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Related Experiment Videos

Fractional dissipative standard map.

Vasily E Tarasov1, M Edelman

  • 1Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, New York 10012, USA.

Chaos (Woodbury, N.Y.)
|July 2, 2010
PubMed
Summary
This summary is machine-generated.

Researchers introduce fractional maps, a novel generalization of the dissipative standard map, exhibiting long-term memory. These fractional maps display unique attractor behaviors influenced by past states, even with slight deviations from integer orders.

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Area of Science:

  • Nonlinear Dynamics
  • Fractional Calculus
  • Chaos Theory

Background:

  • The standard map is a fundamental model in nonlinear dynamics.
  • Dissipative systems exhibit attractors, which are states systems evolve towards.
  • Traditional maps use integer-order derivatives, limiting memory effects.

Purpose of the Study:

  • To generalize the dissipative standard map using fractional calculus.
  • To investigate the impact of long-term memory on system dynamics.
  • To explore novel attractor behaviors in fractional maps.

Main Methods:

  • Utilizing kicked differential equations of motion with non-integer order derivatives.
  • Developing generalized maps termed 'fractional maps'.
  • Analyzing the influence of derivative order on attractor properties.

Main Results:

  • Fractional maps exhibit long-term memory, where current states depend on all past states.
  • Small deviations from integer derivative orders lead to qualitatively new attractor behaviors.
  • A new class of fractional attractors was demonstrated across a range of fractional orders.

Conclusions:

  • Fractional maps offer a powerful framework for studying systems with memory effects.
  • The introduction of fractional derivatives significantly alters attractor dynamics.
  • This work opens new avenues for research in nonlinear dynamics and chaos theory.