Bifurcations in D ifurcations of xed point in D have analogs in D and higher

Bifurcations in D ifurcations of xed point in D have analogs in D and higher - Description

ut action is con64257ned to 1D subspace where bifurcations occur while 64258ow in other D is attraction or repulsion from 1D subspace Saddlenode bifurcation is basic mechanism for creation and de struction of 64257xed points rototypical example for ID: 26486 Download Pdf

74K - views

Bifurcations in D ifurcations of xed point in D have analogs in D and higher

ut action is con64257ned to 1D subspace where bifurcations occur while 64258ow in other D is attraction or repulsion from 1D subspace Saddlenode bifurcation is basic mechanism for creation and de struction of 64257xed points rototypical example for

Similar presentations


Download Pdf

Bifurcations in D ifurcations of xed point in D have analogs in D and higher




Download Pdf - The PPT/PDF document "Bifurcations in D ifurcations of xed poi..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.



Presentation on theme: "Bifurcations in D ifurcations of xed point in D have analogs in D and higher"— Presentation transcript:


Page 1
Bifurcations in 2D ifurcations of fixed point in 1D have analogs in 2D (and higher). ut action is confined to 1D subspace where bifurcations occur, while flow in other D is attraction or repulsion from 1D subspace. Saddle-node bifurcation is basic mechanism for creation and de- struction of fixed points. rototypical example for saddle-node bifurcation in 2D:
Page 2
Saddle-Node Bifurcations in 2D D system with parameter x,y, x,y, ntersections of nullclines are fixed points Saddle-node bifurcations: ullclines pull away as varies, becomes

tangent at ixed points approach each other and collide at ullclines pull apart, no intersections. Fixed points disappear.
Page 3
Example of Saddle-Node Bifurcation in 2D nalyse saddle-node bifurcation for genetic control system ax = ( 1 + by o compute , find where fixed points coalesce ( = 1 ). Unstable manifold trapped in narrow channel between nullclines. Stable manifold separate 2 basins of attraction. It acts as biochemical switch : gene turns on or off, depending on intial values of x,y
Page 4
Transcritical and Pitchfork Bifurcations in 2D igure 8.1.5

is similar to 8.1.1. Saddle node bifurcation is 1D event. Fixed points slide toward each other. rototypical examples of transcritical and Pitchfork bifurcations: x y. x y. onsider pitchfork bifurcation: for < 0, stable node at origin. or = 0, origin is still stable with very slow decay along -axis. or > 0, origin loses stability, gives birth to 2 new stable fixed points at ( ,y ) = ( , 0)
Page 5
Example of Pitchfork Bifurcation in 2D nalyse pitchfork bifurcation for x + sin y. ymmetric w.r.t origin (invariant if x,y ). rigin is

fixed point for all . Stable if < 2, saddle if > , + 2)). Pitchfork may occur at 2. ind 2 new fixed points at 6( + 2) for > 2. Supercritical pitchfork occurs at ero-eigenvalue bifurcations: Saddle-node, transcritical, pitchfork.