Two-Dimensional Flows - Applications

TheoremHartman-Grobman Theorem for Planar Flows

Let $\mathbf{x}^*$ be a hyperbolic fixed point of $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x})$ (i.e., all eigenvalues of the Jacobian $J$ at $\mathbf{x}^*$ have nonzero real parts). Then there exists a neighborhood $U$ of $\mathbf{x}^*$ and a homeomorphism $h: U \to h(U)$ that maps trajectories of the nonlinear system to trajectories of the linearization $\dot{\mathbf{y}} = J\mathbf{y}$ .

Specifically, if $\phi_t$ is the flow of the nonlinear system and $\psi_t$ is the flow of the linearization, then:

$h(\phi_t(\mathbf{x})) = \psi_t(h(\mathbf{x}))$

for all $\mathbf{x} \in U$ and $t$ such that $\phi_t(\mathbf{x}) \in U$ .

The Hartman-Grobman theorem justifies the linear stability analysis performed at fixed points. Near hyperbolic equilibria, the nonlinear flow is topologically equivalent to its linear approximation. This means the phase portrait geometry (number of stable/unstable directions, whether trajectories spiral or approach directly) is determined entirely by the eigenvalues of the Jacobian.

The theorem fails at non-hyperbolic fixed points where eigenvalues have zero real part. Centers are the most important case: linearization gives closed orbits, but the nonlinear system may have a spiral (weak focus) or remain a center, depending on higher-order terms.

TheoremStable Manifold Theorem for Saddles

Let $\mathbf{x}^*$ be a saddle point with eigenvalues $\lambda_s < 0 < \lambda_u$ . Then:

There exist local stable and unstable manifolds $W^s_{\text{loc}}(\mathbf{x}^*)$ and $W^u_{\text{loc}}(\mathbf{x}^*)$ that are smooth curves through $\mathbf{x}^*$ , tangent to the corresponding eigenspaces
These extend to global manifolds: $W^s(\mathbf{x}^*) = \bigcup_{t \leq 0} \phi_t(W^s_{\text{loc}}), \quad W^u(\mathbf{x}^*) = \bigcup_{t \geq 0} \phi_t(W^u_{\text{loc}})$
Trajectories approach $\mathbf{x}^*$ exponentially along $W^s$ as $t \to \infty$ and along $W^u$ as $t \to -\infty$

The global manifolds $W^s$ and $W^u$ can intersect themselves or other manifolds, creating complex phase space geometry.

The stable manifold theorem guarantees the existence and smoothness of invariant manifolds. In planar systems, these are one-dimensional curves that partition phase space into regions. When $W^u$ from one saddle connects to $W^s$ of another saddle (or the same one), we have heteroclinic (or homoclinic) connections that organize global dynamics.

ExampleHeteroclinic Cycles in Competition Models

In the competing species model $\dot{x} = x(1-x-ay)$ , $\dot{y} = y(1-y-bx)$ with $a, b > 1$ (mutual exclusion regime), the phase portrait includes:

Saddle at origin $(0,0)$
Stable nodes at $(1,0)$ and $(0,1)$
Unstable node at interior coexistence point

The unstable manifolds from the origin form the axes $x > 0, y = 0$ and $x = 0, y > 0$ , which are also stable manifolds of $(1,0)$ and $(0,1)$ respectively. This creates a heteroclinic connection between equilibria. Initial conditions determine which species wins: the basin boundary is the stable manifold of the interior saddle point.

Remark

The Hartman-Grobman and stable manifold theorems extend to higher dimensions but are especially transparent in two dimensions where manifolds are curves that can be visualized easily. These results form the foundation for understanding more complex phenomena like horseshoe maps and chaotic invariant sets in higher-dimensional systems, where invariant manifolds become surfaces or higher-dimensional objects that fold and intersect in intricate ways.

ExampleGlycolytic Oscillations

The Sel'kov model for glycolytic oscillations is:

$\dot{x} = -x + ay + x^2y, \quad \dot{y} = b - ay - x^2y$

where $x$ is ADP concentration and $y$ is F6P concentration. For appropriate parameters $a, b$ , the system undergoes a Hopf bifurcation, creating a stable limit cycle representing sustained oscillations in metabolite concentrations. The Hartman-Grobman theorem determines stability near the fixed point, while the Hopf bifurcation theorem (an extension of local analysis) predicts the emergence of oscillations when eigenvalues cross the imaginary axis.

These theorems demonstrate that local analysis near fixed points, combined with global topological constraints from Poincare-Bendixson, provides a complete framework for understanding planar dynamics. Applications range from population biology to chemical kinetics, wherever two interacting quantities evolve over time according to smooth nonlinear laws.