Nonlinear Dynamics and Linearization

ML in Feedback Sys #5

Fall 2025, Prof Sarah Dean

On the Surprising Effectiveness of Spectrum Clipping in Learning Stable Linear Dynamics

What about some nonlinear approx?
I.e. train deep network so $$\hat x_{t+1} = NN(\hat x_t)$$

Can we predict stability of predictions?

unstable

stable

Nonlinear dynamics stability

"What we do"

  • Given: the nonlinear difference equation \(s_{t+1} = F(s_t)\) and a fixed point \(s_{eq}\)
  • Compute the Jacobian $$ J(s) = \begin{bmatrix}\frac{\partial F_1}{\partial s_1} & \dots & \frac{\partial F_1}{\partial s_d} \\ \vdots & \ddots & \vdots \\ \frac{\partial F_d}{\partial s_1} &\dots & \frac{\partial F_d}{\partial s_d}\end{bmatrix}$$
  • Compute the linearized dynamics by evaluating \(J(s_{eq})\)
  • Assess stability: with \(\max_{i=1,..,d} |\lambda_i(J(s_{eq}))|\)
    • \(> 1 \implies\) unstable
    • \(< 1 \implies\) stable
    • \(=1 \implies\) inconclusive

\(\mathbb C\)

stable

unstable

inconclusive

\(1\)

Consider the dynamics of gradient descent on a twice differentiable function \(\ell:\mathbb R^d\to\mathbb R^d\)

\(\theta_{t+1} = \theta_t - \alpha\nabla \ell(\theta_t)\)

Example: gradient descent

  • Equilibria occur when \(\nabla \ell(\theta_{eq}) = 0\), i.e. at critical points
  • Review: critical point classification in terms of second derivatives
    • positive definite: (local) min
    • negative definite: (local) max
    • both positive and negative: saddle point
    • zero: indeterminate or degenerate

global max

local max

global min

local min

saddle

Consider the dynamics of gradient descent on a twice differentiable function \(\ell:\mathbb R^d\to\mathbb R^d\)

\(\theta_{t+1} = \theta_t - \alpha\nabla \ell(\theta_t)\)

Example: gradient descent

  • Equilibria occur when \(\nabla \ell(\theta_{eq}) = 0\), i.e. at critical points
  • Jacobian \(J(\theta) = I - \alpha \nabla^2 \ell(\theta)\)
  • Let \(\{\gamma_i\}_{i=1}^d\) be the eigenvalues of the Hessian \(\nabla^2 \ell(\theta_{eq})\)
  • Then the eigenvalues of the Jacobian are \(1-\alpha\gamma_i\)

    • if any \(\gamma_i\leq 0\), \(\theta_{eq}\) is not strictly stable

      • i.e. saddle, local maximum, or degenerate critical point of \(\ell\)

    • as long as \(\alpha<\frac{1}{\gamma_i}\) for all \(i\), \(\theta_{eq}\) is stable

global max

local max

global min

local min

saddle

Consider the dynamics of gradient descent on a twice differentiable function \(\ell:\mathbb R^d\to\mathbb R^d\)

\(\theta_{t+1} = \theta_t - \alpha\nabla \ell(\theta_t)\)

Example: gradient descent

  • Equilibria occur when \(\nabla \ell(\theta_{eq}) = 0\), i.e. at critical points
  • Only (non-degenerate) local minima are stable

global max

local max

global min

local min

saddle

Outline

"Why we do it"

  1. Linear approximations
  2. Lyapunov stability theory
  3. Limit cycles and chaos

1D Example

Consider linear approximation of nonlinear dynamics

states tend to evolve away from middle equilibrium and towards right (or left)

middle equilibrium has a slope \(>1\) while right/left have slopes \(<1\)

Linearization example

example: discrete-time damped pendulum

\(\theta_{t+1} = \theta_t + h \omega_t\)

\(\omega_{t+1} =\omega_t + h\left(\frac{g}{\ell}\sin\theta_t-d\omega_t\right)\)

angle \(\theta\)

angular velocity \(\omega\)

gravity

length \(\ell\)

 

\(\approx (1-dh)\omega_t + h\frac{g}{\ell}(\sin \theta_{eq}+\cos\theta_{eq}(\theta-\theta_{eq})\)

\(\sin x\approx \sin x_0 + \cos x_0(x - x_0)\)

equilibria at \(\theta=k\pi\) for \(k\in\mathbb N\)

Consider linear approximation of nonlinear dynamics

Linearization example

example: discrete-time damped pendulum

angle \(\theta\)

angular velocity \(\omega\)

gravity

length \(\ell\)

$$\begin{bmatrix}\theta_{t+1}-\theta_{eq}\\ \omega_{t+1}\end{bmatrix} \approx \begin{bmatrix} 1 & h\\ h \frac{g}{\ell}\cos(\theta_{eq})& 1-dh\end{bmatrix}\begin{bmatrix}\theta_{t}-\theta_{eq}\\ \omega_{t}\end{bmatrix} $$

at \(\theta_{eq}=0\), real eigenvalues \(0<\lambda_2<1<\lambda_1\)

at \(\theta_{eq}=\pi\), complex eigenvalues with \(|\lambda|<1\) for small \(d\)

Exercise: work out the details of this analysis (simulation notebook)

Consider linear approximation of nonlinear dynamics

\(\lambda = 1-h\frac{d}{2} \pm h\sqrt{(\frac{d}{2})^2+\frac{g}{\ell}\cos(\theta_{eq})}\)

Linearization example

example: discrete-time damped pendulum

angle \(\theta\)

angular velocity \(\omega\)

gravity

length \(\ell\)

at \(\theta_{eq}=0\), real eigenvalues \(0<\lambda_2<1<\lambda_1\)

at \(\theta_{eq}=\pi\), complex eigenvalues with \(|\lambda|<1\) for small \(d\)

Consider linear approximation of nonlinear dynamics

Linearization via Taylor Series:

\(s_{t+1} = F(s_t) \)

  • \(=F(s_{eq}) + J(s_{eq})  (s_t - s_{eq}) \) + higher order terms
  • \(=s_{eq} + J(s_{eq})  (s_t - s_{eq}) \) + higher order terms
  • \(\implies s_{t+1}-s_{eq} \approx J(s_{eq})(s_t-s_{eq})\)

Vector-valued linearization

The Jacobian \(J\) of \(G:\mathbb R^{n}\to\mathbb R^{m}\) is defined as $$ J(x) = \begin{bmatrix}\frac{\partial G_1}{\partial x_1} & \dots & \frac{\partial G_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial G_m}{\partial x_1} &\dots & \frac{\partial G_m}{\partial x_n}\end{bmatrix}$$

Outline

"Why we do it"

  1. Linear approximations
  2. Lyapunov stability theory
  3. Limit cycles and chaos

Definition: A Lyapunov function \(V:\mathcal S\to \mathbb R\) for \(F\) is continuous and

  • (positive definite) \(V(0)=0\) and \(V(0)>0\) for all \(s\in\mathcal S - \{0\}\)
  • (decreasing) \(V(F(s)) - V(s) \leq 0\) for all \(s\in\mathcal S\)
  • Optionally,
    • (strict) \(V(F(s)) - V(s) < 0\) for all \(s\in\mathcal S-\{0\}\)
    • (global) \(\|s\|_2\to \infty \implies V(s)\to\infty\)

Lyapunov functions

Reference: Bof, Carli, Schenato, "Lyapunov Theory for Discrete Time Systems"

Lyapunov Stability Theory

Theorem (1.2, 1.4): Suppose that \(F\) is locally Lipschitz, \(s_{eq}=0\) is a fixed point, and \(V\) is a Lyapunov function. Then, \(s_{eq}=0\)  is

  • stable
  • asymptotically stable if \(V\) satisfies the strict property
  • globally asymptotically stable if \(V\) satisfies the strict and global properties

Reference: Bof, Carli, Schenato, "Lyapunov Theory for Discrete Time Systems"

Quadratic Lyapunov functions

  • Stable matrices have quadratic Lyapunov functions of the form \(V(s) = s^\top P s\) (Theorem 3.2)
    • For example, \(P = \sum_{t=0}^\infty (F^\top)^t F^t\)
  • Exercise: show that the above is a strict and global Lyapunov function for \(s_{t+1}= F s_t\).
    • positive definite: argue why \(P\succ 0\)
    • decreasing: argue why \(V(Fs) - V(s) \leq 0\) for all \(s\)

Linearization Stability Test

Theorem (3.3): Suppose \(F\) is locally Lipschitz, \(0\) is a fixed point, and let \(\{\lambda_i\}_{i=1}^d\subset \mathbb C\) be the eigenvalues of the Jacobian \(J(0)\). Then \(0\) is

  • asymptotically stable if \(\max_{i\in[d]}|\lambda_i|<1\)
  • unstable if \(\max_{i\in[d]}|\lambda_i|> 1\)

Proof sketch.

  • When Jacobian \(J(0)\) is stable, can define \(P = \sum_{t=0}^\infty (J(0)^\top)^t J(0)^t\). Then use \(s^\top P s\) to construct a strict Lyapunov function for \(s_{t+1} = F(s_t)\).

\(\mathbb C\)

stable

unstable

inconclusive

\(1\)

Outline

"Why we do it"

  1. Linear approximations
  2. Lyapunov stability theory
  3. Limit cycles and chaos
  • Nonlinear dynamics can display complex behaviors beyond stable/unstable equilibria
  • Example: logistic map (simulation notebook)   $$s_{t+1}= rs_t(1-s_t)$$

Complex dynamic behaviors

  • \(r=0.5\) stable at \(s=0\)

  • \(r=1.5\) stable at \(s=\frac{1}{3}\)

  • \(r=3.25\) limit cycle

  • \(r=3.75\) chaotic

Next time: learning nonlinear models

Recap

References: Bof, Carli, Schenato, "Lyapunov Theory for Discrete Time Systems"

  • Nonlinear stability via linearization
  • Theoretical tool: Lyapunov functions

Announcements

  • Second assignment due Thursday
  • My office hours:
    • informally after lecture walking to Gates
    • by appointment after lecture in my office 424 Gates