1:2 Resonance Bifurcation in the E-I Map

1:2 Resonance Bifurcation in the E-I Map

Bifurcation and Continuation

The movie comprises a sequence of phase portraits of the E-I map in the vicinity of the codimension 2 point of a 1:2 resonance bifurcation occurring on a period 2 orbit as a parameter is varied.


1D-spiral

1D-spiral

Patterns and Simulations


3D Manifolds

3D Manifolds

Invariant Manifolds

Heteroclinic tangle for a three dimensional volume preserving mapping.


Bifurcations without parameters

Bifurcations without parameters

Bifurcation and Simulations

 The following pictures illustrate some elements of the nonlinear dynamics near equilibrium manifolds.


Bifurcations without parameters

Bifurcations without parameters

Bifurcation and Simulations


Breathing torus near double Hopf bifurcation

Breathing torus near double Hopf bifurcation

Bifurcation and Continuation


Bursting

Bursting

Patterns and Simulations


Canard Explosion

Canard Explosion

Bifurcation and Simulations

Periodic orbit continuation in the Hindmarsh-Rose model: canard explosion.


Canard explosion

Canard explosion

Bifurcation and Simulations

see more at http://www.mathstat.gsu.edu/~matals/research/neuron/fish/index.htm

Canard explosion in the FitzHugh-Nagumo model

Canard explosion in the FitzHugh-Nagumo model

Bifurcation and Continuation

This animation illustrates the phenomenon of canard explosion in the FitzHugh-Nagumo model. 


Crack patterns

Crack patterns

Real Experiments

Mud cracks when it shrinks and become brittle. If you make a uniform layer of mud and dry it uniformly, you get a pattern of irregular polygons with edges that usually meet at 90 degrees.

Evolution of the EPDiff Equation

Evolution of the EPDiff Equation

Patterns and Simulations


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