Lect_6 - Nonlinear Systems and Control Lecture # 6...

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Unformatted text preview: Nonlinear Systems and Control Lecture # 6 Bifurcation p.1/19 Bifurcation is a change in the equilibrium points or periodic orbits, or in their stability properties, as a parameter is varied Example x 1 = - x 2 1 x 2 =- x 2 Find the equilibrium points and their types for different values of For > there are two equilibrium points at ( , 0) and (- , 0) p.2/19 Linearization at ( , 0) : bracketleftBigg- 2 - 1 bracketrightBigg ( , 0) is a stable node Linearization at (- , 0) : bracketleftBigg 2 - 1 bracketrightBigg (- , 0) is a saddle p.3/19 x 1 = - x 2 1 , x 2 =- x 2 No equilibrium points when < As decreases, the saddle and node approach each other, collide at = 0 , and disappear for < x 1 x 2 x 2 x 1 x 1 x 2 > = 0 < p.4/19 is called the bifurcation parameter and = 0 is the bifurcation point Bifurcation Diagram (a) Saddle-node bifurcation http://www.enm.bris.ac.uk/staff/berndk/chaosweb/saddle.html p.5/19 Example x 1 = x 1- x 2 1 , x 2 =- x 2 Two equilibrium points at (0 , 0) and ( , 0) The Jacobian at (0 , 0) is bracketleftBigg - 1 bracketrightBigg (0 , 0) is a stable node for < and a saddle for > The Jacobian at ( , 0) is bracketleftBigg- - 1 bracketrightBigg...
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Lect_6 - Nonlinear Systems and Control Lecture # 6...

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