Stabilization of prescribed values and periodic orbits with regular and pulse target oriented control

Area of Science:

• Dynamical Systems and Chaos Theory
• Mathematical Modeling
• Population Dynamics

Background:

• Chaos control is crucial for understanding and managing complex systems.
• One-dimensional maps are fundamental models in chaos theory.
• The Allee effect in population dynamics presents challenges for stability and survival.

Purpose of the Study:

• To investigate a novel chaos control method for one-dimensional maps.
• To determine conditions under which stabilization of chaotic systems is possible.
• To explore the application of this control method in population dynamics models.

Main Methods:

• Analysis of one-dimensional maps with state-dependent interventions.
• Mathematical proof of stabilization conditions for increasing and locally Lipschitz maps.
• Study of pulse stabilization with periodic interventions.
• Simulation of population dynamics models with Allee effects.

Main Results:

• Stabilization is proven possible for increasing maps with decreasing slopes and for locally Lipschitz maps.
• Any point in a locally Lipschitz chaotic map can be stabilized.
• Pulse stabilization is demonstrated for the first type of map.
• Control strategies prevent extinction in population models with Allee effects, even without full stabilization.

Conclusions:

• The proposed chaos control method is effective for specific classes of one-dimensional maps.
• The method offers a means to stabilize chaotic behaviors in mathematical models.
• Applications in population dynamics show the potential to mitigate extinction risks and ensure persistent solutions.