babynormalforms - Bifurcations: baby normal forms. Rodolfo...

Info iconThis preview shows pages 1–4. 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

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: Bifurcations: baby normal forms. Rodolfo R. Rosales, Department of Mathematics, Massachusetts Inst. of Technology, Cambridge, Massachusetts, MA 02139 October 10, 2004 Abstract The normal forms for the various bifurcations that can occur in a one dimensional dynamical system ( x = f ( x, r )) are derived via local approximations to the governing equation, valid near the critical values where the bifurcation occurs. The derivations are non-rigorous. Contents 1 Introduction. 3 Necessary condition for a bifurcation. . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Saddle Node bifurcations. 4 General remarks on structural stability. . . . . . . . . . . . . . . . . . . . . . . . . . 4 Saddle node bifurcations are structurally stable. . . . . . . . . . . . . . . . . . . . . 4 Structural stability and allowed perturbations. . . . . . . . . . . . . . . . . . . . . . 5 Normal form for a Saddle Node bifurcation. . . . . . . . . . . . . . . . . . . . . . . 6 Remark on the variable scalings near a bifurcation. . . . . . . . . . . . . . . . . . . 6 Theorem: reduction to normal form. . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Problem 1: formal expansion to reduce to normal form. . . . . . . . . . . . . . . . 7 3 Transcritical bifurcations. 8 Normal form for a transcritical bifurcation. . . . . . . . . . . . . . . . . . . . . . . . 8 Theorem: reduction to normal form. . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Structural stability for transcritical bifurcations. . . . . . . . . . . . . . . . . . . . . 11 Problem 2: formal expansion to reduce to normal form. . . . . . . . . . . . . . . . 11 Problem 3: What about problem 3.2.6 in the book by Strogatz? . . . . . . . . . . 11 1 Rosales Bifurcations: baby normal forms. 2 4 Pitchfork bifurcations. 12 Introduction of the reection symmetry. . . . . . . . . . . . . . . . . . . . . . . . . 12 Simplest symmetry: f ( x, r ) is an odd function of x . . . . . . . . . . . . . . . . . . . 12 Problem 4: normal form for a pitchfork bifurcation. . . . . . . . . . . . . . . . . . 13 Problem 5: formal expansion to reduce to normal form. . . . . . . . . . . . . . . . 13 Problem 6: proof of reduction to normal form. . . . . . . . . . . . . . . . . . . . . 13 5 Problem Answers. 14 Rosales Bifurcations: baby normal forms. 3 1 Introduction. Consider the simple one-dimensional dynamical system dx = f ( x, r ) , (1.1) dt where we will assume that f = f ( x, r ) is a smooth function, and r is a parameter. We wish to study the possible bifurcations for this system, as the parameter r varies. Because the phase portrait for a 1-D system is fully determined by its critical (equilibrium) points , we need only study what happens to the critical points. Bifurcations will (only) occur as these points are created, destroyed, collide, or change stability. For higher dimensional systems, the critical points alone do not determine the phase portrait. However, the bifurcations we study here can still occur, and are important.important....
View Full Document

Page1 / 14

babynormalforms - Bifurcations: baby normal forms. Rodolfo...

This preview shows document pages 1 - 4. Sign up to view the full document.

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