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Dynamical singularities for complex initial conditions and the motion at a real separatrix.

Tamar Shnerb1, K G Kay

  • 1Department of Chemistry, Bar-Ilan University, Ramat-Gan 52900, Israel.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|May 23, 2006
PubMed
Summary

Complex initial conditions in double-well systems create finite-time singularities. These singularities explain unstable motion near the real separatrix, linking complex dynamics to chaos formation.

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Area of Science:

  • Classical dynamics
  • Nonlinear systems
  • Mathematical physics

Background:

  • Double-well systems exhibit complex dynamics.
  • Singularities in classical systems can arise from complex initial conditions.
  • The behavior of systems near separatrices is crucial for understanding stability and chaos.

Purpose of the Study:

  • Investigate finite-time singularities in one-dimensional double-well systems.
  • Relate these singularities to system behavior under real initial conditions.
  • Understand the role of singularities in the onset of chaos.

Main Methods:

  • Analytical treatment of a quartic double-well system.
  • Numerical calculations for multiple double-well systems.
  • Identification of properties of singular trajectories.

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Main Results:

  • A doubly infinite sequence of singularities was found in the quartic double-well system.
  • Singularities converge to real separatrix initial conditions.
  • Numerical evidence confirms numerous singularities approaching the separatrix.
  • The hyperbolic fixed point is critical for singularity formation.

Conclusions:

  • Complex initial conditions lead to finite-time singularities in double-well systems.
  • These singularities explain the unstable motion observed at the real separatrix.
  • The approach of singularities to the real axis is linked to chaos.
  • Understanding these singularities provides insight into the transition to chaotic behavior.