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A chaotic map with two-dimensional offset boosting.

Zhihao Wang1, Chunbiao Li1, Yongxin Li1

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This study introduces a novel chaotic map with two-dimensional offset boosting, demonstrating controllable amplitude and hybrid control. The chaotic map shows improved performance and robustness in particle swarm optimization applications.

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Area of Science:

  • Nonlinear Dynamics
  • Chaos Theory
  • Computational Intelligence

Background:

  • The Lozi map is a foundational chaotic system.
  • Controlling chaotic system amplitude is crucial for applications.
  • Multistability in chaotic systems offers diverse dynamic behaviors.

Purpose of the Study:

  • To exhaustively study a chaotic map with two-dimensional offset boosting.
  • To reveal the mechanism of offset boosting and its effect on dynamics.
  • To explore hybrid control strategies and applications in optimization.

Main Methods:

  • Derivation from the Lozi map.
  • Analysis of offset boosting mechanism via feedback term cancellation.
  • Investigation of multistability with varying initial conditions.
  • Numerical simulations and circuit implementation using microcontrollers.

Main Results:

  • Demonstrated controllability of amplitude via offset boosting.
  • Revealed coexistence of homogeneous and heterogeneous multistability.
  • Identified independent constant terms for amplitude rescaling without altering Lyapunov exponents.
  • Introduced hybrid control by binding amplitude control and offset boosting parameters.
  • Validated theoretical findings through circuit implementation and numerical simulations.

Conclusions:

  • The novel chaotic map offers precise amplitude control and exhibits rich multistability.
  • Hybrid control provides a unique approach to managing chaotic dynamics.
  • The chaotic map demonstrates superior performance and robustness for particle swarm optimization.