Numerical Study Of Grazing Bifurcation And Double Homoclinic Loops In Planar Dynamical Systems

In this thesis we mainly investigates the numerical analysis method of grazing bifurcations for periodic and homoclinic orbit, and the numerical computations of double homoclinic loops in autonomous planar dynamical systems.It is widely studied about numerical computation of periodic orbit [21,48, 5,19]. In recent years, much research effort in applied science and engineering has been focused on piecewise smooth dynamical system (PWS) because of their applicability in a wide class of systems of practical interest including impact oscillators systems. Due to their non-smooth nature, such systems can exhibit rich bifurcation phenomena. Experiments, numerics and theoretical developments have clearly shown that complex transitions in PWS systems are often associated to tangential intersections of orbits with one of the switching manifold, which is called grazing bifurcation. For example, in impact oscillators systems, dynamical behavior from nonimpacting state to impacting state (or vice versa) encounters the grazing situation. Grazing events are known to lead to period-doubling cascades and sudden transitions to a chaotic attractor, which have been observed both analytically and experimentally. Therefore, it is important on how to figure out the grazing bifurcations in applications.The first part of the thesis is focus on the numerical computation method for locating the grazing periodic orbit to a smooth switching manifold. Firstly, we proposed a analytical condition to ensure the periodic orbit quadratically intersect the switching manifold. Secondly, we introduce a new phase condition, which make that there exits a branch of periodic orbits. The nondegenerate condition with respect to its bifurcation parameter is presented, which means the branch of periodic orbits transversally across the switching manifold, so that the defining equation is well posed. Then we define a system locating the grazing bifurcation, and we prove that the periodic grazing orbit and the parameter is a regular solution of the defining system in terms of the exponential dichotomy properties and Fredholm alternative properties. At last, we present three numerical examples to illustrate the theoretical analysis.The connecting orbit plays an important role in determining the global stability and structural stability of the dynamical system. Therefor it is worthwhile to study the grazing point of connecting orbit by numerical means. Then numerical method for computation of connecting orbit is well studied, see [4, 25, 26, 50].The second part of the thesis is focus on the numerical analysis method for locating the grazing connecting orbit to the smooth manifold. Firstly, we proposed a analytical condition to ensure the connecting orbit quadratically intersect the switching manifold. Secondly, we introduce a new phase condition, which make that there exits a branch of connecting orbits. The nondegenerate condition with respect to continuation bifurcation parameter is given, which means the branch of connecting orbits transversally across the switching manifold, so that the defining equation is well posed. Then we define a system locating the grazing bifurcation, and we prove that the connecting grazing orbit and the parameters is a regular solution of the defining system in terms of the exponential dichotomy properties and Fredholm alternative properties. In order to obtain the grazing connecting orbit, we truncate the orbit to a finite interval by virtue of the projection boundary condition. We then investigate the existence of solutions to the truncated problems and their error estimates. At last, we present two numerical examples to illustrate the theoretical analysis.The last part of this thesis is focus on the numerical computation of the double homoclinic loops and analyze the corresponding truncated error. We construct a defining system for computing the double homoclinic loops. In order to obtain the double homoclinic loops, we truncate the orbit on a finite interval by virtue of the projection boundary condition. We also carry out two numerical examples. In each example, we depict the double homoclinic orbit diagram and analyze the truncated error. From the numerical computations, we observe that the convergence rate for both parameters is in superlinear.