Dynamical Analysis Of Some Piecewise Smooth Systems And Singularly Perturbed Systems

Piecewise smooth systems and singularly perturbed systems are important tools for describing nonlinear phenomena in practical fields,such as: engineering technology,circuit design and spring oscillators.The dynamics of these two types of systems are hot and frontier topics in the field of nonlinear dynamics.This paper aims to study the existence and maximum number of limit cycles in two kinds of discontinuous planar piecewise-smooth linear systems,the existence of homoclinic orbits for a three-dimensional continuous piecewise-smooth linear generalized Michelson system,the existence of heteroclinic loops and hyperchaos in a four-dimensional singularly perturbed coupled Chua’s circuit model.The main research contents and innovation work include the following parts:(1)We investigate the maximum number and relative position of limit cycles for the discontinuous planar piecewise linear systems with three zones separated by two parallel straight lines,which is closely related to Hilbert’s 16 th problem.Based on the methods of normal form and Poincar’e map,we present the existence and maximum number of limit cycles in the normal forms of systems with boundary focus-center-boundary focus and boundary focus-center-center types,respectively.Then we show that such discontinuous piecewise linear systems can have at most three limit cycles,being two of them of four intersection points type and the third one of two intersection points type.We also verify the theoretical results by numerical simulations.(2)We study the maximum number and relative position of limit cycles in the discontinuous planar piecewise linear systems separated by a nonregular line and formed by linear Hamiltonian vector fields without equilibria,which is closely related to Hilbert’s 16 th problem.Based on the methods of first integral and Poincar’e map,we prove that such systems have at most two limit cycles,and the limit cycles must intersect the nonregular separation line in two or four points.More precisely,the exact upper bound of limit cycles is two,and this upper bound can indeed be reached: either both intersect the separation line at two points or one intersects the separation line at two points and the other one at four points.Based on Poincar’e map,the stability of various limit cycles is also proved.In addition,we give some concrete examples to illustrate our main results.(3)We investigate the homoclinic orbits for the three-dimensional continuous piecewise linear generalized Michelson systems with one switching plane.Based on the Poincar’e map and invariant manifold theory,we discuss the existence of homoclinic orbits connecting the saddle-focus equilibrium.Furthermore,we provide the precise parameters of homoclinic orbits by symbolic calculation tools and overcome the technical difficulties induced by non-smoothness,and the theoretical results are verified by numerical simulations.(4)We are concerned with the heteroclinic loops and hyperchaos of a fourdimensional singularly perturbed system,which arises from the coupled arrays of Chua’s circuit.By the geometric singular perturbation theory and generalized rotating vector field,we prove that there exists a heteroclinic loop consisting of the traveling front and back waves with the same wave speed.Especially,the expression of corresponding wave speed is also obtained.Furthermore,we show that such heteroclinic loop can lead to hyperchaos by invariant manifold theory and the method of asymptotic expansion.