Exploring the geometry of the bifurcation sets in parameter space.

Area of Science:

• Nonlinear dynamics
• Mathematical biology
• Dynamical systems theory

Background:

• Nonlinear models are crucial for understanding complex phenomena.
• Bifurcation sets in parameter spaces define critical transitions.
• Geometric properties of these sets can reveal underlying dynamics.

Purpose of the Study:

• To introduce and define "geometric bifurcations" as changes determined by the geometry of bifurcation sets.
• To demonstrate the utility of dimensional cuts for detecting these bifurcations.
• To illustrate geometric bifurcations in established models of neuron activity.

Main Methods:

• Studying nonlinear models by analyzing p-dimensional parameter spaces.
• Utilizing q-dimensional cuts to inspect the parameter space.
• Applying the theory of singularities for differentiable mappings and Morse Theory.
• Examining fast-slow systems like the Hindmarsh-Rose and FitzHugh-Nagumo models.

Main Results:

• Geometric bifurcations are detectable through specific dimensional cuts of parameter spaces.
• These bifurcations are independent of the specific nonlinear model, depending only on geometric properties.
• The study successfully illustrates geometric bifurcations in the Hindmarsh-Rose and FitzHugh-Nagumo neuron models.

Conclusions:

• Geometric bifurcations provide a novel perspective on understanding transitions in nonlinear systems.
• The method of dimensional cuts is effective for identifying these geometry-dependent changes.
• This framework enhances the analysis of bifurcation diagrams in fields like computational neuroscience.