Interactive Lecture
Understanding Complex Behaviour in Dynamic Systems
by Dr Arnaldo Delli Carri
Nonlinear systems are those whose behaviour cannot be expressed as a linear combination of their inputs. Unlike linear systems, they can exhibit multiple equilibria, limit cycles, bifurcations, and chaos. Most real-world engineering systems are inherently nonlinear.
Nonlinear systems can behave in ways that linear systems simply cannot. Small changes in parameters or initial conditions can lead to qualitatively different responses, and superposition does not apply — which makes analysis harder but the dynamics far richer.
A phase portrait is a geometric representation of trajectories in phase space (state space). For a second-order system, phase space is two-dimensional with axes representing displacement and velocity.
Phase Space Representation:
For a system described by \(\ddot{x} = f(x, \dot{x}, t)\), define state variables:
\[x_1 = x, \quad x_2 = \dot{x}\]
Then: \[\dot{x}_1 = x_2, \quad \dot{x}_2 = f(x_1, x_2, t)\]
A trajectory in the \((x_1, x_2)\) plane represents the system's evolution over time.
Equilibrium points (fixed points) occur where \(\dot{x}_1 = \dot{x}_2 = 0\). Their stability is determined by linearisation:
The Duffing equation models a spring with cubic nonlinearity:
\[\ddot{x} + \delta\dot{x} + \alpha x + \beta x^3 = \gamma\cos(\omega t)\]
where \(\alpha\) is linear stiffness, \(\beta\) is nonlinear stiffness, \(\delta\) is damping, and the right side is forcing.
Hardening spring: \(\beta > 0\) (stiffness increases with displacement)
Softening spring: \(\beta < 0\) (stiffness decreases with displacement)
Models self-excited oscillations with nonlinear damping:
\[\ddot{x} - \mu(1-x^2)\dot{x} + x = 0\]
where \(\mu > 0\) controls the nonlinearity strength. For small amplitudes, damping is negative (energy added), for large amplitudes, damping is positive (energy dissipated).
The simple pendulum exhibits nonlinear behaviour for large angles:
\[\ddot{\theta} + \frac{g}{L}\sin(\theta) = 0\]
Unlike the linearised version, the period depends on amplitude, and the phase portrait shows separatrices dividing oscillation and rotation regions.
A limit cycle is an isolated closed trajectory in phase space. Unlike the centre in linear systems, limit cycles are structurally stable - nearby trajectories converge to or diverge from them.
Poincaré-Bendixson Theorem: For 2D continuous systems, if a trajectory remains in a bounded region and doesn't settle at a fixed point, it must approach a limit cycle. This theorem doesn't apply in higher dimensions, where chaotic behaviour can occur.
A bifurcation occurs when a qualitative change in system behaviour happens as a parameter varies. Common types include:
Chaotic systems are deterministic yet appear random due to extreme sensitivity to initial conditions. Small differences in initial states lead to exponentially diverging trajectories.
Lyapunov Exponent:
\[\lambda = \lim_{t\to\infty} \frac{1}{t}\ln\frac{|d(t)|}{|d(0)|}\]
where \(d(t)\) is the separation between two initially nearby trajectories and \(d(0)\) their initial separation (a separate symbol from the damping \(\delta\) above). It measures the average rate of divergence of nearby trajectories:
Properties of Chaos: (1) Sensitivity to initial conditions, (2) Topological mixing (phase space gets thoroughly mixed), (3) Dense periodic orbits, (4) Bounded but non-periodic motion. Chaos requires at least three dimensions in continuous systems (Duffing with forcing is effectively 3D).
Explore phase portraits of different nonlinear systems. Watch how trajectories evolve and how changing parameters affects system behaviour.
Click on the phase portrait to add trajectories from different initial conditions
The unforced portraits above conserve or slowly bleed energy, so in two dimensions they can only settle to a fixed point or a limit cycle. Drive the same twin-well spring with a periodic force and a third coordinate — the forcing phase — opens up, which is exactly the room chaos needs. This simulation integrates \(\ddot{x} + \delta\dot{x} - x + x^3 = F\cos(\omega t)\) with RK4 and strobes the motion once every forcing period \(2\pi/\omega\), discarding the first 50 periods as transient.
That strobe is the Poincaré section: it collapses a continuous trajectory to a handful of points. One point means the motion repeats every drive cycle (period-1); two points means it takes two cycles to close (period-2); a smeared cloud that never repeats is chaos. Raise \(F\) and the points split \(1\to2\to4\) in a period-doubling cascade before they scatter — the same cascade the prose describes, now something you can watch. The bifurcation diagram stacks those strobed \(x\)-values against \(F\) so the whole route reads left to right on one axis, with periodic windows punctuating the chaotic band.
Bifurcation diagram not computed yet.
Left: live phase trajectory with persistent Poincaré samples (gold). Right: press the button to trace the cascade; the gold line marks the current F.
Nonlinear dynamics and chaos theory turn up throughout engineering:
| Concept | Equation | Description |
|---|---|---|
| Duffing Oscillator | \(\ddot{x} + \delta\dot{x} + \alpha x + \beta x^3 = \gamma\cos(\omega t)\) | Nonlinear spring with cubic stiffness |
| Van der Pol Oscillator | \(\ddot{x} - \mu(1-x^2)\dot{x} + x = 0\) | Self-excited oscillation with nonlinear damping |
| Nonlinear Pendulum | \(\ddot{\theta} + \frac{g}{L}\sin(\theta) = 0\) | Exact pendulum equation (nonlinear) |
| State Space Form | \(\dot{x}_1 = x_2, \quad \dot{x}_2 = f(x_1, x_2, t)\) | First-order system representation |
| Equilibrium Condition | \(\dot{x}_1 = \dot{x}_2 = 0\) | Fixed points in phase space |
| Lyapunov Exponent | \(\lambda = \lim_{t\to\infty} \frac{1}{t}\ln\frac{|d(t)|}{|d(0)|}\) | Measure of trajectory divergence; \(\lambda > 0\) indicates chaos |
| Jacobian Matrix (2D) | \(J = \begin{bmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} \end{bmatrix}\) | Linearisation for stability analysis |
For the unforced, undamped Duffing equation \(\ddot{x} + x + \beta x^3 = 0\), find all equilibrium points for (a) \(\beta > 0\) and (b) \(\beta < 0\). Discuss their physical meaning.
Show Solution
Equilibrium points satisfy: \(\dot{x} = 0\) and \(\ddot{x} = 0\)
This gives: \(x + \beta x^3 = 0\) or \(x(1 + \beta x^2) = 0\)
(a) For β > 0 (hardening spring):
Since \(1 + \beta x^2 > 0\) for all \(x\), the only real solution is \(x = 0\). Linearising, \(\ddot{x} + (1 + 3\beta x^2)x = 0\) has stiffness \(1 > 0\) at the origin, so the origin is a stable centre (eigenvalues \(\pm i\)); with the potential \(V(x) = \tfrac{1}{2}x^2 + \tfrac{1}{4}\beta x^4\), \(V''(0) = 1 > 0\) confirms it is a minimum.
(b) For β < 0 (softening spring):
\(x = 0\) or \(1 + \beta x^2 = 0 \Rightarrow x^2 = -1/\beta \Rightarrow x = \pm\sqrt{-1/\beta}\). Three equilibria: \(x = 0,\ +\sqrt{-1/\beta},\ -\sqrt{-1/\beta}\).
The linearised stiffness is \(1 + 3\beta x^2\):
Physical Interpretation: There is no pitchfork here — a pitchfork needs the linear stiffness to pass through zero, but it stays fixed at \(+1\). Instead, as \(\beta \to 0^-\) the two saddles \(x = \pm\sqrt{-1/\beta}\) run off to infinity, and they move in from infinity as \(\beta\) becomes more negative. The undamped system is conservative, so nothing "settles" anywhere: small-amplitude trajectories orbit the stable centre at the origin, while the outer saddles and the separatrices through them mark where bounded motion gives way to escape.
Answer: (a) one equilibrium, \(x = 0\), a stable centre. (b) three equilibria: \(x = 0\) (stable centre) and \(x = \pm\sqrt{-1/\beta}\) (unstable saddles) — no bifurcation of the origin occurs.
The Van der Pol oscillator \(\ddot{x} - \mu(1-x^2)\dot{x} + x = 0\) has a limit cycle. Explain why the origin is unstable for small perturbations but trajectories don't escape to infinity.
Show Solution
Analysis near the origin (small x):
For \(|x| \ll 1\), the damping term is approximately \(-\mu\dot{x}\)
The equation becomes: \(\ddot{x} - \mu\dot{x} + x \approx 0\)
This is a linear oscillator with negative damping (\(\mu > 0\)), causing exponential growth of oscillations.
Analysis for large x:
For \(|x| \gg 1\), \((1-x^2) < 0\), so damping term is \(+\mu x^2\dot{x}\)
This provides positive damping (energy dissipation) that limits amplitude growth.
Conclusion: Energy is added to the system at small amplitudes and removed at large amplitudes. This creates a stable limit cycle - a periodic orbit that attracts all trajectories except the unstable origin. The amplitude and frequency of the limit cycle depend on \(\mu\). This behaviour is common in self-excited oscillators like electronic circuits (relaxation oscillators) and biological rhythms (heartbeat, circadian cycles).
Answer: The origin is an unstable focus (negative linear damping \(-\mu\dot{x}\)), so small perturbations grow; at large amplitude the damping \(-\mu(1-x^2)\dot{x}\) becomes positive and dissipative, so trajectories cannot escape. The balance is a single stable limit cycle that every non-origin trajectory approaches.
Sketch the phase portrait of the undamped pendulum \(\ddot{\theta} + \sin(\theta) = 0\). Identify equilibrium points, their stability, and the separatrix.
Show Solution
Equilibrium points: \(\dot{\theta} = 0\) and \(\sin(\theta) = 0\)
\(\theta = n\pi\) where \(n = 0, \pm1, \pm2, ...\)
Stability Analysis:
Linearise near equilibrium: \(\ddot{\theta} + \cos(\theta_0)\theta \approx 0\)
At \(\theta = 0, \pm2\pi, ...\): \(\cos(\theta_0) = 1 > 0\) → Centres (stable)
At \(\theta = \pm\pi, \pm3\pi, ...\): \(\cos(\theta_0) = -1 < 0\) → Saddle points (unstable)
Phase Portrait Features:
Physical Significance: The separatrix corresponds to the minimum energy needed for the pendulum to just reach the inverted position. Trajectories below it oscillate; above it, the pendulum rotates continuously.
Answer: Equilibria at \(\theta = n\pi\); \(\theta = 0, \pm2\pi, \dots\) are centres (stable) and \(\theta = \pm\pi, \pm3\pi, \dots\) are saddles (unstable). The separatrix is the orbit through the saddles at energy \(E = \tfrac{1}{2}\dot{\theta}^2 - \cos\theta = 1\), separating oscillation (\(E < 1\)) from rotation (\(E > 1\)).