Neural-network maps for two-parameter modeling of bistability and codimension-two bifurcations in two-dimensional...

Area of Science:

• Nonlinear Dynamics
• Computational Physics
• Machine Learning

Background:

• The van der Pol oscillator is a fundamental model in nonlinear dynamics.
• Understanding bifurcations, such as the Andronov-Hopf and Bautin bifurcations, is crucial for predicting system behavior.
• Machine learning offers potential for modeling complex dynamical systems.

Purpose of the Study:

• To develop a machine-learning approach to model the behavior of van der Pol oscillators.
• To investigate the ability of a neural network to reproduce subcritical Andronov-Hopf bifurcations and codimension-2 Bautin points.
• To assess the model's capacity to capture dynamical features absent from training data.

Main Methods:

• Development of a neural-network model functioning as a recurrent map.
• Training the neural network on short segments of oscillator trajectories.
• Analysis of the neural network's performance in a two-dimensional parameter space.

Main Results:

• The trained neural network successfully reproduced key dynamical features of the van der Pol oscillators.
• The model captured multistability and various bifurcations (Andronov-Hopf, saddle-node, Bautin) without direct exposure during training.
• The accuracy of reproducing bifurcations depended on the appropriate selection of training data.

Conclusions:

• A simple neural network architecture can effectively model complex nonlinear dynamics.
• Machine learning can generalize and predict phenomena not explicitly present in training datasets.
• Careful data selection is essential for training machine learning models to capture critical dynamical transitions.