Controlling spatiotemporal chaos in active dissipative-dispersive nonlinear systems

Area of Science:

• Dynamical Systems Theory
• Nonlinear Dynamics
• Chaos Theory

Background:

• Infinite-dimensional dynamical systems often exhibit complex spatiotemporal chaos.
• Controlling these systems is challenging due to their infinite dimensionality and chaotic behavior.
• Existing methods may not cover the full spectrum of solutions.

Purpose of the Study:

• To present an alternative methodology for stabilizing and controlling infinite-dimensional dynamical systems.
• To demonstrate the control of diverse solutions, including steady states, traveling waves, and spatiotemporal chaos.
• To validate the proposed method using a paradigmatic model.

Main Methods:

• Development of a novel methodology utilizing time-dependent controls.
• Application of the method to stabilize and control various solution types.
• Exemplification using the generalized Kuramoto-Sivashinsky equation.

Main Results:

• The proposed methodology successfully stabilizes and controls all stable and unstable solutions.
• Demonstrated control over steady solutions, single and multipulse traveling waves, and bound states.
• Effective management of low-dimensional spatiotemporal chaos in the generalized Kuramoto-Sivashinsky equation.

Conclusions:

• The presented methodology offers a robust approach for controlling complex behaviors in infinite-dimensional systems.
• This technique provides a unified framework for stabilizing diverse solutions within chaotic systems.
• The generalized Kuramoto-Sivashinsky equation serves as an effective testbed for this control strategy.