Autonomous Systems and Emergence of Stable Dynamical Patterns

Dr. Constantine Medynets (medynets@usna.edu)

Dr. Irina Popovici (popovici@usna.edu)

United States Naval Academy

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ONR Nonlinear Physics Annual Review, July 27, 2022

Agent Based Systems

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Agent Based Models

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Motivation: The Parabolic Potential Model

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Gazebo Sim: Hector Swarm.

Credit: MIDN Carl Kolon

Research Objectives

• Development of mathematical methods for dimension reduction, control, and stability analysis
• Classification of rare dynamical events occurring in these systems
• Error analysis
• Data-driven classifications of dynamical modes

Areas of applications include: network synchronization, adversary swarm capturing, swarm control, optimal sensor placements (biochemical defense)

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Translating State: nAgents = 60

Ring State: nAgents = 60

Mixed State: nAgents = 3

Stability Analysis of Ring States

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Stability Analysis: Details

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Technical Details

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New Way to Approximate Center Manifold

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Near collapsing ring state

Center Manifold w/ continuum of fixed points

New Way to Approximate Center Manifold

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The Contraction Mapping Principle

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Speed of Convergence. Rare Events

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Convergence to a non collapsing configuration.

Generalized Coupling Matrix

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Generalized Coupling/Communication Matrix

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Going Forward

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• Stability Analysis for Second Order Models in Swarming
• Understanding the Impact of Varying the Network �Configuration on the Limit Cycles of the Collective Dynamics
• Mixed States/Rare Events
• Data-Driven Dynamics and the Study of Aggregate Parameters
• Swarm Capture/Collision �

Midshipman Projects Class of 2022

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• Swarming Dynamics on Riemannian Manifolds, MIDN Cami Herman, Class of 2022.
• Dynamical Systems with Delayed Response, MIDN Rachel Manhertz, Class of 2022.
• Impulsive Differential Equations with Applications to the Pulse Vaccination Strategy for Controlling Disease Outbreaks, MIDN Joe Spirnak, Class of 2022. �