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Unformatted text preview: Syllabus: Bifurcations CDS-140b, 2009 Joris Vankerschaver [email protected] 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 C-equivalent if there exists a homeomorphism φ : 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 / φ U φ U F Y t / U We will only apply this definition on vector fields defined in the plane. Intuitively, the idea is that φ 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 1-norm 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 1-norm 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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This note was uploaded on 01/04/2012 for the course CDS 140b taught by Professor List during the Fall '10 term at Caltech.
- Fall '10