Flows on the Plane
Two-dimensional systems — phase portraits, limit cycles, and bifurcations
A Brief History
Two-Dimensional Systems and Phase Portraits
When two quantities interact — prey and predator, two competing species, cooperators and defectors — the state of the system is a point $(x, y)$ in the phase plane. A two-dimensional autonomous ODE system takes the form:
$$\dot{x} = f(x, y), \qquad \dot{y} = g(x, y)$$At each point $(x, y)$, the functions $f$ and $g$ assign a velocity vector $(\dot{x}, \dot{y})$. The collection of all such vectors is the vector field. A trajectory is a curve in the phase plane that follows the vector field: starting from an initial condition $(x_0, y_0)$, the system traces a path whose tangent at every point equals the local velocity.
A phase portrait is a picture of representative trajectories, revealing the global dynamics at a glance: where do trajectories converge? spiral? separate? The tools developed in this lecture — linearisation, nullclines, and topological constraints — let us sketch phase portraits without solving the equations.
Linear Systems and the Trace–Determinant Classification
Near a fixed point $(x^*, y^*)$ where $f(x^*, y^*) = g(x^*, y^*) = 0$, the dynamics are governed by the Jacobian matrix:
$$J = \begin{pmatrix} \partial f/\partial x & \partial f/\partial y \\ \partial g/\partial x & \partial g/\partial y \end{pmatrix}\bigg|_{(x^*, y^*)}$$The linearised system is $\dot{\mathbf{u}} = J\,\mathbf{u}$ where $\mathbf{u} = (x - x^*, y - y^*)$. The eigenvalues of $J$ determine the local behaviour. For a $2 \times 2$ matrix, they depend on only two quantities:
$$\tau = \text{tr}(J) = a + d, \qquad \Delta = \det(J) = ad - bc$$
The eigenvalues are $\lambda_{1,2} = \frac{\tau \pm \sqrt{\tau^2 - 4\Delta}}{2}$.
The entire classification of a 2D fixed point reduces to the $(\tau, \Delta)$ plane:
- $\Delta < 0$: Saddle. One eigenvalue positive, one negative. Trajectories approach along one direction and diverge along the other.
- $\Delta > 0$, $\tau < 0$: Stable. Both eigenvalues have negative real part. A node if $\tau^2 > 4\Delta$ (real eigenvalues); a spiral if $\tau^2 < 4\Delta$ (complex eigenvalues).
- $\Delta > 0$, $\tau > 0$: Unstable. Same distinction between node and spiral, but trajectories diverge.
- $\Delta > 0$, $\tau = 0$: Centre (in the linear system). Purely imaginary eigenvalues produce closed orbits. In nonlinear systems, this case is delicate: perturbations can turn a centre into a spiral.
- $\Delta = 0$: Degenerate. At least one zero eigenvalue; a line of fixed points exists.
The parabola $\tau^2 = 4\Delta$ separates nodes from spirals. The $\tau$-axis ($\Delta = 0$) separates saddles from nodes/spirals. The $\Delta$-axis ($\tau = 0$) separates stable from unstable. These three curves partition the $(\tau, \Delta)$ plane into distinct dynamical regimes.
Nullclines and Nonlinear Equilibria
For a nonlinear system, we cannot diagonalise a matrix globally. Instead, we use nullclines: curves where one component of the velocity vanishes.
- $x$-nullcline: the set where $f(x, y) = 0$. On this curve, $\dot{x} = 0$, so the flow is purely vertical.
- $y$-nullcline: the set where $g(x, y) = 0$. Here $\dot{y} = 0$ and the flow is purely horizontal.
Fixed points lie at intersections of nullclines — where both components of velocity vanish simultaneously. Between nullclines, the signs of $f$ and $g$ are constant, so the flow direction can be determined by testing a single point in each region. This gives a quick sketch of the global phase portrait.
Once equilibria are located, we classify each by computing the Jacobian $J$ at that point and applying the trace–determinant classification. Linearisation is valid near the fixed point provided neither eigenvalue has zero real part (the Hartman–Grobman theorem).
Conservative Systems and Centres
A system is conservative if there exists a function $H(x, y)$ — a Hamiltonian or first integral — that is constant along trajectories: $\dot{H} = 0$. Trajectories are then confined to level curves of $H$.
The classic example is the Lotka–Volterra predator–prey system (Interacting Species):
$$\dot{N} = rN - aNP, \qquad \dot{P} = eaNP - dP$$This system admits the conserved quantity $H(N, P) = eaN - d\ln N + aP - r\ln P$. Because $H$ is constant, trajectories form closed loops around the coexistence equilibrium — a centre. The oscillations neither grow nor decay.
Centres are structurally unstable: the slightest perturbation — adding logistic self-limitation to the prey, for instance — breaks the conserved quantity and turns the centre into a stable spiral. This is why the Lotka–Volterra model, despite its elegance, is biologically fragile.
Limit Cycles and the Poincaré–Bendixson Theorem
A limit cycle is an isolated closed trajectory: nearby orbits spiral toward it (stable limit cycle) or away from it (unstable limit cycle). Unlike the closed orbits of a centre, a limit cycle is structurally stable — it persists under small perturbations.
If a trajectory of a planar system is confined to a bounded region containing no fixed points, then the trajectory must approach a limit cycle.
This theorem has a profound corollary: chaos is impossible in two-dimensional continuous systems. The long-term behaviour of a planar ODE is restricted to fixed points, limit cycles, or connections between them. Chaos requires at least three dimensions.
To rule out closed orbits, we can use Bendixson's negative criterion: if $\nabla \cdot \mathbf{F} = \partial f/\partial x + \partial g/\partial y$ has one sign throughout a simply connected region, there are no closed orbits in that region. The divergence measures whether the flow expands or contracts area; a closed orbit would require both.
Hopf Bifurcation
The Hopf bifurcation is the birth of a limit cycle from a fixed point. As a parameter $\mu$ varies, a pair of complex conjugate eigenvalues crosses the imaginary axis: their real part changes sign from negative (stable spiral) to positive (unstable spiral), and a limit cycle appears to absorb the trajectories.
The normal form in Cartesian coordinates is:
$$\dot{x} = \mu x - y - x(x^2 + y^2), \qquad \dot{y} = x + \mu y - y(x^2 + y^2)$$In polar coordinates $(r, \theta)$ this simplifies to $\dot{r} = r(\mu - r^2)$ and $\dot{\theta} = 1$. For $\mu \leq 0$: the origin is a stable spiral ($r \to 0$). For $\mu > 0$: the origin becomes unstable, and a stable limit cycle appears at $r = \sqrt{\mu}$. The amplitude grows as $\sqrt{\mu}$ — a supercritical Hopf bifurcation.
Biological example: the paradox of enrichment (Interacting Species). In the Rosenzweig–MacArthur predator–prey model, increasing the prey's carrying capacity $K$ past a critical value destabilises the coexistence equilibrium via a Hopf bifurcation. The stable spiral becomes an unstable spiral surrounded by a stable limit cycle: the system oscillates with growing amplitude, increasing extinction risk.
In a subcritical Hopf bifurcation, an unstable limit cycle exists before the bifurcation and shrinks onto the fixed point as $\mu$ increases, causing a sudden jump to large-amplitude oscillations or another attractor. This mechanism underlies hysteresis in neural and ecological systems.
Explore: Phase Portraits and the Trace–Determinant Plane
Select a system type. The left panel shows the phase portrait with trajectories from multiple initial conditions. The right panel shows the trace–determinant plane with classification regions; a dot marks the current system.
Exercises
Conceptual Questions
- A fixed point has Jacobian with $\tau = -1$ and $\Delta = 3$. Classify the equilibrium and compute the eigenvalues. Will trajectories oscillate as they approach?
- Why can a two-dimensional continuous ODE system not exhibit chaos? State the Poincaré–Bendixson theorem and explain which behaviours are possible.
- In the Lotka–Volterra predator–prey model, the coexistence equilibrium is a centre. Why is this not structurally stable? What qualitative change occurs when prey self-limitation ($N/K$ term) is added?
- Apply Bendixson's negative criterion to the Lotka–Volterra competition model $\dot{N}_1 = r_1 N_1(1 - (N_1 + \alpha_{12}N_2)/K_1)$, $\dot{N}_2 = r_2 N_2(1 - (N_2 + \alpha_{21}N_1)/K_2)$. Compute the divergence $\partial f/\partial N_1 + \partial g/\partial N_2$ in the positive quadrant and determine whether limit cycles are possible.
- Explain the Hopf bifurcation in ecological terms. What happens to a predator–prey system when the environment becomes more productive (increasing carrying capacity $K$)?
Computer Problems
- Linear Phase Portraits. Implement the 2D linear system $\dot{\mathbf{x}} = A\mathbf{x}$ for the five cases: stable node ($A = \begin{pmatrix} -2 & 0 \\ 0 & -1 \end{pmatrix}$), saddle ($A = \begin{pmatrix} 1 & 0 \\ 0 & -2 \end{pmatrix}$), stable spiral ($A = \begin{pmatrix} -0.5 & 2 \\ -2 & -0.5 \end{pmatrix}$), unstable spiral ($A = \begin{pmatrix} 0.5 & 2 \\ -2 & 0.5 \end{pmatrix}$), and centre ($A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$). For each, integrate trajectories from 8 initial conditions and plot the phase portrait. Compute $\tau$ and $\Delta$ and verify the classification.
- Nullcline Analysis. For the Lotka–Volterra competition model with $r_1 = 1$, $r_2 = 0.8$, $K_1 = 100$, $K_2 = 80$, $\alpha_{12} = 0.6$, $\alpha_{21} = 0.8$, plot the nullclines in the $(N_1, N_2)$ plane. Find all equilibria, classify each using the Jacobian, and overlay trajectories from five initial conditions to confirm the global dynamics.
- Hopf Bifurcation. Implement the normal form $\dot{x} = \mu x - y - x(x^2 + y^2)$, $\dot{y} = x + \mu y - y(x^2 + y^2)$. Vary $\mu$ from $-0.5$ to $0.5$ in steps of $0.1$. For each $\mu$, plot the phase portrait (trajectories from $r = 0.1$ and $r = 2.0$). Measure the limit-cycle radius for $\mu > 0$ and verify it equals $\sqrt{\mu}$.
- Paradox of Enrichment. Using the Rosenzweig–MacArthur model ($r = 1$, $a = 0.5$, $e = 0.3$, $d = 0.3$, $h = 0.5$), vary $K$ from 5 to 25. For each $K$, compute the Jacobian at the coexistence equilibrium, find the eigenvalues, and determine whether the equilibrium is a stable spiral or unstable spiral. Identify the critical $K$ at which the Hopf bifurcation occurs. Plot representative phase portraits before and after the bifurcation.
- Trace–Determinant Plane. Write a program that takes any $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ as input, computes $\tau$ and $\Delta$, classifies the equilibrium, plots the phase portrait, and marks the point $(\tau, \Delta)$ on the classification diagram. Test with at least six matrices spanning all regions of the $(\tau, \Delta)$ plane.
References
- Bendixson, I. (1901). Sur les courbes définies par des équations différentielles. Acta Mathematica, 24, 1–88.
- Hopf, E. (1942). Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Ber. Math.-Phys. Kl. Sächs. Akad. Wiss. Leipzig, 94, 1–22.
- Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20, 130–141.
- Lotka, A. J. (1925). Elements of Physical Biology. Williams & Wilkins.
- Rosenzweig, M. L. (1971). Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science, 171, 385–387.
- Strogatz, S. H. (2015). Nonlinear Dynamics and Chaos (2nd ed.). Westview Press.
- Turing, A. M. (1952). The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London B, 237, 37–72.
- Volterra, V. (1926). Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Mem. R. Accad. Naz. Lincei, 2, 31–113.