Discrete-time systems with random switches: From systems stability to networks synchronization

Area of Science:

• Control Theory
• Network Science
• Stochastic Systems

Background:

• Discrete-time systems with random switches present challenges in stability and synchronization analysis.
• Unbounded and continuous-valued distributions in switching matrices add complexity.
• Understanding random connections is crucial for complex dynamical networks.

Purpose of the Study:

• To develop analytical approaches for identifying stability patterns in randomly switching discrete-time systems.
• To investigate and ensure almost sure synchronization in complex dynamical networks with random connections.
• To explore synchronization in networks with nonlinear maps and large population sizes.

Main Methods:

• Analytical identification of essential patterns for almost sure stability.
• Investigation of almost sure synchronization using probability space configurations.
• Numerical simulations to validate theoretical results and demonstrate preserved chaotic dynamics.

Main Results:

• Developed approaches accurately identify stability guarantees for discrete-time systems with random switches, even with unbounded distributions.
• Achieved almost sure synchronization in complex dynamical networks, preserving chaotic dynamics under specific statistical parameters.
• Demonstrated the possibility of achieving synchronization in large-scale randomly switching networks, outperforming deterministic approaches.

Conclusions:

• The developed analytical methods provide robust guarantees for stability and synchronization in complex stochastic systems.
• Random switching and connections can facilitate synchronization in large networks, offering advantages over deterministic configurations.
• The research contributes to the theoretical understanding and practical application of control in complex dynamical networks.