Weakly coupled parametrically forced oscillator networks: existence, stability, and symmetry of solutions

Abstract

In this paper, we discuss existence, stability, and symmetry of solutions for networks of parametrically forced oscillators. We consider a nonlinear oscillator model with strong 2:1 resonance via parametric excitation. For uncoupled systems, the 2:1 resonance property results in sets of solutions that we classify using a combinatorial approach. The symmetry properties for solution sets are presented as are the group operators that generate the isotropy subgroups. We then impose weak coupling and prove that solutions from the uncoupled case persist for small coupling by using an appropriate Poincaré map and the Implicit Function Theorem. Solution bifurcations are investigated as a function of coupling strength and forcing frequency using numerical continuation techniques. We find that the characteristics of the single oscillator system are transferred to the network under weak coupling. We explore interesting dynamics that emerge with larger coupling strength, including anti-synchronized chaos and unsynchronized chaos. A classification for the symmetry-breaking that occurs due to weak coupling is presented for a simple example network. P. Danzl ( ) · J. Moehlis

Metamorphoses of resonance curves for two coupled oscillators: The case of small non-linearities in the main mass frame

“…Dynamic vibration absorber is a typical mechanical model described by (1) [1,2] (in this case m is usually much larger than m e ). Dynamics of coupled periodically driven oscillators is very complicated [3][4][5][6][7][8][9][10][11][12][13]. It is important that experimental study confirms good agreement with theoretical model [6].…”

“…[20] the author outlined a programme to "define and find different branches intersecting at singular points" in onedegree-of-freedom non-linear system and considered also general non-linear dynamical systems. Independently, the Implicit Function Theorem was also successfully applied to bifurcation problems in systems of coupled parametrically forced oscillators in [10]. The authors demonstrated that "provided the periodic orbits for the uncoupled system are hyperbolic, there will be periodic orbits for the weakly coupled system close to the periodic states identified for the uncoupled system" [10].…”

“…Independently, the Implicit Function Theorem was also successfully applied to bifurcation problems in systems of coupled parametrically forced oscillators in [10]. The authors demonstrated that "provided the periodic orbits for the uncoupled system are hyperbolic, there will be periodic orbits for the weakly coupled system close to the periodic states identified for the uncoupled system" [10]. In this work we are working in a more general context of theory of singular points of algebraic curves [21,22].…”

Dynamics of weakly coupled parametrically forced oscillators

“…Danzl and Moehlis [26] considered networks of nonlinear parametrically forced oscillators with weak coupling, used the implicit function theorem to prove the persistence of the uncoupled solution classes, and investigated this numerically. Far from the weak coupling limit, interesting dynamics such as antisynchronized chaotic behavior were found.…”

“…The current paper shares with [26] an emphasis on symmetries and the assumption of weak coupling and with [23,24] an interest in the effect of the forcing phases. As in [23], we utilize a bifurcation theory (unfolding) approach.…”