Flows on the Line
One-dimensional ODEs — fixed points, stability, and bifurcations
A Brief History
What is a Differential Equation?
A differential equation is a rule that prescribes the rate of change of a quantity from its current value. If $x(t)$ represents the state of a system at time $t$, an ordinary differential equation (ODE) takes the form:
$$\dot{x} = f(x)$$where $\dot{x} = dx/dt$ and $f$ is a given function. The equation says: "if you are at state $x$, your velocity is $f(x)$." A solution is a function $x(t)$ whose velocity at each instant matches the rule: $\dot{x}(t) = f(x(t))$ for all $t$.
The equation above is autonomous — $f$ depends only on $x$, not on time directly — and first-order — only the first derivative appears. These are the simplest ODEs, but they already exhibit rich behaviour when $f$ is nonlinear. Almost every model in this course is built from autonomous first-order ODEs.
If $f$ is continuous, solutions to $\dot{x} = f(x)$ exist locally (Peano's theorem). If $f$ is also Lipschitz continuous — roughly, if it has a bounded derivative — then solutions are unique (Picard–Lindelöf theorem). For a one-dimensional ODE, uniqueness has a powerful geometric consequence: trajectories on the real line cannot cross. A trajectory starting to the left of a fixed point can never jump past it. This severely constrains the possible dynamics.
For a linear ODE like $\dot{x} = ax$, the solution $x(t) = x_0 e^{at}$ is elementary. But for a nonlinear equation like $\dot{x} = x - x^3$, closed-form solutions are cumbersome or unavailable. The key insight, due to Poincaré, is that we do not need a formula: we can read the qualitative behaviour — where does $x$ increase? decrease? stop? — directly from the graph of $f$.
Geometric Thinking: Flows on the Line
For a one-dimensional ODE $\dot{x} = f(x)$, the entire dynamics can be visualised by plotting $f(x)$ versus $x$. Where $f(x) > 0$, the state moves to the right (increasing $x$); where $f(x) < 0$, it moves left. Arrows on the $x$-axis indicate the direction of flow.
Fixed points are values $x^*$ where $f(x^*) = 0$: the flow stops. They are the equilibria of the system. The key question is: are nearby trajectories attracted to the fixed point, or repelled?
Consider $f(x) = x - x^3$. The fixed points are $x^* = 0$ and $x^* = \pm 1$. Between $0$ and $1$, $f > 0$, so flow goes right toward $x = 1$. Between $-1$ and $0$, $f < 0$, so flow goes left toward $x = -1$. At $x = 0$, flow goes away in both directions: $x = 0$ is unstable. At $x = \pm 1$, flow converges from both sides: these are stable.
Because trajectories in 1D cannot cross, the long-term fate of every initial condition is fully determined: it converges to a stable fixed point, diverges to $\pm\infty$, or (in the degenerate case) sits exactly on an unstable fixed point. There are no oscillations in one dimension. Periodic orbits require at least two dimensions.
Fixed Points and Stability
To classify stability analytically, write $x = x^* + \eta$ where $\eta$ is a small perturbation. Taylor expanding:
$$\dot{\eta} = f(x^* + \eta) \approx f'(x^*)\,\eta$$since $f(x^*) = 0$. The solution is $\eta(t) = \eta_0\, e^{f'(x^*)\, t}$. If $f'(x^*) < 0$, the perturbation decays exponentially: the fixed point is stable. If $f'(x^*) > 0$, perturbations grow: it is unstable. The borderline case $f'(x^*) = 0$ is marginal and requires higher-order terms to resolve.
For $\dot{x} = f(x)$ with fixed point $f(x^*) = 0$:
$$f'(x^*) < 0 \;\Rightarrow\; \text{stable}, \qquad f'(x^*) > 0 \;\Rightarrow\; \text{unstable}$$
Example: the logistic equation. $\dot{x} = rx(1 - x/K)$ has fixed points at $x^* = 0$ and $x^* = K$. The derivative is $f'(x) = r - 2rx/K$. At $x^* = 0$: $f'(0) = r > 0$ (unstable for $r > 0$). At $x^* = K$: $f'(K) = -r < 0$ (stable). Any positive initial condition is attracted to $K$.
Potentials and the Landscape Metaphor
For any one-dimensional ODE $\dot{x} = f(x)$, define the potential:
$$V(x) = -\int f(x)\, dx$$Then $\dot{x} = f(x) = -V'(x)$: the system slides downhill on the potential landscape, like a ball rolling in a valley. Stable fixed points correspond to local minima of $V$; unstable fixed points correspond to local maxima.
The potential decreases along trajectories: $\dot{V} = V'(x)\,\dot{x} = -[f(x)]^2 \leq 0$. This makes $V$ a Lyapunov function — its monotonic decrease rules out oscillations and guarantees convergence to a fixed point (or divergence to infinity). The potential viewpoint is powerful but special to one dimension; in two or more dimensions, a global potential generally does not exist.
Bifurcations in One Dimension
A bifurcation is a qualitative change in the dynamics — a change in the number or stability of fixed points — as a parameter varies. Three canonical types appear in one-dimensional systems.
Saddle-node bifurcation
Two fixed points collide and annihilate. The normal form is:
$$\dot{x} = r - x^2$$For $r > 0$: two fixed points at $x^* = \pm\sqrt{r}$ (the positive root is stable, the negative unstable). At $r = 0$: they merge into a single half-stable point. For $r < 0$: no fixed points exist; all trajectories escape to $-\infty$.
Biological example: a harvested population $\dot{N} = rN(1 - N/K) - h$ undergoes a saddle-node bifurcation at the maximum sustainable yield $h = rK/4$. Beyond this threshold, the population collapses — the two equilibria (high and low population) merge and vanish.
Transcritical bifurcation
Two fixed points exchange stability as they cross. Normal form:
$$\dot{x} = rx - x^2$$Fixed points at $x^* = 0$ and $x^* = r$. For $r < 0$: $x = 0$ is stable and $x = r$ is unstable. As $r$ increases through $0$, they swap: $x = 0$ becomes unstable and $x = r$ becomes stable.
Biological example: the logistic equation $\dot{N} = rN(1 - N/K)$. At $r = 0$ the carrying capacity equilibrium and the extinction equilibrium exchange stability — the population transitions from inevitable decay to growth.
Pitchfork bifurcation
A single fixed point splits into three. The supercritical normal form is:
$$\dot{x} = rx - x^3$$For $r < 0$: only $x^* = 0$ exists (stable). For $r > 0$: $x = 0$ becomes unstable, and two new stable branches $x^* = \pm\sqrt{r}$ appear symmetrically. This is symmetry-breaking: the system must "choose" one of two equivalent stable states.
The pitchfork requires the symmetry $f(-x) = -f(x)$. Breaking this symmetry by adding a small constant (imperfect bifurcation) replaces the pitchfork with a saddle-node, showing that the pitchfork is structurally unstable — a fragile organising centre rather than a robust mechanism.
Explore: One-Dimensional Flows and Bifurcations
Select a system below and drag the parameter slider. The left panel shows the vector field $f(x)$ versus $x$ with flow arrows; the right panel shows the bifurcation diagram — how fixed points and their stability change with $r$. Solid curves are stable, dashed are unstable.
Exercises
Conceptual Questions
- For $\dot{x} = \sin x$, sketch the phase portrait on the real line. Identify all fixed points and classify their stability. What is the biological interpretation if $x$ represents the phase of a biological oscillator coupled to an external signal?
- Explain why oscillations (periodic orbits) are impossible for a one-dimensional autonomous ODE $\dot{x} = f(x)$. Use the uniqueness theorem to argue that a trajectory starting at $x_0$ cannot return to $x_0$ at a later time.
- The logistic equation $\dot{N} = rN(1 - N/K)$ undergoes a transcritical bifurcation at $r = 0$. Describe what happens to the fixed points and their stability as $r$ passes from negative to positive. What is the biological meaning of $r < 0$?
- Consider the harvested population $\dot{N} = rN(1 - N/K) - h$. Using the potential $V(N) = -\int f(N)\, dN$, sketch the potential landscape for $h$ below, at, and above the maximum sustainable yield $h_{\text{MSY}} = rK/4$. Explain geometrically why the population collapses.
- The supercritical pitchfork $\dot{x} = rx - x^3$ requires the symmetry $f(-x) = -f(x)$. Add a small perturbation $\epsilon$ to get $\dot{x} = \epsilon + rx - x^3$ and sketch the bifurcation diagram for $\epsilon = 0.1$. What happens to the pitchfork? Why is the saddle-node more "robust"?
Computer Problems
- Phase Portraits for Normal Forms. Implement the three normal forms (saddle-node $\dot{x} = r - x^2$, transcritical $\dot{x} = rx - x^2$, pitchfork $\dot{x} = rx - x^3$). For each, plot $f(x)$ vs $x$ for $r = -1, 0, 1$ and draw flow arrows indicating direction. Identify fixed points graphically.
- Bifurcation Diagrams. For each normal form, vary $r$ from $-2$ to $2$ in steps of $0.01$. At each $r$, numerically find fixed points (roots of $f(x, r) = 0$) and classify their stability using $f'(x^*)$. Plot the bifurcation diagram with stable branches solid and unstable branches dashed.
- Potential Landscapes. Compute and plot the potential $V(x) = -\int f(x)\,dx$ for the Allee-effect model $\dot{N} = rN(N/A - 1)(1 - N/K)$ with $r = 1$, $A = 50$, $K = 200$. Plot $V(N)$ for several values of $A$ (e.g., $10, 50, 100$) and show how the potential wells change.
- Numerical Integration Near a Saddle-Node. Integrate $\dot{x} = r - x^2$ using RK4 with $r = 0.01$ and initial condition $x_0 = 0$. Observe that the trajectory lingers near $x = 0$ ("ghost" of the vanished fixed point) before escaping. Measure the passage time as a function of $r$ and show it scales as $T \sim 1/\sqrt{r}$.
- Imperfect Bifurcation. For $\dot{x} = \epsilon + rx - x^3$, compute the bifurcation diagram in the $(r, x)$ plane for $\epsilon = 0$, $0.05$, and $0.2$. Show that the imperfection breaks the pitchfork into a saddle-node plus a smooth curve. Plot the cusp in the $(r, \epsilon)$ parameter plane where the saddle-node bifurcation occurs.
References
- Andronov, A. A. & Pontryagin, L. S. (1937). Systèmes grossiers. Dokl. Akad. Nauk SSSR, 14, 247–250.
- Lyapunov, A. M. (1892). The General Problem of the Stability of Motion. Kharkov. (English translation by Fuller, A. T., Taylor & Francis, 1992.)
- May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261, 459–467.
- Newton, I. (1687). Philosophiae Naturalis Principia Mathematica. London.
- Poincaré, H. (1881). Mémoire sur les courbes définies par une équation différentielle. J. Math. Pures Appl. (3e série), 7, 375–422.
- Strogatz, S. H. (2015). Nonlinear Dynamics and Chaos (2nd ed.). Westview Press.
- Verhulst, P.-F. (1838). Notice sur la loi que la population suit dans son accroissement. Correspondance Mathématique et Physique, 10, 113–121.