Complex dynamics in simple systems with periodic parameter oscillations.
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 epsilon-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.
Juárez
L Héctor
LH
Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, Apdo. Postal 55-534, 09340 México DF, Mexico.
Kantz
Holger
H
Martínez
Oscar
O
Ramos
Eduardo
E
Rechtman
Raúl
R
D000222:Adaptation, Physiological / Q000502:physiology*; D001683:Biological Clocks; D003198:Computer Simulation; D008954:Models, Biological; D017711:Nonlinear Dynamics; D010507:Periodicity; D011673:Pulsatile Flow / Q000502:physiology*