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HOPF AND SUBHARMONIC BIFURCATION OF VIBRO-IMPACT SYSTEM IN ORDER 4 RESONANCE CASE

MA Yong-jing

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Abstract

A three-degree-of-freedom vibro-impact system is considered in this paper. Based on the solutions of differential equations between impacts, impact conditions and match conditions of periodic motion, the six-dimensional Poincare maps of n ?1periodic motion are established. The two-parameter unfoldings of local dynamical behavior in resonance are investigated. By numerical simulation, as two controlling parameters varying on the two-parameter plane, the topology regions of parameter plane are divided. An invariant torus via Hopf bifurcation and period 4 ? 4 motions via subharmonic bifurcation are analyzed, which are characterized by “four-square” and “four-leaf ” different lodge orbits and exist near the critical point, and the route from order 4 subharmonic bifurcation to chaos is further analyzed.

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What this paper is about

A three-degree-of-freedom vibro-impact system is considered in this paper. Based on the solutions of differential equations between impacts, impact conditions and match conditions of periodic motion, the six-dimensional Poincare maps of n ?1periodic motion are established. The two-parameter unfoldings of local dynamical behavior in resonance are investigated. By numerical simulation, as two controlling parameters varying on the two-parameter plane, the topology regions of parameter plane are divided. An invariant torus via Hopf bifurcation and period 4 ? 4 motions via subharmonic bifurcation are analyzed, which are characterized by “four-square” and “four-leaf ” different lodge orbits and exist near the critical point, and the route from order 4 subharmonic bifurcation to chaos is further analyzed.

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Available abstract

A three-degree-of-freedom vibro-impact system is considered in this paper. Based on the solutions of differential equations between impacts, impact conditions and match conditions of periodic motion, the six-dimensional Poincare maps of n ?1periodic motion are established. The two-parameter unfoldings of local dynamical behavior in resonance are investigated. By numerical simulation, as two controlling parameters varying on the two-parameter plane, the topology regions of parameter plane are divided. An invariant torus via Hopf bifurcation and period 4 ? 4 motions via subharmonic bifurcation are analyzed, which are characterized by “four-square” and “four-leaf ” different lodge orbits and exist near the critical point, and the route from order 4 subharmonic bifurcation to chaos is further analyzed.

Key concepts: Saddle-node bifurcation, Bifurcation, Hopf bifurcation, Mathematics, Transcritical bifurcation, Bifurcation diagram, Torus, Period-doubling bifurcation

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