Nonlinear Systems and Phase-Plane Models

Definition An autonomous system in the plane is

where $f$ and $g$ do not depend on $t$ and have continuous first partial derivatives on an open set $U$. In vector form $\mathbf{x}' = \mathbf{F}(\mathbf{x})$ with $\mathbf{F} = \langle f, g \rangle$. Under these hypotheses, through each point of $U$ and each starting time there passes exactly one solution, defined on some open time interval. If $\mathbf{x}(t)$ is a solution, so is $\mathbf{x}(t - c)$ for every constant $c$.

Definition A point $(x^*, y^*)$ in $U$ is an equilibrium if $f(x^*, y^*) = 0$ and $g(x^*, y^*) = 0$. Then the constant function $\mathbf{x}(t) = (x^*, y^*)$ is a solution. The curves $f = 0$ and $g = 0$ are the nullclines, and the equilibria are exactly the points where an $x$ nullcline meets a $y$ nullcline.

Definition Let $\mathbf{F}$ have continuous first partial derivatives near an equilibrium $\mathbf{x}^*$. Its Jacobian matrix there is

Writing $\mathbf{u} = \mathbf{x} - \mathbf{x}^*$, differentiability of $\mathbf{F}$ gives $\mathbf{u}' = J\mathbf{u} + \mathbf{r}(\mathbf{u})$ with $\lVert \mathbf{r}(\mathbf{u}) \rVert / \lVert \mathbf{u} \rVert \to 0$ as $\mathbf{u} \to \mathbf{0}$. The linear system $\mathbf{u}' = J\mathbf{u}$ is the linearization at $\mathbf{x}^*$.

Definition An equilibrium $\mathbf{x}^*$ is hyperbolic if no eigenvalue of $J$ has zero real part. Linearization theorem (Hartman-Grobman). If $\mathbf{F}$ has continuous first partial derivatives on an open set containing a hyperbolic equilibrium $\mathbf{x}^*$, then there is a homeomorphism, a continuous map with continuous inverse, from a neighborhood of $\mathbf{x}^*$ onto a neighborhood of $\mathbf{0}$ that carries trajectories of the nonlinear system onto trajectories of the linearization and preserves the direction of time. In particular

• if every eigenvalue has negative real part, $\mathbf{x}^*$ is asymptotically stable (a sink),
• if every eigenvalue has positive real part, $\mathbf{x}^*$ is a source, and solutions starting near it move away,
• if the eigenvalues are real with opposite signs, $\mathbf{x}^*$ is a saddle, which is unstable.

A homeomorphism can bend straight approaches into spirals, so this theorem alone does not separate a node from a spiral. A finer result in the plane does. Suppose $\mathbf{F}$ has continuous second partial derivatives near $\mathbf{x}^*$. If the eigenvalues of $J$ are real, distinct and of the same sign, then $\mathbf{x}^*$ is a node, meaning every trajectory that tends to it, as $t \to \infty$ for a sink or as $t \to -\infty$ for a source, does so tangent to a line through it. If they are $\alpha \pm i\beta$ with $\alpha \ne 0$ and $\beta \ne 0$, then $\mathbf{x}^*$ is a spiral, meaning nearby trajectories wind around it infinitely often. A repeated real eigenvalue still gives a sink or a source, but if $\mathbf{F}$ is only $C^1$ the trajectories near it may spiral, so the linearization does not settle the finer type in that case.

Definition If some eigenvalue of $J$ has zero real part, the equilibrium is nonhyperbolic, and the linearization alone does not determine stability or type. In particular a linear center may be a nonlinear center, a stable spiral or an unstable spiral. Other tools are then needed, such as a conserved quantity, a polar or radial equation, or a Lyapunov function.

Definition A trajectory, or orbit, is the set of points $\{\mathbf{x}(t)\}$ traced by one solution over its interval of existence, drawn with an arrow for increasing $t$. Under the hypotheses above, two distinct trajectories never meet, and each trajectory is an equilibrium point, a closed curve (a periodic solution), or a curve that never crosses itself. Where $f \ne 0$, trajectories satisfy

Definition The Lotka-Volterra model for prey $x \ge 0$ and predators $y \ge 0$ is

with constants $a, b, c, d > 0$. Its equilibria are $(0, 0)$, a saddle, and $\bigl(\frac{c}{d}, \frac{a}{b}\bigr)$, where the linearization has eigenvalues $\pm i\sqrt{ac}$, a nonhyperbolic center. In the open quadrant $x > 0$, $y > 0$ the function

is constant along every trajectory and has a strict minimum at the interior equilibrium. Hence every trajectory in the open quadrant other than that equilibrium is a closed curve, and the populations are periodic.

Definition Linearization at an equilibrium describes trajectories only in some neighborhood of it, whose size the theorem does not give. Global questions, such as which equilibrium a given starting point approaches, whether solutions exist for all $t \ge 0$, or whether closed orbits exist far from equilibria, need the whole phase portrait. The basin of attraction of an asymptotically stable equilibrium is the set of initial points whose solutions tend to it.