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Smoothing tautologies, hidden dynamics, and sigmoid asymptotics for piecewise smooth systems
1Engineering Mathematics, University of Bristol, Merchant Venturer's Building, Bristol BS8 1UB, United Kingdom.
Chaos (Woodbury, N.Y.)
|November 2, 2015
Summary
Modeling switches as instantaneous events can miss crucial nonlinear effects. This study explores "hidden terms" in switching systems, revealing how nonlinearities significantly alter electronic circuit behavior and create hidden attractors.
Area of Science:
- Nonlinear Dynamics
- Control Systems Engineering
- Theoretical Physics
Background:
- Real-world switches manifest in diverse forms, including physical impacts, electronic relays, biological processes like mitosis, and decision-making algorithms.
- Modeling these switches as instantaneous events can lead to the loss of critical information, necessitating the consideration of 'hidden terms'.
Purpose of the Study:
- To investigate the impact of 'hidden terms' in nonlinear switching systems.
- To develop a generalized expression for switches using sigmoid functions.
- To extend Filippov's method of sliding modes to analyze these complex systems.
Main Methods:
- Development of a generalized switch expression using a series of sigmoid functions.
- Extension of Filippov's method for sliding mode analysis in nonlinear systems.
- Analysis of electronic control circuits to identify nonlinear effects.
Main Results:
- Identified 'hidden terms' that vanish asymptotically but are encoded in nonlinear switching functions.
- Demonstrated that even minor nonlinear effects can drastically change electronic control circuit behavior.
- Revealed the existence of 'hidden attractors' within the switching surface.
Conclusions:
- Instantaneous switch modeling is an oversimplification that neglects significant nonlinear dynamics.
- Nonlinear switching terms and their associated 'hidden terms' are crucial for accurately describing system behavior.
- The presence of 'hidden attractors' highlights complex dynamics within switching surfaces that warrant further investigation.
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