Smale Horseshoes and Symbolic Dynamics in the Buck-Boost DC-DC Converter

• Published in: IEEE transactions on industrial electronics (1982) Vol. 65; no. 1; pp. 800 - 809
• Main Authors: Wang, Yanfen, Yang, Ru, Zhang, Bo, Hu, Wei
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.01.2018; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Buck–boost converter; Chaos; Circuit stability; DC-DC power converters; Dependence; Dynamical systems; Electric power distribution; Electric power generation; Fixed points (mathematics); Invariants; Manifolds; Nonlinear dynamics; Nonlinear systems; Numerical stability; Poincare maps; Power converters; smale horseshoe; Stability criteria; symbolic dynamics; Voltage converters (DC to DC)
• ISSN: 0278-0046; 1557-9948
• DOI: 10.1109/TIE.2017.2703676

DC-DC converter, which is used to change and control voltage magnitude, is a really important part of the power electronic conditioning systems. Study of nonlinear dynamics and chaos in power electronics are becoming increasingly significant in the power quality and stability of distributed power generation systems. In this paper, the mechanisms that give rise to chaotic dynamical in the buck- boost power converter is studied. The Poincare map of a buck-boost converter system possesses the invariant manifolds having a delightfully complicated structure. Based on the Smale-Birkhoff homoclinic theorem, the structure of invariant manifolds of a fixed point indicates the presence of Smale horseshoe. The horseshoe map of the system is topologically conjugate with the shift map of the symbolic dynamics. The system can be identified with the Smale horseshoe by means of its symbolic descriptions. Furthermore, since the Smale horseshoe map implies sensitive dependence on the initial condition, homoclinic intersection of stable and unstable manifolds may be applied to illustrate the existence of chaos.