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Periodically kicked hard oscillators.

G. A. Cecchi1, D. L. Gonzalez, M. O. Magnasco

  • 1Departamento de Fisica, U.N.L.P., ArgentinaSezione di Cinematografia Scientifica, C.N.R., BolognaThe James Franck Institute, The University of Chicago, Chicago, Illinois 60637Department of Physics and Atomic Sciences, Drexel University, Philadelphia, Pennsylvania 19104Mathematics Department, Queen Mary College, LondonDepartamento de Fisica, U.N.L.P., Argentina.

Chaos (Woodbury, N.Y.)
|January 1, 1993
PubMed
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This study presents a hard oscillator model, revealing four distinct dynamic regions under periodic forcing. Novel regions intrinsic to hard oscillators emerge with intermediate forcing amplitudes.

Area of Science:

  • Nonlinear Dynamics
  • Theoretical Physics
  • Mathematical Modeling

Background:

  • Hard oscillators exhibit unique behaviors not seen in soft oscillators.
  • Understanding the dynamics of periodically forced systems is crucial in various scientific fields.

Purpose of the Study:

  • To present an analytically solvable model of a hard oscillator.
  • To investigate the complex dynamics of this hard oscillator under periodic forcing.
  • To identify and characterize novel dynamical regions specific to hard oscillators.

Main Methods:

  • Development of a hard oscillator model with an analytic solution.
  • Derivation of a closed-form stroboscopic map for periodic forcing.
  • Analysis of the system's behavior across different forcing amplitudes.

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Main Results:

  • Identification of four distinct dynamical regions for the hard oscillator.
  • Outer regions resemble soft oscillator behaviors (degree one and zero circle maps).
  • Two novel regions emerge at intermediate forcing amplitudes: Region 3 (discontinuous circle maps) and Region 2 (branched manifolds).

Conclusions:

  • The behavior of hard oscillators under periodic forcing is complex and exhibits unique characteristics.
  • Region 3, with discontinuous circle maps, is intrinsic to hard oscillators at moderate-high forcing.
  • Region 2, featuring branched manifolds, is also intrinsic to hard oscillators at low-moderate forcing, differing from soft oscillator dynamics.