Dynamical system is the study of the evolution of states. While dynamical systems are abstract objects, they can be studied via their representations, e.g. differential equations. Governing Equations of Dynamic System is a closed set of differential equations.

## Concepts

Dynamical system $(G, X, \phi)$ is a continuous action of a topological group on a topological space: $(G, ∗, \mathcal{T})$, $(X, \mathcal{T})$, $\phi \in C(G \times X, X)$. Continuous-time dynamical system or flow $(\mathbb{R}, X, \phi)$ is a continuous action of the real group. Discrete-time dynamical system, map, or cascade $(\mathbb{Z}, X, \phi)$ is a continuous action of the integer group. If the action is on a semi-group, i.e. without identiy and inverse elements, we add "semi-" to its name, e.g. semi-flow and semi-cascade.

For a dynamical system, we call the topological space its phase space or state space, the first variable its evolution parameter or time, the second variable its state. Evolution map $\phi_t(x)$ at a time is the transformation on the state space defined by the action with time fixed: $\phi_t(x) = \phi(t, x)$. Time-one map $\phi_1(x)$ is the evolution map at time one. Trajectory or solution curve $\phi_x(t): G \to X$ through a point is the parameterized curve defined by the action with state fixed at the point: $\phi_x(t) = \phi(t, x)$. I will use superscripts + and - to denote forward and backward entities, respectively, e.g. forward trajectory $\phi_x^+$. Orbit $\gamma_x$ through a point is the image of the trajectory through the point: $\gamma_x = \phi_x(G)$. Phase portrait $\{\phi_x: x \in X\}$ of a dynamical system is its collection of trajectories. Orbit structure refers to the topological invariant properties of the phase portrait. Fixed point, stationary point, or equilibrium of a dynamical system is a point in its state space where the action has no effect: $\phi_x^{-1}(t) = x$. Periodic orbit or closed orbit is the image of a nonconstant periodic trajectory: $\exists t \in G \setminus \{e\}$, $\phi_x^{-1}(x) = \{i t : i \in \mathbb{Z}\}$. An orbit in a dynamical system on a Hausdorff space is compact if and only if it is a fixed point or a periodic orbit.

By definition, a dynamical system cannot have memory effect, i.e. the outcome of a dynamical system at each moment of time is uniquely determined by its initial state and not any previous state. Autonomous dynamical system is a dynamical system whose outcome only depends on the duration since a given state, regardless of the moment the given state is registered: $\forall (t', x') \in \phi^{-1}(x)$, $\phi(t' + t, x') = \phi(t, x)$; or equivalently, its evolution map at any time equals its time-one map repeated for that many times: $\phi_t = \phi^t$, i.e. $\phi(t, x) = \phi^t(x)$, the superscript is interpreted as an exponent for cascades, and as integral bound for flows. An autonomous cascade $(\phi^t)_{t \in \mathbb{Z}}$ is determined by the time-one map. An autonomous flow $(\phi^t)_{t \in \mathbb{R}}$ is also called a one-parameter group of homeomorphisms. Unless explicitly stated, any dynamical system here is assumed to be autonomous.

Topological conjugacy between automorphisms (discrete dynamical systems) of homeomorphic topological spaces is a homeomorphism between the spaces that reproduces the automorphisms: $f: X \cong X$, $g: Y \cong Y$, $h: X \cong Y$, $g(h(x)) = h(f(x))$. Flow map $h: X \mapsto Y$ from a flow to another on separate topological spaces is a continuous map between the underlying topological spaces such that with a positive scaling of time it reproduces the other flow on its range: $\phi \in C(\mathbb{R} \times X, X)$, $\psi \in C(\mathbb{R} \times Y, Y)$, $h \in C(X, Y)$, $\exists a > 0$: $\psi(a t, h(x)) = h(\phi(t, x))$. Topological equivalence between flows on homeomorphic topological spaces is a flow map that is also a homeomorphism. We call two flows on homeomorphic topological spaces topologically equivalent if there is a flow equivalence between them.

### Derived Dynamical Systems

Embedded cascade in a flow is a cascade whose time-one map equals an evolution map of the flow: $\exists t \in \mathbb{R}$: $\psi^1 = \phi^t$. A homeomorphism on a topological space can be embedded in a flow on the space only if it is isotopic to the identity map, because any such flow is an isotopy between its time-zero map (the identity map) and its time-one map.

Suspension of a cascade on a topological space is the flow on the quotient space of the product space of the real line and the space, which identifies every time-state pair with inverse integer differences and iterated states, and the flow maps between the equivalence classes by time addition: $\psi^1 \in \text{Homeo} X$, $Y = (\mathbb{R} \times X) / \sim$, where $[(\tau, x)] = {(\tau - i, \psi^i (x)) : i \in \mathbb{Z}}$, $\phi(t, [(\tau, x)]) = [(\tau + t, x)]$. Every cascade is embedded in the restriction of its suspension to any cross section in time: $\forall \tau \in \mathbb{R}$, $\phi^1([(\tau, x)]) = [(\tau, \psi^1 x)]$.

## Topics

### Measurable Dynamics

Ergodic theory or measurable dynamics is the study of statistical properties of measurable dynamical systems. Many properties of invariant measures can be seen as statistical counterparts to topological invariants. Ergodic hypothesis by Boltzmann asserts that for certain dynamical systems the time average of their properties is equal to the average over the entire space.

Invariant measure for a measurable transformation on a metric space with Borel sigma-algebra is a measure on the space that is preserved under preimage of the map: $(X, d(\cdot, \cdot), \mathcal{T}, \mathcal{B(T)}, \mu)$, $\psi: X \mapsto X$, $\mu = \mu \circ \psi^{-1}$. We call the invariant measure a $\psi$-invariant measure. Krylov—Bogolubov Theorem: Any continuous map on a metrizable compact space has an invariant Borel probability measure. A dynamical system may have lots of invariant measures, many possess some complex behavior. Ergodic measure is an invariant measure that is irreducible, i.e. there is no partition of the underlying space into subspaces of positive measure where the measure remains to be invariant for the transformation.

Time average $I_x(f)$ of a real-valued function on a metrizable space on a forward trajectory of a dynamical system is the limit of the average of the image of the trajectory, if exists: $f: X \mapsto \mathbb{R}$, $x \in X$, for a cascade, $I_x(f) := \lim_{n \to \infty} 1/n \sum_{i=0}^{n-1} f(\psi^i(x))$; for a flow, $I_x(f) := \lim_{t \to \infty} 1/t \int_0^t f(\phi_x(t)) dt$. By Riesz representation theorem, time averages of continuous functions on a forward trajectory, if exist, can be written as the integral of the functions w.r.t. a unique Borel probability measure: $\exists! \mu_x$: $\forall f \in C(X)$, $I_x f = \int_X f d \mu_x$; this is an invariant measure. Asymptotic measure or asymptotic distribution $\mu_x$ of a trajectory of a dynamical system is the invariant Borel probability measure w.r.t. which integrals of continuous functions equal their time averages. Birkhoff Ergodic Theorem: If the measure of a probability space is invariant for some measurable dynamical system, the time average of every Lebesgue integrable function exists on a full-measure set. Every continuous map on a metrizable compact space has an ergodic invariant Borel probability measure.

Sinai-Ruelle-Bowen measure [@Young2002] for a dynamical system on a metrizable space is an invariant measure supported on a hyperbolic attractor such that the time averages of every continuous real-valued function on the space on forward trajectories started from almost all points (w.r.t. the Lebesgue measure) in a neighborhood of the attractor equals the integral of the function w.r.t. the measure [@Guckenheimer1983]: $\exists U(A)$, $\forall f \in C(X)$: $\{x \in U: I_x(f) = \int_A f d\mu\} \approx U$. Sinai-Ruelle-Bowen Theorem [@Sinai1972; @Bowen and Ruelle, 1975]: Every $C^2$ dynamical system has a unique SRB measure.

### Topological Dynamics

Topological dynamics is the study of topological transformation groups. Topological transformation group $(G, X, \alpha)$ is an action $\phi: G \times X \mapsto X$ of a topological group $(G, ∗, \mathcal{T})$ on a topological space $(X, \mathcal{T})$ such that it is equivalent to a continuous homomorphism of the topological group to a topological group $(\text{Homeo} X, \circ, \mathcal{T})$ based on the homeomorphism group on the space [@Irwin1980]: $\alpha(g)(x) = \phi(g, x)$, $\alpha \in \text{Hom}(G, \text{Homeo} X)$. To make the homeomorphism group a topological group, one may use the compact-open (C.O.) topology if the underlying space is locally-connected locally-compact Hausdorff or compact Hausdorff, or use the Arens G-topology if it is just locally-compact Hausdorff. Lie transformation group is an action of a Lie group on a smooth manifold such that it is equivalent to a smooth homomorphism of the Lie group to a Lie group based on the diffeomorphism group on the space. But it is not straightforward to give the diffeomorphism group a Lie group structure [@Leslie1967].

### Differentiable Dynamics

Smooth dynamical system $(G, M, \phi)$ is a smooth action of a Lie group on a smooth manifold. $(G, ∗, \mathcal{T, A})$, $(M, \mathcal{T, A})$, $\phi \in C^\infty(G \times M, M)$. Differentiable dynamical system is analogous to smooth dynamical system, but only requires continuous differentiability up to a certain order, e.g. a $C^k$ dynamical system is one that is a $C^k$ map: $\phi \in C^k(G \times M, M)$, $k \in \mathbb{N}_+$. Every differentiable flow determines a vector field on its state space at every time instance as the velocity of state variation: $v = \frac{\partial \phi}{\partial t}$.

### Symplectic Dynamics

Symplectic dynamics or Hamiltonian dynamics is the study of groups of diffeomorphisms on a symplectic manifold that preserve the symplectic form. It generalizes the study of Hamiltonian differential equations of classical mechanics.

Hamilton's canonical equations are written as: $\begin{cases} \dot q = \frac{\partial H}{\partial p} \\ \dot p = -\frac{\partial H}{\partial q} \end{cases}$, where $H \in C^r(U)$, $r \ge 2$, $U \subset \mathbb{R}^{2n}$, $(q, p) \in U$; or in a more compact notation, $\dot x = B d H$, where $x = (q, p)$, $B = \begin{pmatrix} 0, I_n \\ -I_n, 0 \end{pmatrix}$. Hamiltonian vector field...

Integrable system...

## Ordinary Differential Equations

Ordinary differential equation (ODE) is an equation relating an unknown univariate function and its derivatives of various orders: $f(t, x^{(i)})_{i=0}^k = 0$, $x \in C^k(\mathbb{R}, \mathbb{R}^n)$. Order of a differential equation is the order of the highest derivative that occurs explicitly in the equation. The vector fields determined by a differentiable flow can be written as a system of ODEs. For an autonomous flow on a Euclidean space, its velocity field can be written as a system of first-order ODEs without explicit time variable: $\dot x = f(x)$, $f \in C(\mathbb{R}^n, \mathbb{R}^n)$.

The theory of linear ODEs is relatively complete. Perturbation methods can be used for weakly nonlinear problems but may fail when the series diverge. Qualitative Theory of Differential Equations

### Linear ODEs

A system of linear first-order ODEs can be written as $\dot x = A x$, $A \in M_n(\mathbb{R})$. The solution of a linear first-order ODE system, given any initial value, is the exponential of the time-scaled coefficient matrix acting on the initial value: $x(t) = e^{t A} x(0)$. Linear flow $e^{t A} x$ associated with a linear first-order ODE system with coefficient matrix $A$ is the flow defined by the action of the exponential of the time-scaled coefficient matrix.

General Theory: Existence and uniqueness: Every locally Lipschitz vector field on a manifold has a unique local integral curve at every point. Every continuously differentiable vector field on a compact $C^1$ manifold has global integral curves at every point, i.e. it generates a global flow. Continuity and differentiability. Continuous dependence: The difference between integral curves of a Lipschitz vector field on a manifold is exponentially bounded by the Lipschitz constant: $\frac{|\phi_x(t) - \phi_y(t)|}{|x-y|} \le e^{Lt}$.

Solutions: 1-st order ODEs; Singular solutions; n-th order & system of ODEs, generalized eigenvector, fundamental solution matrix; Solution of some common ODEs. General solutions are not necessarily all the solutions. For example, ODE $y^2 + y'^2 = 1$ has general solution $y = \sin(x+c)$ and extra solutions (singular solutions) $y=\pm 1$.

Hyperbolic functions are analogs of trigonometric/circular functions: $\sinh x = (e^x - e^{-x}) / 2$, $\cosh x = (e^x + e^{-x}) / 2$. Relations to circular functions: $\cos(ix) = \cosh x$, $\sin(ix) = i \sinh x$. Differentiation: $(\sinh x)' = \cosh x$, $(\cosh x)' = \sinh x$. Complex functions: let $z = x + iy$, then $\cos z = \cosh y \cos x - i \sinh y \sin x$, $\sin z = \cosh y \sin x + i \sinh y \cos x$. The general solution to the ODE $\ddot{y} - y = 0$ are linear combinations of exponential functions: $y = c_1 e^x + c_2 e^{-x}$, which can be written as linear combinations of hyperbolic functions, $y = A \sinh x + B \cosh x$.

## Partial Differential Equations

Non-autonomous dynamical systems and partial differential equations. Partial differential equation (PDE) is an equation relating an unknown function and its derivatives of various orders: $f(u, \frac{\partial x}{\partial u^I})_{|I|=0}^k = 0$, $x \in C^k(\mathbb{R}^m, \mathbb{R}^n)$.

Classification of 2nd Order PDE

Solution methods:

Special functions: Bessel functions & Legendre polynomials

Reaction-diffusion system: Notes on Reaction-diffusion system

Discontinuity: Notes on Hyperbolic Conservation Laws

## References

1. [@Hirsch and Smale, 1974] Differential Equations, Dynamical Systems, and Linear Algebra.
• w/ Devaney, 2003/2013. Differential Equations, Dynamical Systems, and an Introduction to Chaos.
2. [@Irwin1980] Smooth Dynamical Systems.
3. [@Arnold1983] Geometrical Methods in the Theory of Ordinary Differential Equations.
4. [@Guckenheimer1983] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields.
5. [@Wiggins1990] Introduction to Applied Nonlinear Dynamical Systems and Chaos.
6. [@Kuznetsov1995] Elements of Applied Bifurcation Theory.
7. [@Katok1995] Introduction to the Modern Theory of Dynamical Systems.