cds140b-bif

cds140b-bif - Syllabus: Bifurcations CDS-140b, 2009 Joris...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Syllabus: Bifurcations CDS-140b, 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 C-equivalent 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 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....
View Full Document

Page1 / 6

cds140b-bif - Syllabus: Bifurcations CDS-140b, 2009 Joris...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online