Nonlinear Systems and Autonomous Equations

Nonlinear Systems and the Linearization Method

From Linear to Nonlinear

The world is predominantly nonlinear. A pendulum with large oscillations is described by the equation θ'' + (g/L) sin θ = 0—a nonlinear one. The predator–prey system follows the Lotka–Volterra equations—which are nonlinear. The Navier–Stokes equations of hydrodynamics are nonlinear. How do we analyze nonlinear systems?

The main instrument is linearization near an equilibrium point. The idea: in a small neighborhood of equilibrium, a nonlinear system “resembles” a linear one. We already know how to analyze the dynamics of linear systems via eigenvalues.

Equilibrium Points and Their Determination

An equilibrium point (fixed point) of the system x' = f(x) is a point x*, where f(x*) = 0, that is, the derivative vanishes. At equilibrium, the system “stands” and does not change over time.

Example—the pendulum: θ'' = −(g/L) sin θ. In matrix form: x₁ = θ, x₂ = θ'. System: x₁' = x₂, x₂' = −(g/L) sin x₁. Equilibrium points: x₂ = 0 and sin x₁ = 0, that is, x₁ = nπ. Two types: θ = 0 (hanging pendulum) and θ = π (inverted pendulum).

Jacobian Matrix and Linearization

Let us expand f(x) by Taylor series near x*: f(x* + δx) ≈ f(x*) + J(x*) δx = J(x*) δx (since f(x*) = 0).

The Jacobian matrix J = ∂fᵢ/∂xⱼ evaluated at point x* is the “derivative of the vector field.” The linearized system: δx' = J(x*) δx.

The behavior of the nonlinear system near x* is determined by the eigenvalues of J(x*)—assuming they are “noncritical” (not on the imaginary axis).

Hartman–Grobman Theorem

If all eigenvalues of J(x*) have nonzero real part (a hyperbolic equilibrium), then the nonlinear system near x* is topologically equivalent to its linearization: there exists a continuous invertible transformation mapping phase portraits of one to the other.

Practically: in the case of hyperbolic equilibrium, the type of singular point (node, focus, saddle) is determined by linearization. If there are eigenvalues with zero real part—not so: nonlinear analysis is required.

Example: Pendulum with Friction

Equation: θ'' + bθ' + (g/L) sin θ = 0. System matrix: x' = (x₂, −bx₂ − (g/L) sin x₁).

Jacobian matrix: J = [[0, 1], [−(g/L) cos x₁, −b]].

At point x = (0, 0)* (lower equilibrium): J = [[0, 1], [−g/L, −b]].

tr J = −b < 0 (if b > 0), det J = g/L > 0. For small b: focus (damped oscillations around lower position). For critical b = 2√(g/L): node. ✓ Physically accurate!

At point x = (π, 0)* (upper equilibrium): J = [[0, 1], [g/L, −b]].

det J = −g/L < 0 → saddle (for all b). The upper position of the pendulum is always unstable—as intuition suggests.

Quantitative Example: Damped Pendulum

Data: g/L = 4 (rad/s)², b = 1 (s⁻¹). J(0,0) = [[0, 1], [−4, −1]].

Eigenvalues: λ = (−1 ± √(1−16))/2 = −0.5 ± i√(15)/2 ≈ −0.5 ± 1.94i.

Type: stable focus. The pendulum with initial conditions θ(0) = 0.5 rad, θ'(0) = 0 executes damped oscillations with frequency ≈ 1.94 rad/s and damping time 1/0.5 = 2 s.

Bifurcation of the "Pitchfork" Type

As a parameter changes, an equilibrium point may lose stability and “split.” Simplest example: x' = μx − x³.

For μ < 0: single equilibrium x* = 0, stable. At μ = 0: bifurcation point. For μ > 0: x* = 0 becomes unstable, two new equilibria x* = ±√μ appear—both stable.

Physical example: Loss of stability in a straight beam under longitudinal load (Euler’s problem of longitudinal buckling). Upon reaching the critical load μ = μcr, the straight state becomes unstable, and the beam bends to one of two sides.

Question for reflection: As θ → π, the pendulum becomes an “unstable inverted pendulum.” How can it be stabilized in the inverted position using vibrations of the pivot point (Kapitza’s paradox)? What type of bifurcation occurs during stabilization?

The Significance and Limitations of Linearization

Linearization is a powerful but limited tool. It fully describes system behavior near hyperbolic singular points (Hartman–Grobman theorem). However, for centers, linearization is unreliable: nonlinear terms can turn a center into a stable or unstable focus. For such critical cases, the Lyapunov function method or calculation of Lyapunov exponents—invariants that determine “weak nonlinearity”—are necessary.

The Poincaré–Bendixson Theorem and Limit Cycles

In planar systems (ℝ²), the Poincaré–Bendixson theorem states: any bounded trajectory as t → ∞ tends either to a singular point, or to a limit cycle, or to a set composed of singular points and trajectories. Consequence: chaos is impossible in autonomous two-dimensional systems. Bendixson’s criterion: if div f = ∂f₁/∂x₁ + ∂f₂/∂x₂ does not change sign in a simply connected domain D, then D contains no closed trajectories. This is a convenient way to exclude limit cycles. Neural oscillators: The Hodgkin–Huxley model for neuron spikes contains a limit cycle—a stable periodic solution in four-dimensional phase space, responsible for rhythmic discharges.