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Updated: Jul 15, 2026
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Published on: September 23, 2025
Resonant forcing of multidimensional chaotic map dynamics
Glenn Foster1, Alfred W Hübler, Karin Dahmen
1Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA.
Summary
Researchers identified optimal forcing functions for chaotic dynamics. These resonant functions conserve trajectory displacement and mirror the system
Area of Science:
- Nonlinear dynamics
- Chaos theory
- Statistical mechanics
Background:
- Chaotic systems exhibit sensitive dependence on initial conditions.
- Understanding and controlling chaotic behavior is a key challenge in nonlinear dynamics.
Purpose of the Study:
- To determine the additive forcing function that elicits the largest response in chaotic map dynamics.
- To analyze the properties and effectiveness of resonant forcing functions.
Main Methods:
- Utilizing the calculus of variations to derive the optimal forcing function.
- Analyzing the relationship between forcing functions and trajectory separation.
- Comparing optimal forcing with random forcing.
Main Results:
- Resonant forcing functions conserve the product of trajectory displacement and forcing.
- The periodicity of resonant forcing matches that of displacement dynamics.
- Resonant forcing functions decrease exponentially at a rate determined by the largest Lyapunov exponent.
Conclusions:
- Optimal resonant forcing is highly effective when the largest Lyapunov exponent dominates.
- For very large Lyapunov exponents, optimal forcing effectiveness diminishes, approaching that of single-push forcing.
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