Introduction to Dynamic Bifurcation.
BROWN UNIV PROVIDENCE RI LEFSCHETZ CENTER FOR DYNAMICAL SYSTEMS
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Dynamic bifurcation theory in differential equations is concerned with the changes that occur in the structure of the limit sets of solutions as parameters in the vector field are varied. For example, if the vector field is the gradient of a function with a finite number of critical points, then the omega-limit set of each orbit is an equilibrium point. Thus, one must be concerned with how the number of equilibrium points changes with the parameters this is usually called static bifurcation theory, how the stability properties of the equilibrium points change and the manner in which the equilibrium points are connected to each other by orbits. If the vector field is not the gradient of a function, then other types of limiting motions can occur for example, periodic orbits, invariant tori, homoclinic and heteroclinic orbits. The purpose of these notes is to give an introduction to the methods used in determining how these more complicated limit sets change as parameters vary. Author
- Theoretical Mathematics