Mode interaction in horses, tea, and other nonlinear oscillators: The universal role of symmetry

Abstract

This paper is about mode interaction in systems of coupled nonlinear oscillators. The main ideas are demonstrated by means of a model consisting of two coupled, parametrically driven pendulums. On the basis of this we also discuss mode interaction in the Faraday experiment ͑as observed by Ciliberto and Gollub͒ and in running animals. In all these systems the interaction between two modes is seen to take place via a third mode: This interaction mode is a common daughter, born by means of a symmetry breaking bifurcation, of the two interacting modes. Thus, not just any two modes can interact with each other, but only those that are linked ͑in the system's group-theoretical hierarchy͒ by a common daughter mode. This is the quintessence of mode interaction. In many cases of interest, the interaction mode is seen to undergo further bifurcations, and this can eventually lead to chaos. These stages correspond to lower and lower levels of symmetry, and the constraints imposed by group theory become less and less restrictive. Indeed, the precise sequence of events during these later stages is determined not so much by group-theoretical stipulations as by the accidental values of the nonlinear terms in the equations of motion.

Asymptotic analysis of oscillating parametric integrals and ordinary boundary value problems at resonance

“…It allows us to prove that the solution set of the auxiliary equation is connected and consequently we find more precise information on the range of (1.3). The result is inspired from paper [15] and, as in [2] for the case of an scalar equation, it involves some restrictions on the derivatives of f and g. To the best of our knowledge, Dirichlet boundary conditions for problems like (1.3) have not been previously considered in the literature (see [8,12,16,17,24] for the periodic boundary value problem, which seems to be different in many aspects from the Dirichlet one).…”

“…A concrete case is the motion of two identical pendulums coupled by a nonlinear torsion elastic force. This problem originates systems of the form [12,16,17,21,24] …”

In Search for the Neural Mechanisms of Individual Development: Behavior-Driven Differential Hebbian Learning

“…The role of symmetries of the system for the pertinent motion patterns has been extensively studied in the literature, see Schöner et al (1990), Collins and Stewart (1993), Strogatz and Stewart (1993), Golubitsky et al (1998, 1999), van der Weele and Banning (2001, Golubitsky (2012), and Tero et al (2013). In those works, gaits are driven by central pattern generators (CPGs) that are constructed of non-linear oscillators.…”

“…For instance, stotting may be associated with the group of maximum symmetry. Transitions between gaits can be associated with symmetry breaking bifurcations (Collins and Stewart, 1993;van der Weele and Banning, 2001). So, the approach leads to natural hierarchies of gaits, ordered by symmetry, and to natural sequences of gait bifurcations.…”

“…In the bifurcation scenario discussed in those papers, symmetry breaking was induced by changing the controller parameters, such as the amplitude and frequency of the driving force in Collins and Stewart (1993) and van der Weele and Banning (2001), from outside: when crossing the bifurcation point, the system becomes instable and the system state jumps into one of the emerging alternatives. With BDDHL, we have a self-referential system (Der and Martius, 2012), a dynamical system that changes its parameters by itself, driven by the BDDHL mechanism.…”