-node bifurcation. Hopf bifurcation and Bogdanov-Takens bifurcation.