Nonlinear ODEs and Qualitative Analysis

1925-1926. Alfred Lotka in Baltimore and Vito Volterra in Rome independently derive the predator-prey equations. Volterra is asked by his son-in-law, the biologist d'Ancona, who noticed something strange: after World War I, when Adriatic fishing collapsed, the share of predator fish in the catch grew. Volterra showed that periodic population swings are the inevitable consequence of nonlinear two-species interaction. Since then phase portraits, limit cycles, and bifurcations have been the language of cardiology (heart fibrillation, FitzHugh-Nagumo), climatology (paleoclimate bifurcations), epidemiology (R₀ in SIR), and chaos theory (Feigenbaum's δ = 4.6692, universal across all period-doubling cascades).

Ecology: predator-prey population oscillations (Lotka-Volterra)
Cardiology: heart rhythm as a nonlinear oscillator (van der Pol)
Climatology: bifurcation between glacial and interglacial regimes
Neuroscience: neural oscillations and synchronization (FitzHugh-Nagumo)
Control engineering: stability of closed-loop systems (Lyapunov stability boundary)

Цели урока

Linearize a nonlinear ODE system near equilibrium and use the Jacobian matrix for stability analysis
Build a phase portrait and classify trajectories by the type of fixed point
Identify bifurcations (fold, pitchfork, Hopf) as a parameter varies

Предварительные знания

Linear ODE systems and matrix exponential
Eigenvalues and eigenvectors of matrices
Geometry of vector fields

Linearization and the Jacobian matrix

For the system $\dot{x} = f(x)$ near an equilibrium $x^*$ where $f(x^*)=0$: $\dot{\xi} = Jf(x^*)\xi + O(\xi^2)$, with $J_{ij} = \frac{\partial f_i}{\partial x_j}$. Hartman-Grobman theorem: when $J$ has no zero or purely imaginary eigenvalues, the nonlinear phase portrait is homeomorphic to the linearized portrait near $x^*$.

Lyapunov stability and bifurcations

A Lyapunov function $V(x) > 0$ with $\dot{V} = \nabla V \cdot f \leq 0$ proves stability without solving the ODE. Bifurcations: fold - collision of stable and unstable equilibria at a zero eigenvalue; pitchfork - one equilibrium splits into three; Hopf - birth of a limit cycle as eigenvalues cross the imaginary axis.

Linearization Near Equilibria

Nonlinear systems x' = f(x) are hard to solve analytically, but their behavior near equilibria (f(x*) = 0) is fully determined by the Jacobian J = Df(x*).

Linearization: expand f(x) in a Taylor series around x*: f(x) ≈ f(x*) + J(x-x*) = J·ξ, where ξ = x-x*. The nonlinear system behaves like the linear system ξ' = Jξ near x* (Hartman-Grobman theorem, provided Re(λᵢ) ≠ 0).

Equilibrium classification by Jacobian eigenvalues: both Re(λ) < 0 - stable node/spiral; both Re(λ) > 0 - unstable node/spiral; λ of opposite signs - saddle; Im(λ) ≠ 0 - spiral or center.

The Hartman-Grobman theorem fails when Re(λ) = 0 (center in the linear system). In this case nonlinear terms determine the actual behavior - Lyapunov analysis or normal form reduction is needed.

What determines the local behavior of a nonlinear system near an equilibrium x*?

Phase Plane and Limit Cycles

The phase plane is the state space (x₁, x₂). System trajectories are curves in this plane. Nullclines: curves where f₁=0 and f₂=0; their intersections are equilibria.

Van der Pol oscillator: x'' - μ(1-x²)x' + x = 0. For μ > 0 there exists a limit cycle (attractor). This illustrates stable nonlinear oscillations: trajectories from different initial conditions converge to the same cycle.

What is a limit cycle in a nonlinear system?

Lyapunov Stability

Why do we need yet another stability method when we already have linearization? Because linearization only answers at a hyperbolic equilibrium (Hartman-Grobman theorem). On the boundary - purely imaginary λ - it is silent. Lyapunov's method solves this elegantly: instead of solving the ODE or linearizing it, find an 'energy' function V(x) that decreases monotonically along trajectories. Once V is found, stability is proved without an explicit solution. This is exactly how engineers prove stability of spacecraft controllers, neural-network controllers, and convergence theorems in ML (e.g. Robbins-Monro stochastic approximation).

A Lyapunov function V(x) is a generalized 'energy' of the system. If V > 0 in a neighborhood of x* (positive definite) and dV/dt ≤ 0 along trajectories, the equilibrium is stable. If dV/dt < 0 strictly, it is asymptotically stable.

Applications of Lyapunov functions: battery state-of-charge (SOC) estimation - V(x) = (x-x_desired)², power system stability analysis, control loop verification - all without solving the ODE explicitly.

Lyapunov's direct (second) method: prove stability without solving - find V(x) > 0 with V'(x) ≤ 0. No general algorithm for finding V exists, but for polynomial systems Sum-of-Squares (SOS) programming works.

What does 'asymptotic stability' of an equilibrium mean?

Bifurcations

A bifurcation is a qualitative change in dynamics as a parameter μ varies smoothly. Basic types: saddle-node (fold), transcritical, pitchfork, and Hopf. Bifurcations explain why a climate system can flip from glacial to interglacial, why a laser 'turns on' above a critical pump rate, and why a heart can switch from sinus rhythm to fibrillation.

Bifurcation	Normal form	What happens	Example
Saddle-node (fold)	x'=r+x²	Birth/annihilation of equilibrium pair	Power grid voltage collapse
Transcritical	x'=rx-x²	Exchange of stability between two equilibria	SIR model at R₀=1
Pitchfork	x'=rx-x³	One equilibrium splits into three (symmetry)	Buckling of a column
Hopf	r'=μr-r³	Equilibrium births a limit cycle	Van der Pol, cardiac rhythm

Mitchell Feigenbaum and the universality of chaos (1975-1978)

In 1975, working at Los Alamos with an HP-65 calculator, Feigenbaum was computing bifurcation points of the logistic map x_{n+1} = rx_n(1-x_n). He noticed that successive period-doubling spacings shrank by a constant ratio δ ≈ 4.6692. In 1976 he checked: the same δ appears in the sine map x_{n+1} = a·sin(πx_n). It was a universal constant, like π. Four years later Albert Libchaber measured δ experimentally in liquid-helium convection to four-digit accuracy. Chaos theory ceased to be mathematical exotica.

Feigenbaum's universality showed that very different physical systems near the onset of chaos obey the same laws, much like phase transitions in condensed matter.

Which type of bifurcation describes the birth of a limit cycle from an equilibrium point?

Lotka-Volterra: equilibrium analysis

$\dot{x} = x(\alpha - \beta y)$, $\dot{y} = y(\delta x - \gamma)$. Equilibrium $(x^*, y^*) = (\gamma/\delta, \alpha/\beta)$. Jacobian: $J = \begin{pmatrix} 0 & -\beta x^* \\ \delta y^* & 0 \end{pmatrix}$. Eigenvalues $\pm i\sqrt{\alpha\gamma}$ - purely imaginary: a center (nonlinear analysis required for confirmation).

Итоги

Linearization $\dot{\xi} = J(x^*)\xi$ + Hartman-Grobman: nonlinear portrait homeomorphic to linear at a hyperbolic equilibrium
Lyapunov function $V > 0$, $\dot{V} \leq 0$ proves stability without explicit solutions
Bifurcations (fold, pitchfork, Hopf) - qualitative changes in the phase portrait as a parameter varies

Вопросы для размышления

The Hartman-Grobman theorem does not apply when $J$ has zero or imaginary eigenvalues. What tools are available in those cases?
A Lyapunov function for the Lotka-Volterra system exists but is not obvious. How can one be found systematically?
A Hopf bifurcation creates a limit cycle. Under what conditions is the cycle stable (supercritical bifurcation) and when is it not?

Связанные уроки

dyn-01