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N-dimensional dynamical systems exploiting instabilities in full.

J. Rius1, M. Figueras, R. Herrero

  • 1Departament de Fisica, Universitat Autonoma de Barcelona, 08193 Bellaterra, Spain.

Chaos (Woodbury, N.Y.)
|June 5, 2003
PubMed
Summary

Complex dynamics in N-dimensional systems arise from nonlinear mode mixing. This "full instability behavior" emerges from successive bifurcations, creating complex attractors and irregular oscillations.

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Area of Science:

  • Nonlinear Dynamics
  • Complex Systems
  • Thermo-optical Systems

Background:

  • N-dimensional dynamical systems can exhibit complex behaviors.
  • Understanding the mechanisms behind complex time evolutions is crucial.

Purpose of the Study:

  • To investigate the emergence of complex time evolutions in N-dimensional dynamical systems.
  • To characterize the phenomenon of "full instability behavior" through experimental and numerical studies.

Main Methods:

  • Experimental studies using thermo-optical systems with dimensions from 1 to 6.
  • Numerical simulations employing N-dimensional vector fields with nonlinear functions.
  • Analysis of successive Hopf bifurcations and invariant set formation.

Main Results:

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  • Demonstrated complex time evolutions resulting from nonlinear combinations of N-1 oscillation modes.
  • Observed "full instability behavior" linked to successive Hopf bifurcations in saddle-node pairs.
  • Identified nonlinear mode mixing and the influence of saddle sets on attractor formation, likely involving invariant tori.

Conclusions:

  • The study presents a generalized Landau scenario for complex behavior emergence.
  • Nonlinear superposition of oscillatory motions is a key mechanism for irregular dynamics.
  • Full instability behavior provides a framework for understanding complex dynamics in high-dimensional systems.