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Unformatted text preview: Syllabus: Bifurcations CDS140b, 2009 Joris Vankerschaver jv@caltech.edu 1 Structural Stability Recall that a homeomorphism is a continuous bijection with a continuous inverse. Definition 1.1. Let U be a domain in R n . Consider vector fields X,Y X 1 ( U ) with flows F X ,F Y : I U U , where I is an open interval in R such that I . The dynamics of X and Y is said to be Cequivalent if there exists a homeomorphism : U U with the following property: for all t I there exists a t I such that the following diagram commutes: U F X t / U U F Y t / U We will only apply this definition on vector fields defined in the plane. Intuitively, the idea is that maps the phase plane of X into the phase plane of Y , possibly distorting the trajectories (in a continuous way). Definition 1.2. The C 1norm of a vector field X X 1 ( U ) is defined as k X k 1 = sup x U k X ( x ) k + sup x U k DX ( x ) k . If U is compact, then k X k 1 < + . The C 1norm provides us with a notion of nearness in the space of vector fields, making it into a Banach space. Using this notion, we say that a vector field is structurally stable if all nearby vector fields are topologically equivalent to the original vector field. Definition 1.3. A vector field X X 1 ( U ) is said to be structurally stable if there exists an > such that the following property holds: for all Y X 1 ( U ) and K U a compact set, if k X Y k 1 < on K , then X and Y are topologically equivalent on K . This definition is different from the one used in Perko [ 2001 ] but it agrees with Holmes et al. [ 1996 ]. However, Perko [ 2001 ] uses it implicitly, see for instance section 4.1, example 1 and 2 (p. 319). Example 1. The harmonic oscillator x = y , y = x is structurally unstable. Indeed, it is a linear system with eigenvalues on the imaginary axis, so arbitrarily small linear perturbations will turn the origin into a source or a sink. 1 1 Structural Stability 2 Example 2. Consider the following perturbation of the harmonic oscillator: x = y y = x + x 2 . The origin is still a center but the nonlinear perturbation term introduces a new, hyperbolic fixed point at (1 / , 0) and a homoclinic orbit encircling the origin. This is another example showing that the harmonic oscillator is structurally unstable. However, for this to be a small perturbation of the harmonic oscillator, we need to check that it is close to the harmonic oscillator on compact subsets of R 2 . Indeed, note that the term x 2 is unbounded on R 2 , no matter how small . Generic Properties. A property is typically said to be generic if almost all systems have that property. It remains to define almost all: in measure theory, this can be done by saying that the complement of the generic set has measure zero. In topology and dynamical system, a slightly different definition is used. We stick with Perko [ 2001 ] and define a set to be generic if it contains an open and dense subset.if it contains an open and dense subset....
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