Lecture 5: 2-d models and phase-plane analysis. references: Gerstner & Kistler, Ch 3 Koch, Ch 7. Reduced models. In HH, m is much faster than n, h. Reduced models. In HH, m is much faster than n, h. Reduced models. In HH, m is much faster than n, h. Try 2-d system:.

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Lecture 5: 2-d models and phase-plane analysis

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linearization set Find eigenvectors v and eigenvalues l

linearization set Find eigenvectors v and eigenvalues l Stability if l1, l2 both < 0

linearization set Find eigenvectors v and eigenvalues l Stability if l1, l2 both < 0 i.e., l1+ l2 = trM< 0(Fu + Gw < 0) and l1l2 = det M > 0 (FuGw – FwGu > 0)

linearization set Find eigenvectors v and eigenvalues l Stability if l1, l2 both < 0 i.e., l1+ l2 = trM< 0(Fu + Gw < 0) and l1l2 = det M > 0 (FuGw – FwGu > 0) If det M < 0, both eigenvalues are real, one >0, the other <0:

linearization set Find eigenvectors v and eigenvalues l Stability if l1, l2 both < 0 i.e., l1+ l2 = trM< 0(Fu + Gw < 0) and l1l2 = det M > 0 (FuGw – FwGu > 0) If det M < 0, both eigenvalues are real, one >0, the other <0: Saddle point

Poincare-Bendixson theorem If • You have a repulsive fixed point • (both eigenvalues have positive real parts)

Poincare-Bendixson theorem If • You have a repulsive fixed point • (both eigenvalues have positive real parts) (2) There is a “bounding box” with the property that all flow across it is inward

Poincare-Bendixson theorem If • You have a repulsive fixed point • (both eigenvalues have positive real parts) (2) There is a “bounding box” with the property that all flow across it is inward

Poincare-Bendixson theorem If • You have a repulsive fixed point • (both eigenvalues have positive real parts) (2) There is a “bounding box” with the property that all flow across it is inward Then There must be a limit cycle in between