Analysis of Bifurcations from a Node on a Graph

Understanding the Setup

The context involves bifurcations in dynamical systems where the state space is a graph (network). This typically means differential equations on a graph structure, where each node hosts a dynamical system and edges represent coupling between them.

The phrase "bifurcation from a node on a graph to another point" suggests:

A symmetric configuration where all nodes initially reside at the same equilibrium (synchronized state). As a parameter varies, this synchronized state loses stability, and a new pattern emerges where some nodes transition to one state while others remain or move to another—this is a symmetry-breaking bifurcation. The description "from a node to another point" indicates that one node's state diverges from others onto a different equilibrium branch.

Differentiability of Bifurcating Branches

In classical bifurcation theory (e.g., in \(\mathbb{R}^n\)), when a steady state undergoes a steady-state bifurcation (saddle-node, transcritical, pitchfork), the new branch of equilibria is governed by the implicit function theorem everywhere except at the bifurcation point itself.

At a simple eigenvalue crossing through zero, the new branch is typically smooth (\(C^k\) if the vector field is \(C^k\)) away from the bifurcation point. At the bifurcation point, the branch may not be differentiable if it's a saddle-node, or it may be differentiable but with a vertical tangent in parameter-state space, making the derivative with respect to the parameter infinite.

Example: Pitchfork Bifurcation in All-to-All Coupled System

Consider two nodes with symmetric coupling:

\(\dot{x}_1 = f(x_1, \alpha) + d (x_2 - x_1)\)
\(\dot{x}_2 = f(x_2, \alpha) + d (x_1 - x_2)\)

The synchronized manifold is \(x_1 = x_2 = x^*(\alpha)\), where \(f(x^*(\alpha), \alpha) = 0\). Linear stability analysis reveals eigenvalues in the synchrony direction and anti-synchrony direction. When the eigenvalue in the anti-synchrony direction crosses zero, a bifurcation occurs, creating a branch where \(x_1 \neq x_2\).

This branch, \(x_1(\alpha), x_2(\alpha)\), is typically a smooth function of \(\alpha\) near the bifurcation point. At the bifurcation point itself, the branch might be differentiable (as in a supercritical pitchfork), though the derivative \(\frac{dx}{d\alpha}\) may be infinite when plotting state versus parameter.

General Graph Case

For a graph with \(N\) nodes and diffusive coupling:

\(\dot{u}_i = f(u_i, \alpha) + d \sum_{j} L_{ij} u_j\)

The synchronized state \(u_i = u^*(\alpha)\) undergoes linear stability analysis where eigenvalues take the form \(f_x(u^*,\alpha) + d \mu_k\), with \(\mu_k\) being Laplacian eigenvalues. Bifurcation occurs when \(f_x + d \mu_k = 0\) for some \(k \neq 0\).

The bifurcating branch corresponds to patterns shaped by the eigenvector associated with \(\mu_k\). Through equivariant bifurcation theory, the reduced equation on the center manifold is smooth, ensuring the branch is smooth in the parameter, except at degenerate bifurcation points.

Interpreting "Differentiatable"

If "differentiatable" refers to whether the function \(\alpha \mapsto \text{(state of a node on the bifurcating branch)}\) is differentiable, then yes, typically it is differentiable. Even at the bifurcation point for pitchfork or transcritical bifurcations, the branch is smooth, though the derivative \(\frac{dx}{d\alpha}\) might be infinite. For saddle-node bifurcations, each branch is smooth but they do not extend through the bifurcation point.

Special Case: Discontinuous Bifurcations

If the new state "jumps" to a distant attractor on the graph (a non-local bifurcation), differentiability may fail. However, for local symmetry-breaking induced by a zero eigenvalue, the new branch remains smooth.

Final Answer

In the standard setting of smooth dynamical systems on graphs, bifurcating branches arising from a local steady-state bifurcation are smooth (hence differentiable) with respect to parameters, except possibly exactly at the bifurcation point where the derivative of the state variable with respect to the parameter may be infinite. The bifurcation from a node to another point in state space is typically differentiable along the branch.