class: center, middle, inverse, title-slide .title[ # Fundamentals of dynamical system ] .author[ ### Wenbo Lyu ] .date[ ### 2026/10/29 ] --- <style> .remark-slide-content { font-family: "Arial", sans-serif; color: #243238; background-color: #fbfcfc; font-size: 25px; padding: 1em 2em; } h1, h2, h3 { color: #1c5253; font-family: "Arial", sans-serif; font-weight: 600; } .title-slide { background-color: #1c5253; color: #ffffff; } .title-slide h1, .title-slide h2, .title-slide h3 { color: #ffffff; } .small { font-size: 0.72em; } .tiny { font-size: 0.62em; } .math { font-size: 1.05em; } .remark-slide-number { color: #66777a; } </style> A **dynamical system** describes the evolution of a system's state over time, governed by a specified rule that may or may not depend explicitly on time. -- **Continuous time: differential equation** $$ \frac{d\mathbf{x}(t)}{dt} = \mathbf{f}(\mathbf{x}(t),t) $$ -- **Discrete time: iterated map** $$ \mathbf{x}_{t+1} = \mathbf{F}_t(\mathbf{x}_t) $$ -- - `\(\mathbf{x} \in \mathbb{R}^n\)`: the system state. - `\(\mathbf{f}\)`: the instantaneous rate of change. - `\(\mathbf{F}_t\)`: the rule mapping the current state to the next state. -- > *The evolution rule may or may not depend explicitly on time, distinguishing autonomous from non-autonomous systems.* --- **Autonomous system:** the evolution rule does not explicitly depend on time. -- $$ \frac{d\mathbf{x}}{dt}=\mathbf{f}(\mathbf{x}) \qquad \mathbf{x}_{t+1}=\mathbf{F}(\mathbf{x}_t) $$ -- **Non-autonomous system:** the evolution rule explicitly depends on time. -- $$ \frac{d\mathbf{x}}{dt}=\mathbf{f}(\mathbf{x},t) \qquad \mathbf{x}_{t+1}=\mathbf{F}_t(\mathbf{x}_t) $$ -- - Autonomous dynamics use the same rule at every time point. -- - Non-autonomous dynamics can change with time, for example because of seasonal forcing or a changing environment. -- > *Here, we focus on autonomous systems for simplicity.* -- Discrete-time dynamics often describe a continuous-time system that is sampled discretely in time, so that `\(\mathbf{x}_{t} = \mathbf{x}(t\Delta t)\)` with sampling time `\(\Delta t\)`. --- **Linear dynamics:** the next state is a linear function of the current state. -- $$ x_{t+1}=a x_t $$ -- **Nonlinear dynamics:** the next state depends nonlinearly on the current state. -- $$ x_{t+1}=F(x_t) $$ -- Nonlinear dynamics can produce richer behaviors, including multiple equilibria, oscillations, and deterministic chaos. -- **Linear map** $$ x_{t+1}=0.5x_t,\qquad x_0=0.8 $$ $$ 0.8\rightarrow0.4\rightarrow0.2\rightarrow0.1\rightarrow\cdots $$ -- **Nonlinear map: logistic map** $$ x_{t+1}=r x_t(1-x_t),\qquad r=3.8,\quad x_0=0.2 $$ --- Consider a more general iterated map: $$ `\begin{equation} x_{t+1}=F(x_t,x_{t-1},\ldots,x_{t-p}) \end{equation}` $$ -- In practice: - ** `\(F\)` is often unknown.** We may only have measurements of the system's dynamics. - **Even when `\(F\)` is known,** nonlinear systems may lack explicit analytical solutions, making their long-term behavior difficult to characterize. -- ### How can we study such dynamics? -- - **State-space-based methods:** reconstruct and analyze the latent dynamics directly from observations. -- - **Koopman operator theory:** reformulate nonlinear dynamics through the evolution of observables under a linear, generally infinite-dimensional operator. --- The evolution rule `\(\mathbf{F}\)` may be unknown or difficult to model directly. -- **Koopman operator theory:** Instead of tracking the state itself, we **lift** the state into a space of observable functions. $$ \mathbf{z}_t = \boldsymbol{\psi}(\mathbf{x}_t) $$ where `\(\boldsymbol{\psi}\)` represents a set of observable functions (the "lifting map"). -- The Koopman operator `\(\mathcal{K}\)` governs the evolution of these observables: $$ `\begin{equation} (\mathcal{K}\boldsymbol{\psi})(\mathbf{x}_t) = \boldsymbol{\psi}(\mathbf{F}(\mathbf{x}_t)) = \boldsymbol{\psi}(\mathbf{x}_{t+1}) \end{equation}` $$ -- Under a finite-dimensional approximation, this yields a linear system: $$ `\begin{equation} \mathbf{z}_{t+1} \approx \mathbf{K}\mathbf{z}_t \end{equation}` $$ -- - **Nonlinear in the original state, linear in the observable space.** - Although exactly infinite-dimensional, the Koopman operator enables characterization of nonlinear dynamics without specifying `\(\mathbf{F}\)`. - **Neural networks** can learn useful observables `\(\boldsymbol{\psi}\)` and approximate the linear evolution matrix `\(\mathbf{K}\)` directly from data. --- **State space:** a space in which each point represents the system's state, and system evolution traces a trajectory through this space. - Each coordinate represents a state variable. - The trajectory reveals the geometric structure of the dynamics. -- **Embedding:** represent observations as points in a reconstructed state space. When only a single variable `\(x_t\)` is observed, delayed measurements can be used to construct delay coordinates: -- $$ `\begin{equation} \mathcal{M}_x = \left[x_t,\;x_{t-\tau},\;\ldots,\;x_{t-(E-1)\tau}\right] \end{equation}` $$ - `\(E\)`: embedding dimension. - `\(\tau\)`: step of time delay. -- Under suitable conditions, the reconstructed space can preserve key geometric and dynamical properties of the underlying system. -- > the system evolves according to a well-defined dynamical law; the reconstructed space has enough dimensions to unfold the underlying attractor; the data sufficiently resolve the underlying dynamics. --- The Lorenz equations are a classic continuous-time nonlinear system: $$ `\begin{aligned} \dot{x} &= \sigma(y-x),\\ \dot{y} &= x(\rho-z)-y,\\ \dot{z} &= xy-\beta z. \end{aligned}` $$ -- For `\(\sigma=10\)`, `\(\rho=28\)`, and `\(\beta=8/3\)`, trajectories approach a bounded, aperiodic structure known as the **Lorenz attractor**. -- <div class="plotly 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