The family gives a has using the has the same

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Unformatted text preview: act, for corresponding and , the vector fields are the same.) ■ Example. Consider the family and the constant map family of vector fields induced by using the map is is clearly simpler than the family . ■ Let be the flow of and be the flow of vector fields and are conjugate for corresponding values of diagram . The . The family where the and . Then we have the 7 Ch8Lecs , where and . Then the flows are related by . [1] If is a diffeomorphism we can also find a relationship between the corresponding vector fields. Differentiate [1] with respect to : Set to obtain the desired relationship. , where and Example. Let [2] . Compare with Eq. 4.34 in Meiss. and . Then , where and [2] gives verified by making the indicated substitutions for and . This can be and . ■ Unfolding Vector Fields Consider a vector field that has a degenerate orbit. This could be a degenerate equilibrium point or, for example, the homoclinic orbit in the homoclinic bifurcation. Then we say fulfills a singularity condition. A family of vector fields is an unfolding of if . Example. The singularity associated with the saddle-node bifurcation is degenerate because it has a zero eigenvalue. Two unfolding of are and .■ . is 8 Ch8Lecs An unfolding of is versal if it contains all possible qualitative dynamics that can occur near to . This means that every other unfolding in some neighborhood of will have the same dynamics as some family induced by . Example. Let and . We will show below that is a versal unfolding of the saddle node bifurcation. However, is not versal. For example, there is no value of for which the system has two equilibrium points. An unfolding is miniversal if it is a versal unfolding with the minimum number of parameters. Example. Let and . The flows of are diffeomorphic. To see this complete the square in : and . The change of variables and converts into . Note that is a diffeomorphism. Since may be any real number, the families of vector fields and have the same dynamics. However, has two parameters and has only one.. Then is a versal unfolding, but not a miniversal unfolding of the saddle node bifurcation. ■ 8.4 Saddle-Node Bifurcation in One Dimension Consider the ODE on Theorem 8.3 Suppose that origin, , . , and that with a nonhyperbolic equilibrium at the satisfies the nondegeneracy condition . Then there is a such that when that there is a unique extremal value , there is an open interval , containing . There are two equilibria in when . Proof. Let , one when and zero when . The singularity and nondegeneracy conditions imply . such 9 Ch8Lecs Since form , the higher order term satisfies . A general unfolding of , where will h...
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This document was uploaded on 02/24/2014 for the course MATH 512 at Washington State University .

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