Chaotic bubbles and phase locking for a shaker system in the vicinity of three coexisting critical points

Abstract In this paper, the dynamical model of the shaker system, Poincare maps, Jacobian matrix and power spectrum are established. Different phase-locking phenomena and chaotic bubbles are investigated in the vicinity of three coexisting critical points including Hopf–Hopf bifurcation point, 1:3 resonance point and 1:4 resonance point. In two strong resonance cases, phase-locking dynamics and associated bifurcations are easily to occur. Coexisting attractors have also been introduced to provide mechanisms for chaotic bubbles with connections between pieces. The occurrence of phase locking on doubling torus to multi-period leads to interruption of torus-doubling bifurcation. Isolated chaotic bubbles are birth via period-doubling bifurcation of such a multi-period. Phase-locking phenomena on T2 torus are also observed in such a neighborhood of critical points. The number of periods on torus by phase locking can be identified by power spectrum methods. The system parameters may be optimized by studying of phase-locking dynamics of this system.

[1]  Marten Scheffer,et al.  Chaos in a long-term experiment with a plankton community , 2008, Nature.

[2]  G. Iooss.  Bifurcation of maps and applications , 1979 .

[3]  M. A. Safonova,et al.  Bifurcations and transition to chaos through three-dimensional tori , 1994 .

[4]  E. Ott.  Chaos in Dynamical Systems: Contents , 1993 .

[5]  Yongxiang Zhang,et al.  Multifarious intertwined basin boundaries of strange nonchaotic attractors in a quasiperiodically forced system , 2009 .

[6]  G. Wen,et al.  CODIMENSION-2 HOPF BIFURCATION OF A TWO-DEGREE-OF-FREEDOM VIBRO-IMPACT SYSTEM , 2001 .

[7]  F. Peterka,et al.  Bifurcations and transition phenomena in an impact oscillator , 1996 .

[8]  F. Peterka.  Comments on "Periodic and Chaotic Behaviour of a Threshold-Limited Two-Degree-of-Freedom System" , 1993 .

[9]  Toshio Okada,et al.  A numerical analysis of chaos in the double pendulum , 2006 .

[10]  G. Luo,et al.  Period-doubling bifurcations and routes to chaos of the vibratory systems contacting stops , 2004 .

[11]  J. M. T. Thompson,et al.  Chaos after period-doubling bifurcations in the resonance of an impact oscillator , 1982 .

[12]  Y. Yue,et al.  Lyapunov exponents and coexistence of attractors in vibro-impact systems with symmetric two-sided rigid constraints , 2009 .

[13]  Steven W. Shaw,et al.  Periodically forced linear oscillator with impacts: Chaos and long-period motions , 1983 .

[14]  Horst Malchow,et al.  Experimental demonstration of chaos in a microbial food web , 2005, Nature.

[15]  U. Feudel,et al.  Renormalization of correlations and spectra of a strange non-chaotic attractor , 1996 .

[16]  T. Chai,et al.  Adaptive synchronization between two different chaotic systems with unknown parameters , 2006 .

[17]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[18]  Xie Jian-hua,et al.  Codimension two bifurcations and hopf bifurcations of an impacting vibrating system , 1996 .

[19]  Jianhua Xie,et al.  Dynamical analysis of a two-parameter family for a vibro-impact system in resonance cases , 2005 .

[20]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[21]  J. Aidanpää,et al.  Periodic and chaotic behaviour of a threshold-limited two-degree-of-freedom system , 1993 .

[22]  Grebogi,et al.  Critical exponents for crisis-induced intermittency. , 1987, Physical review. A, General physics.

[23]  Amitabha Nandi,et al.  Coexisting attractors in periodically modulated logistic maps. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  J. Yorke,et al.  Crises, sudden changes in chaotic attractors, and transient chaos , 1983 .

[25]  Arkady Pikovsky,et al.  Correlations and spectra of strange non-chaotic attractors , 1994 .

[26]  Y. Chu,et al.  A special type of attractor and transitions in a delayed system , 2008 .

[27]  Arne Nordmark,et al.  Non-periodic motion caused by grazing incidence in an impact oscillator , 1991 .

[28]  Huaguang Zhang,et al.  Adaptive Synchronization Between Two Different Chaotic Neural Networks With Time Delay , 2007, IEEE Transactions on Neural Networks.

[29]  Gamal M. Mahmoud,et al.  Strange attractors and chaos control in periodically forced complex Duffing's oscillators , 2001 .

[30]  Periodic-impact motions and bifurcations of a dual component system , 2006 .

[31]  Yongxiang Zhang,et al.  Critical curves and coexisting attractors in a quasiperiodically forced delayed system , 2009 .

[32]  Daniel Thiel,et al.  System dynamics modeling and simulation of a particular food supply chain , 2000, Simul. Pract. Theory.

[33]  T. Charnley,et al.  Normal modes of the modern English church bell , 1983 .

[34]  Guanwei Luo,et al.  Codimension two bifurcations of fixed points in a class of vibratory systems with symmetrical rigid stops , 2007 .

[35]  G. S. Whiston,et al.  Singularities in vibro-impact dynamics , 1992 .

[36]  Yongxiang Zhang,et al.  Bistability and Dynamical Transitions in a Phase-Modulated Delayed System , 2009 .

[37]  A. P. Ivanov,et al.  Bifurcations in impact systems , 1996 .

[38]  P. Holmes,et al.  A periodically forced piecewise linear oscillator , 1983 .