Bifurcation Of Periodic Solutions Of Non-Smooth Dynamical Systems

As the non-smooth dynamical systems are becoming more and more important in engineering fields,they draw attention of many scientific workers and engineers.The non-smooth dynamical systems have been one of the hot spot of non-linear dynamics study in recent years.Due to non-smooth characteristic of vector field of non-smooth dynamical systems the conventional methods in analyzing stability and bifurcation of periodic solution of smooth systems don't apply,and the new methods need be developed to analyze the dynamic behaviors of periodic solution of non-smooth systems,which is a big challenge for theoretical research.The main respects of the research are followings:1.The computation method of invariant circles for high dimensional maps is addressed.The bifurcating conditions of invariant circles are analyzed.A necessary condition for the existence of a solution of a kind of equations,which rises in computing invariant circle,is presented through Fredholm method,then the variable of phase angle is expanded as a Fourier series and the expressions of computing invariant circles are given by identification of the Fourier coefficients. The computation method of invariant circles for mapping is realized in a three-dimensional map finally.2.The transversal periodic solutions of impacting systems are addressed.The linearized matrix is derived by means of zero-time discontinuity mapping.The stroboscopic Poincarémap is established to analyze the period-doubling bifurcation of periodic motions in the single-degree-of-freedom impacting system. The theoretical result is verified.3.The local dynamics behavior of grazing orbit of impacting systems is investigated.The conditions for the existence of grazing orbit are analyzed.The Zero-time discontinuity mapping and the Poincaré-section discontinuity mapping are deduced in detail.The parameter space is included to construct a compound piecewise smooth map by composing previous discontinuity mappings with a Poincarémap for smooth periodic solution.The grazing bifurcation of periodic motions in the single-degree-of-freedom impacting system is investigated.The grazing bifurcation point is found in terms of the conditions for the existence of grazing orbit.The bifurcation diagrams are gained through previous piecewise smooth map and numerical calculation,they are similar structurally.A saltation arises to cause the discontinuity bifurcation when the parameter is a transitional point from one periodic orbit area to the other one and the grazing phenomenon happens at the transitional point.This bifurcation phenomenon is different from continuity period-doubling bifurcation of smooth systems.4.Hopf-pitchfork bifurcation and Hopf-Hopf bifurcation of high dimensional smooth maps obtained in impacting systems are investigated.The high dimensional map is reduced to a three-dimensional map by the center manifold theorem.The reduced map is further transformed into its normal form by theory of normal forms.The two parameter unfoldings of the map near the point of Hopf-pitchfork bifurcation is investigated theoretically.The obtained result is applied to analyzing the Hopf-pitchfork bifurcation of periodic motion in a three-degree-of freedom vibro-impact system and is verified by numerical work. The high dimensional map is reduced to a four-dimensional normal form map by the center manifold theorem and theory of normal forms.The two-parameter unfoldings of local dynamical behavior,near the point of Hopf-Hopf bifurcation, is investigated theoretically.Hopf-Hopf bifurcation in a three-degree-of freedom vibro-impact system is further investigated by means of numerical simulations. Numerical simulation results indicate that the vibro-impact system presents interesting and complicated dynamical behavior.5.With regard to transversal periodic solutions of piecewise smooth systems, the stroboscopic Poincarémap is established.The saltation matrix is deduced by zero-time discontinuity mapping method and then the linearized matrix of the map is given.The dynamics behaviors of transversal periodic motions of a single-degree-of-freedom piecewise linear system is studied.The period-doubling bifurcation of periodic motions of the piecewise linear system is investigated by the saltation matrix and the Floquet theory.The period-doubling bifurcations and chaotic behaviors in the non-smooth system are further investigated by means of numerical simulations.The bifurcation and chaos of transversal periodic motions of a two-degree-of-freedom non-smooth system with piecewise-linearity is investigated.The saltation matrix is given out at the switching boundaries and the Neimark-Sacker bifurcation point and period-doubling bifurcation point of periodic motions of the system are investigated by the numerical calculation. Numerical results demonstrate the existence of Neimark-Sacker bifurcation, period-doubling bifurcation and subharmonic bifurcation in the non-smooth system.6.The local dynamics behaviors of grazing orbit of piecewise smooth systems are investigated.The conditions for the existence of grazing orbit are analyzed.Zero-time discontinuity mapping is deduced in detail.A compound piecewise smooth map is given by composing the discontinuity mapping with a Poincarémap for no intersection periodic solution.The grazing bifurcation of periodic motions in the single-degree-of-freedom piecewise linear system is investigated.The grazing bifurcation point is found in terms of the conditions for the existence of grazing orbit.The bifurcation diagrams are gained through the previous piecewise smooth map and numerical calculation,they are similar structurally.Then the grazing behavior is further analyzed by phase portraits.The numerical results indicate that the piecewise linear non-smooth system presents period 1 motion→period-doubling bifurcation→period 2 motion→grazing bifurcation→period 2 motion→grazing bifurcation→period 4 motion→period-doubling bifurcation→chaos with the increase of parameter and period 1 motion→period-doubling bifurcation→period 2 motion→grazing bifurcation→period 3 motion→grazing bifurcation→chaos with the decrease of parameter.7.The corner-collision bifurcation of piecewise smooth systems is addressed. The external and internal corner collisions geometry and the conditions for the existence of corner collision orbit are investigated.The Zero-time discontinuity mapping is deduced in detail and the compound piecewise smooth map of corner-collision bifurcation is given.The single-degree-of-freedom piecewise linear system with two non-smooth switching boundaries is studies.The numerical results indicate that the piecewise linear system presents saddle-node bifurcation and comer-collision bifurcation.8.The sliding orbit geometry and the conditions for the existence of sliding orbit are investigated.Four types of sliding bifurcations and their bifurcation conditions are introduced.The single-degree-of-freedom dry friction system is considered.The crossing-sliding bifurcation and switching-sliding bifurcation are shown by numerical simulations.The numerical results also indicate that the dry friction system presents the special period-doubling bifurcation accompanied by grazing bifurcation.