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Give the definition of a Lyapunov function. Consider the differential equation d dt x(t) = ax+y=x√√x² + y² d dt y(t) = =ay-x-y√√√x²+ y², First, multiply yx and xy to show that the origin is the only equilibrium. Then, determine the local stability of the origin as a function of the model parameter a. Next, determine which values of a the function V(x,y) = x² + y² is a Lyapunov function. Finally, for a > 0, write the differential equation in polar co-ordinates and show that there is a stable limit cycle.

Give the definition of a Lyapunov function. Consider the differential equation d dt x(t) = ax+y=x√√x² + y² d dt y(t) = =ay-x-y√√√x²+ y², First, multiply yx and xy to show that the origin is the only equilibrium. Then, determine the local stability of the origin as a function of the model parameter a. Next, determine which values of a the function V(x,y) = x² + y² is a Lyapunov function. Finally, for a > 0, write the differential equation in polar co-ordinates and show that there is a stable limit cycle.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Give the definition of a Lyapunov function. Consider the differential equation d dt x(t) = ax+y=x√√x² + y² d dt y(t) = =ay-x-y√√√x²+ y², First, multiply yx and xy to show that the origin is the only equilibrium. Then, determine the local stability of the origin as a function of the model parameter a. Next, determine which values of a the function V(x,y) = x² + y² is a Lyapunov function. Finally, for a > 0, write the differential equation in polar co-ordinates and show that there is a stable limit cycle.
Transcribed Image Text:Give the definition of a Lyapunov function. Consider the differential equation d dt x(t) = ax+y=x√√x² + y² d dt y(t) = =ay-x-y√√√x²+ y², First, multiply yx and xy to show that the origin is the only equilibrium. Then, determine the local stability of the origin as a function of the model parameter a. Next, determine which values of a the function V(x,y) = x² + y² is a Lyapunov function. Finally, for a > 0, write the differential equation in polar co-ordinates and show that there is a stable limit cycle.
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