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4D Printed Bifurcated Stents with Kirigami-Inspired Structures

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Bifurcation phenomena in two-dimensional piecewise smooth discontinuous maps.

Biswambhar Rakshit¹, Manjul Apratim, Soumitro Banerjee

¹Department of Mathematics and Centre for Theoretical Studies, Indian Institute of Technology, Kharagpur 721302, India.

Chaos (Woodbury, N.Y.)

|October 5, 2010

Summary

This study introduces a new theory for analyzing border collision bifurcations in discontinuous 2D maps, crucial for understanding complex switching dynamical systems and electrical power systems.

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Area of Science:

Dynamical Systems and Control Theory
Nonlinear Dynamics
Electrical Engineering

Background:

Existing theories for border collision bifurcations apply to continuous piecewise smooth maps.
Many real-world switching systems generate discontinuous maps, limiting current bifurcation analysis.
Understanding bifurcations in discontinuous maps is vital for analyzing complex dynamical systems.

Purpose of the Study:

To develop a systematic approach for analyzing bifurcation phenomena in two-dimensional discontinuous maps.
To extend the theory of border collision bifurcations to discontinuous systems.
To provide a theoretical framework applicable to various physical systems with discontinuous maps.

Main Methods:

Developed a piecewise linear approximation near the border for discontinuous maps.
Formulated a new theoretical framework for analyzing bifurcations in these systems.
Applied the theory to the static VAR compensator in electrical power systems.

Main Results:

Successfully analyzed bifurcation phenomena in discontinuous two-dimensional maps.
Integrated observed bifurcation behavior of the static VAR compensator with the new theory.
Demonstrated the applicability of the developed theory to similar systems.

Conclusions:

The proposed theory offers a systematic method for analyzing bifurcations in discontinuous 2D maps.
This framework enhances understanding of nonsmooth phenomena in switching dynamical systems.
The approach is generalizable to other systems exhibiting discontinuous map behavior.