Non-Markovian processes on heteroclinic networks
Gray Manicom1, Vivien Kirk1, Claire Postlethwaite1
1Department of Mathematics, The University of Auckland, Auckland 1142, New Zealand.
Chaos (Woodbury, N.Y.)
|March 12, 2024
Summary
This study explores how noise-induced lift-off in heteroclinic networks affects Markov chain dynamics. We analyze the persistence of lift-off effects and propose methods to determine Markov chain order and transition probabilities.
Area of Science:
- Dynamical Systems and Chaos Theory
- Stochastic Processes
- Computational Neuroscience
Background:
- Heteroclinic networks feature saddle equilibria connected by trajectories.
- Noise can induce phenomena like 'lift-off' affecting state transitions.
- The sequence of visited saddles can be modeled as a stochastic process.
Purpose of the Study:
- Investigate the impact of lift-off at one saddle on subsequent saddle dynamics.
- Determine the order of the associated Markov chain of states.
- Calculate transition probabilities for the Markov chain.
Main Methods:
- Review and extend Bakhtin's methods for saddle dynamics with noise.
- Analyze the persistence of lift-off effects across saddles.
- Develop a method to find a lower bound for Markov chain order.
- Numerically simulate results for various noise amplitudes.
Main Results:
- Lift-off at one saddle can influence dynamics at the next saddle.
- Conditions for the persistence of lift-off effects were identified.
- A method for bounding the Markov chain order was proposed.
- Numerical simulations validated theoretical predictions across noise levels.
Conclusions:
- Lift-off is a critical phenomenon influencing the Markovian or non-Markovian nature of state sequences in heteroclinic networks.
- The study provides tools to analyze and predict the behavior of these systems under noise.
- Understanding these dynamics is crucial for applications in fields like computational neuroscience.
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