Complex dynamics in simple systems with periodic parameter oscillations
Juarez, L. H., Kantz, H., Martinez, O., Ramos, E. and Rechtman, R. (2004) Complex dynamics in simple systems with periodic parameter oscillations. Physical Review E, 70 (5). 056202. ISSN 1539-3755
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To link to this item DOI: 10.1103/PhysRevE.70.056202
We study systems with periodically oscillating parameters that can give way to complex periodic or nonperiodic orbits. Performing the long time limit, we can define ergodic averages such as Lyapunov exponents, where a negative maximal Lyapunov exponent corresponds to a stable periodic orbit. By this, extremely complicated periodic orbits composed of contracting and expanding phases appear in a natural way. Employing the technique of ϵ-uncertain points, we find that values of the control parameters supporting such periodic motion are densely embedded in a set of values for which the motion is chaotic. When a tiny amount of noise is coupled to the system, dynamics with positive and with negative nontrivial Lyapunov exponents are indistinguishable. We discuss two physical systems, an oscillatory flow inside a duct and a dripping faucet with variable water supply, where such a mechanism seems to be responsible for a complicated alternation of laminar and turbulent phases.