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Return maps of folded nodes and folded saddle-nodes
1Mathematics Department, Cornell University, Ithaca, New York 14853, USA.
Chaos (Woodbury, N.Y.)
|April 2, 2008
Summary
Folded nodes in slow-fast dynamical systems are crucial for understanding complex behaviors like mixed mode oscillations. This study reveals their role in generating chaos and complex dynamics, particularly near folded saddle-nodes.
Area of Science:
- Dynamical Systems Theory
- Nonlinear Dynamics
- Computational Mathematics
Background:
- Folded nodes are key features in slow-fast dynamical systems with two slow variables.
- Their impact on global dynamics, beyond local trajectory twisting, remains under-explored.
- Mixed mode oscillations in chemical and neural systems are linked to folded node phenomena.
Purpose of the Study:
- To systematically investigate the return maps of trajectories interacting with folded nodes.
- To explore the role of folded nodes as a mechanism for chaos generation.
- To analyze the complexity of dynamics near folded saddle-node bifurcations.
Main Methods:
- Numerical simulation of return maps for trajectories near folded nodes.
- Approximation of return maps using rank-1 maps.
- Analysis of a variant of the forced van der Pol system.
Main Results:
- Return maps near folded nodes exhibit multiple turning points due to local trajectory twisting.
- Folded nodes can act as a mechanism for generating chaos.
- Systems near folded saddle-nodes display more complex return maps than those far from them.
Conclusions:
- Folded nodes significantly influence global dynamics in slow-fast systems.
- The study provides novel insights into chaos generation via folded nodes.
- Folded saddle-nodes introduce enhanced complexity in dynamical system behavior.
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