Stability and Phase Portraits

Phase portrait analysis, fixed equilibrium points, and Lyapunov stability for physical dynamical systems.

What it is

The geometric and qualitative analysis of continuous autonomous dynamic systems: $\frac{d\mathbf{x}}{dt} = \mathbf{f}(\mathbf{x}), \quad \mathbf{x} \in \mathbb{R}^n$ determines global asymptotic asymptotic behavior without requiring closed-form analytic quadratures. An equilibrium point (fixed point) $\mathbf{x}^*$ is a stationary state where velocities vanish identically: $\mathbf{f}(\mathbf{x}^*) = \mathbf{0}$

Near an equilibrium, the non-linear system is approximated via first-order Taylor expansion (linearization): $\frac{d\mathbf{\xi}}{dt} = \mathbf{J}(\mathbf{x}^*) \mathbf{\xi}, \quad \mathbf{\xi} = \mathbf{x} - \mathbf{x}^*$ where $\mathbf{J}(\mathbf{x}^*) = \left.\frac{\partial f_i}{\partial x_j}\right|_{\mathbf{x}^*}$ is the Jacobian matrix.

By the Hartman-Grobman theorem, if an equilibrium is hyperbolic (no eigenvalues with zero real parts, $\operatorname{Re}(\lambda_i) \neq 0$), the local orbit structure of the non-linear system is topologically conjugate to that of its linearization.

In 2D phase space $(x, y)$, trace $\tau = \operatorname{tr}(\mathbf{J})$ and determinant $\Delta = \det(\mathbf{J})$ fully classify local dynamics:

1. $\Delta < 0$: Saddle point (unstable, real eigenvalues of opposite sign).
2. $\Delta > 0$ and $\tau^2 - 4\Delta > 0$: Node:
   • Stable (attractor/sink) if $\tau < 0$.
   • Unstable (repeller/source) if $\tau > 0$.
3. $\Delta > 0$ and $\tau^2 - 4\Delta < 0$: Spiral focus:
   • Stable focus if $\tau < 0$ (converging inward spiral).
   • Unstable focus if $\tau > 0$ (diverging outward spiral).
4. $\Delta > 0$ and $\tau = 0$: Center (pure imaginary roots $\pm i\omega$, concentric closed neutral periodic orbits).

Why it appears in physics

Most differential equations in physical science are intrinsically non-linear and lack elemental closed solutions (finite-amplitude pendulums, Rayleigh-Bénard convection cells, astrophysical stellar models, turbulence). Phase space portraits yield immediate physical answers:

• Is an unforced equilibrium robust against perturbations?
• Does the system converge to rest, sustain an isolated limit cycle, or wander chaotically?
• At what operating threshold will the system undergo a qualitative regime transition (bifurcation)?

Visual or geometric intuition

A phase portrait is a complete topological map of all candidate system trajectories across state space. Nullclines are geometric curves along which a single velocity coordinate vanishes:

• $x$-nullcline: Curve $f(x, y) = 0$ (trajectories cross vertically).
• $y$-nullcline: Curve $g(x, y) = 0$ (trajectories cross horizontally). Intersection points of nullclines are the equilibrium states. Trajectories flow along vector arrows: stable and unstable manifolds form dynamic barriers (separatrices) dividing state space into distinct basins of attraction.

Minimal symbolic language

• Lyapunov Stability Criterion: An equilibrium $\mathbf{x}^* = \mathbf{0}$ is stable if there exists a continuously differentiable scalar Lyapunov function $V(\mathbf{x})$ such that:
  1. $V(\mathbf{0}) = 0$ and $V(\mathbf{x}) > 0$ for all $\mathbf{x} \neq \mathbf{0}$ (positive definite).
  2. $\dot{V}(\mathbf{x}) = \nabla V \cdot \mathbf{f}(\mathbf{x}) \le 0$ across the neighborhood. If $\dot{V}(\mathbf{x}) < 0$ strictly for $\mathbf{x} \neq \mathbf{0}$, the equilibrium is asymptotically stable.
• Poincaré-Bendixson Theorem: In bounded regions of the continuous plane $\mathbb{R}^2$ containing no fixed points, any bounded trajectory must asymptotically approach a closed periodic orbit (limit cycle). Continuous chaos requires phase space dimension $n \ge 3$.

What the result means

1. Structural stability: Hyperbolic nodes, foci, and saddles persist under small parameter shifts.
2. Limit cycles: Represent self-sustained non-linear oscillations with fixed amplitude and frequency independent of initial disturbances (e.g. heartbeat rhythms, organ pipe acoustics).
3. Bifurcations: Critical parameter boundaries where eigenvalues cross the imaginary axis, creating or annihilating equilibria (saddle-node, pitchfork, Hopf bifurcations).

When not to use it

• Linearizing neutral centers ($\tau = 0, \Delta > 0$): Higher-order nonlinear terms can turn a linearized center into a weakly stable or unstable spiral. Lyapunov or Hamiltonian energy methods are required.
• Expecting chaos in 2D continuous systems: Poincaré-Bendixson strictly rules out strange attractors in autonomous planar systems ($n = 2$).
• Non-autonomous explicit time-dependent fields $\mathbf{f}(\mathbf{x}, t)$: Planar trajectories can intersect because field arrows change in time; augment state dimension by treating time as a coordinate ($z = t$).

Operational summary

1. Set velocities to zero: $f_1(x, y) = 0$ and $f_2(x, y) = 0$ to find fixed points $(x^*, y^*)$.
2. Sketch nullclines to establish directional flow quadrants.
3. Compute the symbolic Jacobian matrix $\mathbf{J}(x, y)$.
4. Evaluate $\mathbf{J}$ at each equilibrium point and compute trace $\tau$ and determinant $\Delta$.
5. Classify stability (saddle if $\Delta < 0$; sink/source node or focus based on $\tau^2 - 4\Delta$ and $\operatorname{sgn}(\tau)$).
6. For centers, verify orbit closedness via conservative energy first integrals.

Procedure: Constructing and Analyzing Phase Portraits

Before calculating

1. Verify system autonomy: Ensure explicit time dependencies are absent: $\dot{x} = f(x, y), \quad \dot{y} = g(x, y)$
2. Check for symmetries: Symmetries under $(x, y) \to (-x, -y)$ or time reversal $(t \to -t)$ indicate reflective phase space patterns.
3. Mark physical boundaries: In biology or chemistry, restrict domain to positive quadrants ($x \ge 0, y \ge 0$).

Step 1: Identify the mathematical object

Locate all fixed equilibrium points by solving: $\begin{cases} f(x, y) = 0 \\ g(x, y) = 0 \end{cases}$ Record all coordinate pairs $(x_i^*, y_i^*)$.

Step 2: Choose the operation

2A. Plot nullclines

• Draw $f(x, y) = 0$ ($x$-nullcline): $\dot{x} = 0$, vector arrows point purely vertically.
• Draw $g(x, y) = 0$ ($y$-nullcline): $\dot{y} = 0$, vector arrows point purely horizontally.

2B. Jacobian linearization

Compute the general Jacobian: $\mathbf{J}(x, y) = \begin{pmatrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \\ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} \end{pmatrix}$ At each fixed point, evaluate trace and determinant: $\tau = \operatorname{tr}(\mathbf{J}), \quad \Delta = \det(\mathbf{J})$

Step 3: Conserve units and signs

• If $\Delta < 0$: Saddle point (unstable).
• If $\Delta > 0$:
  • $\tau^2 - 4\Delta > 0$: Node (stable sink if $\tau < 0$, source if $\tau > 0$).
  • $\tau^2 - 4\Delta < 0$: Spiral focus (stable if $\tau < 0$, unstable if $\tau > 0$).
  • $\tau = 0$: Linear center (requires nonlinear energy verification).

Step 4: Calculate invariant manifold directions

For saddle points:

1. Compute eigenvector for $\lambda_s < 0$: tangent to incoming stable manifold.
2. Compute eigenvector for $\lambda_u > 0$: tangent to outgoing unstable manifold.

Step 5: Interpret the result

1. Smoothly interpolate flow lines through nullcline sectors.
2. Search for limit cycles surrounding unstable foci via Poincaré-Bendixson theorem.
3. Trace separatrices to map basins of attraction.

Self-check checklist

• Do nullcline crossings match the computed equilibrium points?
• Did I double check trace and determinant evaluations?
• Do trajectories cross nullclines horizontally/vertically as required?
• In conservative undamped systems, are sinks/sources absent?
• Did I ensure phase trajectories never cross each other?

Worked Examples: Stability and Phase Portraits

Step-by-step guided example

Problem

Consider the non-linear simple pendulum with weak viscous damping: $\ddot{\theta} + \gamma \dot{\theta} + \omega_0^2 \sin\theta = 0$ with $\omega_0 = 2\,\text{rad}/\text{s}$ and $\gamma = 0.5\,\text{s}^{-1}$.

1. Cast the second-order ODE into a first-order autonomous system in state variables $(x, y) = (\theta, \dot{\theta})$.
2. Determine all fixed equilibrium points on $\theta \in [-\pi, \pi]$.
3. Evaluate the Jacobian matrix and classify local stability.
4. Sketch the qualitative phase portrait topology.

Detailed solution

Step 1: First-order system reduction Set $x = \theta$ (angular displacement) and $y = \dot{\theta}$ (angular velocity): $\begin{cases} \dot{x} = y \\ \dot{y} = -4 \sin x - 0.5 y \end{cases}$

Step 2: Equilibrium points Setting velocities to zero: $y = 0, \quad -4 \sin x = 0 \implies x \in \{0, \pm \pi\}$ Fixed points:

• $P_1 = (0, 0)$ (downward stable rest).
• $P_2 = (\pi, 0)$ and $P_3 = (-\pi, 0)$ (upright inverted rest).

Step 3: Jacobian matrix evaluation $\mathbf{J}(x, y) = \begin{pmatrix} 0 & 1 \\ -4 \cos x & -0.5 \end{pmatrix}$

• At $P_1 = (0, 0)$: $\mathbf{J}(0, 0) = \begin{pmatrix} 0 & 1 \\ -4 & -0.5 \end{pmatrix}$ $\tau = -0.5, \quad \Delta = 4$ Discriminant: $\tau^2 - 4\Delta = 0.25 - 16 = -15.75 < 0$. Since $\Delta > 0, \tau < 0$, eigenvalues are complex with negative real part: $\lambda = -0.25 \pm 1.984 i$ $(0, 0)$ is an asymptotically stable spiral focus (sink).
• At $P_2 = (\pi, 0)$: $\mathbf{J}(\pi, 0) = \begin{pmatrix} 0 & 1 \\ 4 & -0.5 \end{pmatrix}$ $\tau = -0.5, \quad \Delta = -4 < 0$ Roots are real with opposite signs: $\lambda_1 \approx 1.766, \lambda_2 \approx -2.266$. $(\pi, 0)$ is an unstable saddle point.

Step 4: Phase portrait geometry

• Orbits near $(0, 0)$ spiral clockwise into the origin attractor.
• The stable manifolds of $(\pm \pi, 0)$ form separatrices demarcating librational oscillations from whole-revolution rotations.

Inverse example

Problem

An experimental planar system exhibits an isolated fixed point at $(0, 0)$. Trajectories expand outward in spiral paths away from the origin and asymptotically converge onto a circle of radius $R = 2$. Trajectories initiated outside at $r_0 > 2$ spiral inward to the same circle. Reconstruct an autonomous 2D system in Cartesian coordinates reproducing this exact behavior.

Detailed solution

Step 1: Polar coordinates design The observed phase portrait corresponds to a stable circular limit cycle of radius $R = 2$. We choose an uncoupled radial ODE with a stable attractor at $r = 2$ and unstable repeller at $r = 0$: $\dot{r} = r(4 - r^2), \quad \dot{\theta} = 1$

Step 2: Converting to Cartesian coordinates With $x = r \cos\theta, y = r \sin\theta$: $\dot{x} = \dot{r} \cos\theta - r \dot{\theta} \sin\theta = x(4 - r^2) - y$ $\dot{y} = \dot{r} \sin\theta + r \dot{\theta} \cos\theta = y(4 - r^2) + x$ Using $r^2 = x^2 + y^2$: $\begin{cases} \dot{x} = x(4 - x^2 - y^2) - y \\ \dot{y} = y(4 - x^2 - y^2) + x \end{cases}$ Generates an unstable focus at the origin surrounded by an asymptotically stable limit cycle along $x^2 + y^2 = 4$.

Common Errors: Stability and Phase Portraits

Error 1: Declaring a true center based solely on Jacobian linearization

Asserting that an equilibrium is a periodic center merely because $\tau = 0$ and $\Delta > 0$ in the linearized Jacobian.

• Correction: Hartman-Grobman does not apply to non-hyperbolic points. Higher-order nonlinear terms often break closed orbits, converting the center into a spiral sink or source.

Error 2: Drawing intersecting trajectories in autonomous phase portraits

Sketching phase trajectories that cross each other at regular points.

• Consequence: Directly violates Picard-Lindelöf uniqueness. Trajectories can only meet asymptotically at fixed equilibrium points.

Error 3: Conflating nullclines with actual phase paths

Assuming state points slide along nullcline curves.

• Correction: Nullclines are reference marker curves where trajectory velocity vectors align purely vertically or horizontally.

Error 4: Proposing indefinite Lyapunov functions

Employing a candidate $V(x, y)$ that satisfies $\dot{V} \le 0$ but fails positive definiteness ($V \le 0$ in certain sectors).

• Correction: $V$ must form a strict topological bowl with a unique minimum $V(\mathbf{0}) = 0$.

Error 5: Expecting continuous chaos in 2D autonomous systems

Searching for strange attractors in planar ODEs.

• Correction: Poincaré-Bendixson strictly bounds 2D planar trajectories to fixed points or limit cycles. Continuous chaos demands dimension $n \ge 3$.

Quick control rule

To determine the rotational direction of a focus: test the velocity vector on the positive $x$-axis $(x > 0, y = 0)$. If $\dot{y} > 0$, the flow curls counter-clockwise; if $\dot{y} < 0$, it curls clockwise.

Physical Applications: Stability and Phase Portraits

1. The Van der Pol oscillator and relaxation electronics

Originally designed to model vacuum triode oscillation, the Van der Pol equation satisfies: $\ddot{x} - \mu (1 - x^2) \dot{x} + x = 0$ In the $(x, \dot{x})$ phase plane, the origin is an unstable source focus for $\mu > 0$. For $|x| < 1$, negative damping feeds energy into the system; for $|x| > 1$, positive dissipation drains energy. The trajectory contracts onto a stable limit cycle attractor, explaining cardiac pacemaker rhythms, clock pulse generators, and FitzHugh-Nagumo neural spike dynamics.

2. Column elastic buckling and pitchfork bifurcations

A slender structural column loaded axially under compressive load $P$ obeys: $\ddot{\theta} + \left(\frac{P - P_c}{E I}\right) \theta + \alpha \theta^3 = 0$ For subcritical loads $P < P_c$, the undeflected state $\theta = 0$ is stable. When axial load exceeds Euler's buckling limit $P > P_c$, the straight equilibrium destabilizes into a saddle, birthing two symmetric buckled stable states $\theta^* = \pm \sqrt{\frac{P - P_c}{\alpha E I}}$ via a supercritical pitchfork bifurcation.

3. Semiconductor laser rate equations and Hopf bifurcations

Carrier density $N(t)$ and photon density $S(t)$ inside a diode laser cavity obey: $\dot{N} = \frac{I}{e V} - \frac{N}{\tau_s} - g (N - N_0) S, \quad \dot{S} = \Gamma g (N - N_0) S - \frac{S}{\tau_p}$ Ramping pump current $I$ past the stimulated emission threshold triggers a Hopf bifurcation, where the spontaneous emission branch destabilizes and births coherent optical limit-cycle lasing.

4. Orbital astrodynamics and Lagrange libration points

In the circular restricted three-body problem (Earth-Moon-spacecraft), rotating-frame dynamics contain five zero-velocity equilibria: Lagrange points $L_1$ through $L_5$. Jacobian linearization reveals that collinear points $L_1, L_2, L_3$ are saddle points requiring stationkeeping (e.g. James Webb Space Telescope at Sun-Earth $L_2$), whereas triangular points $L_4, L_5$ are stabilized by Coriolis acceleration, housing Trojan asteroid clusters.

5. Atmospheric dynamics: the Lorenz attractor and chaos

Truncating Navier-Stokes Rayleigh-Bénard thermal convection to three spatial modes yields the Lorenz system: $\dot{x} = \sigma (y - x), \quad \dot{y} = x(\rho - z) - y, \quad \dot{z} = x y - \beta z$ For $\rho > 24.74$, all fixed points are unstable; trajectories stay bounded on a fractal strange attractor exhibiting sensitive dependence on initial conditions (the butterfly effect), establishing deterministic bounds on weather forecasting.

Interdisciplinary connection

Phase portraits and bifurcation theory form the core analytical tools of financial bubble crash dynamics, epidemiological disease eradication thresholds, and developmental biology (Waddington's epigenetic landscape).

Guided Practice: Stability and Phase Portraits

Exercise 1: Lotka-Volterra predator-prey dynamics

Dimensionless prey $x(t) \ge 0$ and predator $y(t) \ge 0$ populations follow: $\dot{x} = x(3 - y), \quad \dot{y} = y(x - 2)$

1. Determine the two system fixed points.
2. Classify the trivial extinction equilibrium $(0, 0)$.
3. Prove that coexistence state $(2, 3)$ is a linear center and establish the conserved invariant $V(x, y)$.

Solution

1. Fixed points: $\dot{x} = 0 \implies x=0 \text{ or } y=3; \quad \dot{y} = 0 \implies y=0 \text{ or } x=2$ Points: $(0, 0)$ and $(2, 3)$.
2. Jacobian at origin: $\mathbf{J}(0, 0) = \begin{pmatrix} 3 & 0 \\ 0 & -2 \end{pmatrix} \implies \Delta = -6 < 0 \quad (\text{Saddle point})$
3. At $(2, 3)$: $\mathbf{J}(2, 3) = \begin{pmatrix} 0 & -2 \\ 3 & 0 \end{pmatrix} \implies \tau = 0, \Delta = 6 > 0 \implies \lambda = \pm i\sqrt{6}$ Separating variables: $\frac{3 - y}{y} dy = \frac{x - 2}{x} dx \implies V(x, y) = x - 2\ln x + y - 3\ln y = C$ Level curves form closed orbits; populations cycle perpetually.

Exercise 2: Saddle-node bifurcation in gene regulatory activation

A genetic switch follows: $\dot{x} = \mu + x^2$

1. Solve for fixed points versus parameter $\mu$.
2. Classify stability across $\mu$.
3. Identify the bifurcation threshold.

Solution

1. Equilibrium states: $x^* = \pm \sqrt{-\mu}$.
   • $\mu > 0$: No real fixed points.
   • $\mu = 0$: Single half-stable fixed point at $x^* = 0$.
   • $\mu < 0$: Two fixed points $x_{1,2}^* = \pm \sqrt{-\mu}$.
2. Evaluating $f'(x) = 2x$:
   • $x_1^* = -\sqrt{-\mu} \implies f' < 0$ (Stable sink).
   • $x_2^* = +\sqrt{-\mu} \implies f' > 0$ (Unstable source).
3. At $\mu = 0$, both equilibria coalesce and vanish in a classic saddle-node bifurcation.

Exercise 3: Global non-linear Lyapunov function stability

Consider: $\dot{x} = -x + y^2, \quad \dot{y} = -y - x y$

1. Confirm $(0, 0)$ is an equilibrium.
2. Linearize at the origin.
3. Use Lyapunov candidate $V(x, y) = \frac{1}{2}(x^2 + y^2)$ to prove global asymptotic stability.

Solution

1. Clear: $\mathbf{f}(0, 0) = \mathbf{0}$.
2. $\mathbf{J}(0, 0) = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \implies \lambda_1 = \lambda_2 = -1 < 0$.
3. Evaluating $\dot{V}$: $\dot{V} = x(-x + y^2) + y(-y - xy) = -x^2 + xy^2 - y^2 - xy^2 = -(x^2 + y^2)$ Strictly negative definite everywhere outside the origin $\implies$ origin is globally asymptotically stable.