Localizing Bifurcations in Non-Linear Dynamical Systems via Analytical and Numerical Methods

Abstract

In this paper, we study the bifurcations of non-linear dynamical systems. We continue to develop the analytical approach, permitting the prediction of the bifurcation of dynamics. Our approach is based on implicit (approximate) amplitude-frequency response equations of the form FΩ,A;c̲=0, where c̲ denotes the parameters. We demonstrate that tools of differential geometry make possible the discovery of the change of differential properties of solutions of the equation FΩ,A;c̲=0. Such qualitative changes of the solutions of the amplitude-frequency response equation, referred to as metamorphoses, lead to qualitative changes of dynamics (bifurcations). We show that the analytical prediction of metamorphoses is of great help in numerical simulation.

Asymmetric Duffing oscillator: jump manifold and border set

“…Therefore, solving Eqs. (11) numerically for several values of F 0 we easily find that for F 0 = 0.301 007 there is indeed a double root: Ω = 0.597 114, A 0 = 0.679 284 and Ω = 0.597 114, A 0 = 1. 411 787.…”

“…Working in the implicit function framework [10,11], we have computed in Subsection 4.1, using the (approximate) steady-state solution obtained in [1,6,9], the jump manifold 13, comprising information about all jumps in the dynamical system 1. Our formalism, described in Section 4 -built on an idea to use a differential condition to detect vertical tangencies, due to Kalmár-Nagy and Balachandran [8] -can be applied to arbitrary steady-state solution.…”

“…where equation (11b) is the condition for a vertical tangency (note that equations (11) are Eqs. (7a), (7c)).…”

“…For example, let γ = 0.0783, ζ = 0.025, F = 0.1. To find a value of F 0 for which this happens we have to find a double root of Ω of equations (11). Therefore, solving Eqs.…”

“…In the next Section we describe the steady state solution (2) [1,6,9] which is an implicit function of A 0 and Ω (5) and compute an implicit function of A 1 and Ω (6a), (6b). Working in the framework developed in our earlier papers [10,11] we compute the jump manifold in Section 4 containing information about all possible jumps which is the main achievement of this work. We summarize our results in the last Section.…”