New topological tool for multistable dynamical systems

Area of Science:

• Dynamical Systems Theory
• Nonlinear Dynamics
• Mathematical Physics

Background:

• Investigating dynamical systems often requires complex, full-dimensional analysis.
• Identifying hidden oscillations and understanding attractor geometry can be challenging.
• Existing methods may lack efficiency in extracting comprehensive system dynamics from equations alone.

Purpose of the Study:

• To introduce a novel method for analyzing dynamical systems based on their governing equations.
• To develop a simplified procedure for localizing hidden oscillations.
• To enhance the understanding of attractor geometry in complex dynamical systems.

Main Methods:

• Utilizing critical surfaces defined by zero velocity/acceleration fields.
• Implementing dimension reduction within the phase space.
• Comparing the new method with standard approaches on example systems.

Main Results:

• The critical surfaces method effectively extracts information about system dynamics from equations.
• Dimension reduction offers a computational advantage over full-dimensional analysis.
• The method successfully localizes hidden oscillations and clarifies attractor geometry.

Conclusions:

• The critical surfaces approach provides a powerful and efficient tool for dynamical systems analysis.
• This method offers new insights into attractor geometry, especially for multistable and hidden attractors.
• The technique has broad applicability in science and engineering for studying complex systems.