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Boundary-equilibrium bifurcations in piecewise-smooth slow-fast systems
1Manchester Metropolitan University, School of Computing, Mathematics and Digital Technology, Manchester M1 5GD, United Kingdom. piotr.kowalczyk@manchester.ac.uk
Chaos (Woodbury, N.Y.)
|July 5, 2011
Summary
This study analyzes piecewise-smooth slow-fast systems, revealing phase space topology to understand bifurcations. It uncovers four phase portraits and applies findings to thermohaline circulation models.
Area of Science:
- Dynamical Systems Theory
- Mathematical Modeling
- Oceanography
Background:
- Piecewise-smooth systems, also known as singularly perturbed systems, exhibit complex dynamics due to discontinuities.
- Understanding the phase space topology is crucial for analyzing the qualitative behavior of these systems.
- Previous studies often simplified the analysis by assuming continuity or neglecting discontinuities.
Purpose of the Study:
- To investigate the qualitative dynamics of everywhere continuous piecewise-smooth slow-fast systems.
- To analyze the phase space topology of systems with one-dimensional slow and fast dynamics.
- To uncover boundary-equilibrium bifurcations in planar slow-fast systems and apply these findings to climate models.
Main Methods:
- Analysis of phase space topology for piecewise-smooth slow-fast systems.
- Study of reduced systems with piecewise-continuous slow manifolds.
- Investigation of local dynamics near switching surfaces, focusing on boundary-equilibrium bifurcations.
- Application of theoretical findings to a box model of thermohaline circulation.
Main Results:
- The slow manifold of the reduced system is piecewise-continuous, with lost differentiability at the switching surface.
- The full system exhibits an O(ɛ) discontinuity across the switching manifold, which does not qualitatively alter dynamics.
- Four distinct qualitative phase portraits were uncovered for planar slow-fast systems with an equilibrium point on the switching surface.
- A boundary-equilibrium bifurcation of a fold type was identified in a thermohaline circulation box model.
Conclusions:
- The revealed phase space topology provides a framework for understanding the qualitative dynamics of piecewise-smooth slow-fast systems.
- Boundary-equilibrium bifurcations play a significant role in the dynamics of these systems, particularly when equilibria lie on switching surfaces.
- The study demonstrates the applicability of these theoretical insights to complex real-world systems like thermohaline circulation.
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