Characterization of events in the Rössler system using singular value decomposition

Abstract

Chaotic systems change their behavior through topological events such as creation/annihilation and bifurcation. These can be characterized by defining a tangent phase space which measures the first-order response of stable limit cycles to a change in an external variable. If the period of the limit cycle is constant, then the tangent phase space response can be formulated as a boundary-value problem, which is dependent upon a previously calculated limit cycle. If the period is not constant, the tangent phase space will contain an unknown linear drift in time. This can be analytically removed by transforming into a time-dependent coordinate system in which one variable points in the direction of the instantaneous velocity in phase space. The remaining variables can then be decoupled from this motion, and will satisfy a linear system of differential equations subject to periodic boundary conditions. The solutions of these equations at bifurcation events can be analyzed using singular value decomposition of two matrices, one of which contains interactions within the limit cycle, while the other contains interactions with the changing external variable. Collectively, these two decompositions allow us to uniquely characterize any topological event. The method is applied to period-doubling and turning-points of limit cycles in the Rössler system, where it confirms previous work done on the Zeeman Catastrophe Machine. It is also applied to bifurcations of equilibria in the Rössler system, where it allows us to distinguish between Andronov-Hopf and fold-Hopf bifurcations.