Bifurcation Control: Theories, Methods, and Applications

Abstract

Bifurcation control deals with modification of bifurcation characteristics of a parameterized nonlinear system by a designed control input. Typical bifurcation control objectives include delaying the onset of an inherent bifurcation, stabilizing a bifurcated solution or branch, changing the parameter value of an existing bifurcation point, modifying the shape or type of a bifurcation chain, introducing a new bifurcation at a preferable parameter value, monitoring the multiplicity, amplitude, and/or frequency of some limit cycles emerging from bifurcation, optimizing the system performance near a bifurcation point, or a combination of some of these objectives. This article offers an overview of this emerging, challenging, stimulating, and yet promising field of research, putting the main subject of bifurcation control into perspective.

Switched Systems With Multiple Invariant Sets

“…However, most control strategies view such behavior as damaging and try to mitigate it [1]. Relatively less work actively inserts bifurcations as part of a control strategy.…”

A Novel Parametric Controller Design Method for Hopf Bifurcation System

“…In order to calculate the parameters of 2 ( , ) u X a , evaluating the Jacobian of system (9) at the equilibrium yields: …”

“…(16). In order to simplify the process, 2 2 c  is chosen in this case. So the three-dimensional segmentation problem is converted into a two-dimensional space segmentation problem.…”

“…The parameters 0 c and 1 c in each region are examined to identify the set of values that satisfy the criteria. Finally, region Q is found, which is surrounded by 3 2 L is part of the curves of g  (from point B to C ). Point A , B , and C are three specific points on 01 cc  plane, where A is the crossover point of conditions (20) and (23).…”

“…In the Hopf bifurcation control, the basic task is to change the system dynamics in order to achieve the desired properties, such as delaying the onset of an inherent bifurcation, changing the parameter value of an existing bifurcation point, stabilizing a bifurcated branch, modifying the shape or type of a bifurcation chain, optimizing the system performance near a bifurcation point, or a combination of some of these objectives [1][2].…”