Controlling multistability in coupled systems with soft impacts

Abstract In this paper we present an influence of discontinuous coupling on the dynamics of multistable systems. Our model consists of two periodically forced oscillators that can interact via soft impacts. The controlling parameters are the distance between the oscillators and the difference in the phase of the harmonic excitation. When the distance is large two systems do not collide and a number of different possible solutions can be observed in both of them. When decreasing of the distance, one can observe some transient impacts and then the systems settle down on non-impacting attractor. It is shown that with the properly chosen distance and difference in the phase of the harmonic excitation, the number of possible solutions of the coupled systems can be reduced. The proposed method is robust and applicable in many different systems.

[1]  T Kapitaniak,et al.  Experimental observation of ragged synchronizability. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Anil K. Bajaj,et al.  Resonant dynamics of an autoparametric system : A study using higher-order averaging , 1996 .

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

[4]  Arkady Pikovsky,et al.  A universal concept in nonlinear sciences , 2006 .

[5]  C. Budd,et al.  Review of ”Piecewise-Smooth Dynamical Systems: Theory and Applications by M. di Bernardo, C. Budd, A. Champneys and P. 2008” , 2020 .

[6]  P. Brzeski,et al.  Basin stability approach for quantifying responses of multistable systems with parameters mismatch , 2016 .

[7]  J. Kurths,et al.  From Phase to Lag Synchronization in Coupled Chaotic Oscillators , 1997 .

[8]  B. Brogliato.  Nonsmooth Mechanics: Models, Dynamics and Control , 1999 .

[9]  Damian Giaouris,et al.  Vanishing singularity in hard impacting systems , 2011 .

[10]  J. Kurths,et al.  Phase synchronization of chaotic oscillations in terms of periodic orbits. , 1997, Chaos.

[11]  W. Goldsmith,et al.  Impact: the theory and physical behaviour of colliding solids. , 1960 .

[12]  Antonio Politi,et al.  From synchronization to Lyapunov exponents and back , 2006 .

[13]  Kevin D. Murphy,et al.  Multi-modal analysis on the intermittent contact dynamics of atomic force microscope , 2011 .

[14]  Tomasz Kapitaniak,et al.  Lyapunov exponents of impact oscillators with Hertz׳s and Newton׳s contact models , 2014 .

[15]  Ekaterina Pavlovskaia,et al.  Complex Dynamics of Bilinear oscillator Close to Grazing , 2010, Int. J. Bifurc. Chaos.

[16]  Przemyslaw Perlikowski,et al.  Synchronous states of slowly rotating pendula , 2014 .

[17]  Tomasz Kapitaniak,et al.  Antiphase synchronization of chaos by noncontinuous coupling: two impacting oscillators , 2001 .

[18]  Kurths,et al.  Phase synchronization of chaotic oscillators. , 1996, Physical review letters.

[19]  Christopher George,et al.  A driven system of impacting pendulums: Experiments and simulations , 2014 .

[20]  Przemyslaw Perlikowski,et al.  Multistability and Rare attractors in van der Pol-Duffing oscillator , 2011, Int. J. Bifurc. Chaos.

[21]  Jürgen Kurths,et al.  Synchronization - A Universal Concept in Nonlinear Sciences , 2001, Cambridge Nonlinear Science Series.

[22]  F. Peterka,et al.  Behaviour of impact oscillator with soft and preloaded stop , 2003 .

[23]  Tomasz Kapitaniak,et al.  Hard versus soft impacts in oscillatory systems modeling , 2010 .

[24]  T. Carroll,et al.  Master Stability Functions for Synchronized Coupled Systems , 1998 .

[25]  Soumitro Banerjee,et al.  Dangerous bifurcation at border collision: when does it occur? , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  M. Lakshmanan,et al.  Transition from anticipatory to lag synchronization via complete synchronization in time-delay systems. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  C. Mirasso,et al.  Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers. , 2001, Physical review letters.

[28]  R Sevilla-Escoboza,et al.  Generalized synchronization in relay systems with instantaneous coupling. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  Peter J. Menck,et al.  How basin stability complements the linear-stability paradigm , 2013, Nature Physics.

[30]  Soumitro Banerjee,et al.  Border collision bifurcations in a soft impact system , 2006 .

[31]  Edward Ott,et al.  Theoretical mechanics: crowd synchrony on the Millennium Bridge. , 2005 .

[32]  Ugo Andreaus,et al.  Numerical simulation of the soft contact dynamics of an impacting bilinear oscillator , 2010 .

[33]  Inna Sharf,et al.  Literature survey of contact dynamics modelling , 2002 .

[34]  Ilʹi︠a︡ Izrailevich Blekhman,et al.  Synchronization in science and technology , 1988 .

[35]  Finn Macleod,et al.  Rocking Newton’s cradle , 2004 .

[36]  H. Fujisaka,et al.  Stability Theory of Synchronized Motion in Coupled-Oscillator Systems , 1983 .