Qualitative (or geometric) theory of ordinary differential equations, started by Henri Poincaré in 1881-1885, is the study of diffeomorphic invariants of the dynamical systems they represent. The phase space of second order ODEs is the Euclidean plane, the simplest manifold with nontrivial trajectories. Plane analysis of second order ODEs is the study of their fixed points and peoriodic orbits. Notes on qualitative theory of ODE, boundary value problem.

Every linear first-order ODE system on the plane with a nonsingular coefficient matrix has a unique fixed point at the origin: $\dot y = A y$, $A \in \text{GL}(2, \mathbb{R})$. If the coefficient matrix has two real eigenvalues, e.g. $A = \text{diag}(\lambda_1, \lambda_2)$, then the ODE system has general solution: $y = \text{diag}(e^{\lambda_1 t}, e^{\lambda_2 t}) y_0$. If the coefficient matrix has a pair of complex eigenvalues, e.g. $A = \begin{pmatrix} a & b \\ -b & a \end{pmatrix}$, then the ODE system has general solution: $y_1 = c e^{at} \sin(bt+\phi)$, $y_2 = c e^{at} \cos(bt+\phi)$.

Linearization of an ODE system $\dot x = f(x)$: find fixed points, $f(x) = 0$; compute Jacobian matrix at the fixed points, $A = \nabla f(x_0)$; Jordan decompose the matrix, $A = P J P^{-1}$; solve the linear ODE system, $\dot \xi = J \xi$ where transformed coordinates $\xi = P^{-1} (x - x_0)$ and thus $x= P \xi - x_0$.

**Node** of a differentiable flow is a fixed point
where the differential of the vector field has all positive or negative eigenvalues.
**Saddle** is analogous to a node, but has both positive and negative eigenvalues and nothing else.
**Focus** of a differentiable 2-dimensional flow is a fixed point
where the differential of the vector field has a pair of complex eigenvalues with nonzero real parts.
Every focus is topologically equivalent to a node.
**Center** of a differentiable even-dimensional flow is a fixed point
where the differential of the vector field has all purely imaginary nonzero eigenvalues.
For linear systems, peoriodic orbits exist around a center and have the same period.
For a nonlinear system, a center found in its linearized form may not be an actual center;
but symmetry of the original system about y-axis suffice to preserve this property.

Figure: Topological classification of fixed points of dynamical systems on 2-manifolds.

Hyperbolicity in dynamical systems theory means nondegeneracy.
**Hyperbolic fixed point** of a differentiable flow
is a fixed point where the differential has no eigenvalues on the imaginary axis.
A special case of hyperbolic flow corresponds to area-preserving vector fields on the plane,
which have determinant one and the trajectories are the axes and standard hyperbolas
$xy = c$, which is why such dynamical systems are called hyperbolic.
**Hyperbolic automorphism** of a vector space
is a bijiective linear transformation on the space with no eigenvalues on the complex unit circle:
$T \in \text{GL}(V)$, $\sigma(T) \cap \mathbb{S}^1 = \emptyset$.
The set of all hyperbolic automorphisms of a vector space is denoted by $\text{HL}(V)$.
**Hyperbolic fixed point** of a differentiable cascade
is a fixed point where the differential is a hyperbolic automorphism.
*Topological classification of hyperbolic fixed points*:
Differentiable flows on n-manifolds are locally topologically equivalent at hyperbolic fixed points
if and only if the differentials of the vector fields at the fixed points
have the same number of eigenvalues with negative (or positive) real parts;
the theorem is analogous for differentiable cascades, but for their own differential,
and is determined by the number of eigenvalues outside (or within) the complex unit circle.

**Elliptic fixed point** of a symplectic dynamical system is a fixed point
where the differential of the Hamiltonian vector field has all purely imaginary nonzero eigenvalues.
A large class of elliptic fixed points are Lyapunov stable.

**Transverse fixed point** of a differentiable flow is a fixed point
where the differential of the vector field does not have eigenvalue zero.
**Transverse fixed point** of a differentiable cascade is a fixed point
where the graph intersects transversely with the diagonal:
$(x, x) \in \Gamma{\psi} \pitchfork I$, where $I = \{(x, x) : x \in M\}$.
A fixed point of a differentiable cascade is transverse if and only if
the differential of the map at the point does not have eigenvalue one.
Hyperbolic fixed points and elliptic fixed points are transverse.

Orbits of the Duffing equation

Homoclinic orbits

**Heteroclinic orbit** is a trajectory that connect two fixed points.
**Homoclinic orbit** is a trajectory that connect the same fixed point.
Homo- and hetero-clinic orbits typically approach the fixed points in infinite time,
forward and backward, because in an eigenspace at a hyperbolic fixed point
the solutions are exponential: $\dot y = \lambda y$, $y = e^{\lambda t}$.
**Duffing equation** can be written as $\ddot{u} + u - u^3 = 0$;
or as a first-order ODE system, $\begin{cases} \dot u = v \\ \dot v = -u + u^3 \end{cases}$.
The duffing equation has a center at the origin and two saddles at $(\pm 1,0)$;
it has heteroclinic orbits.
The following system has homoclinic oribts: $\begin{cases}
\dot x = y + y(1-x^2)[y^2-x^2(1-\frac{x^2}{2})] \\ \dot y = x(1-x^2) - y[y^2-x^2(1-\frac{x^2}{2})]
\end{cases}$.
It also has three fixed points: a saddle at the origin and two unstable foci at $(\pm 1,0)$.

Poincare-Bendixson theorem

**First return function** $\rho: U \mapsto \mathbb{R}$
for an open disc $Y$ transverse to a peoriodic orbit $\gamma_x$ in a dynamical system
is the continuous real-valued function on a neighbourhood of the disc
that gives the first time for every point in the neighbourhood to return to the disc:
${\phi(\rho(y), y) : y \in U} \subset Y$,
$\rho(x) = \tau$ which is the period of the peoriodic orbit.
**Poincare map**, **first return map**, or **section map** $f: U \mapsto Y$
for an open disc transverse to a peoriodic orbit in a dynamical system
is the map that gives the location of first return for every point in a neighbourhood of the disc:
$f(y) = \phi(\rho(y), y)$.
The differential of a Poincare map is linearly conjugate
the differential of the period-one transformation restricted to an invariant tangent hyperplane.

**Limit cycle** is a peoriodic orbit that any trajectory nearby either goes towards it, or moves away.
**Sink** is a periodic orbit where all Lyapunov exponents are negative.
Van der Pol equation has a sink.
*Poincare-Bendixson Theorem*:
For a planar dynamical system,
if a ring-shaped region between two simple closed curves has no fixed point within,
and any trajectory that intersects with the two boundary curves enters the region,
then the region has at least one limit cycle.
The conclusion holds if the inner boundary curve shrinks into an unstable node.

**Hyperbolic peoriodic orbit** in a dynamical system is a peoriodic orbit such that
the differential of the transformation at one period of the orbit
restricts to a hyperbolic automorphism on a tangent hyperplane at a point of the orbit:
$T_x \phi^t = H \oplus \hat v_x \hat v_x^T$, $H \in \text{HL}(\mathbb{R}^{n-1})$,
$\hat v_x = v_x / \|v_x\|$.
A peoriodic orbit is hyperbolic if and only if
a point on the orbit is a hyperbolic fixed point of a Poincare map at the point:
$d f_x \in \text{HL}(\mathbb{R}^{n-1})$.
**Hyperbolic periodic point** of a differentiable cascade
is a periodic point where the differential of the period-one map is a hyperbolic automorphism:
$T_x \phi^k \in \text{HL}(\mathbb{R}^n)$, where $k$ is the period.
The orbit of a hyperbolic periodic point is called a **hyperbolic periodic orbit**.

**Transverse periodic orbit** of a differentiable flow is a fixed point
where the differential of the period-one map has one as a simple eigenvalue;
or equivalently, it is a transverse fixed point for a Poincare map.

Local stability of fixed points of nonlinear dynamical systems can be defined in various ways.
**Lyapunov stable** fixed point is one such that trajectories started near it stay nearby forever:
[@Lyapunov1892] $\forall \varepsilon > 0$, $\exists \delta \in (0, \varepsilon)$:
$d(x, x^∗ ) < \delta$ then $\sup_{t \ge 0} d(\phi_x(t), x^∗) < \varepsilon$.

**(Locally) asymptotically stable** fixed point is one
that is Lyapunov stable and (locally) **attractive**,
i.e. trajectories started near it converge to it:
$\exists \delta > 0$: $d(x, x^∗) < \delta$ then $\lim_{t \to \infty} d(\phi_x(t), x^∗) = 0$.
Notice that Lyapunov stability is distinct from local attractiveness.
**Basin of attraction** or **basin** $B(x)$ of an asymptotically stable fixed point
of a dynamical system is the union of all trajectories that tend toward the fixed point.
**Globally asymptotically stable** fixed point is one that is Lyapunov stable
and **globally attractive**, i.e. all trajectories converge to it.

**Exponentially stable** fixed point is one that is Lyapunov stable and
(locally) exponentially attractive, i.e., trajectories started near it converge to it exponentially:
$\exists \delta, \alpha, \beta > 0$: $d(x, x^∗) < \delta$ then
$d(\phi_x(t), x^∗) / d(x, x^∗) \le \alpha e^{-\beta t}$.

Liapunov exponent...

**Lyapunov function** $V(x)$ for a differentiable flow with a fixed point at the origin
is a locally positive-definite, continuously differentiable function
with locally negative-semidefinite time derivative:
$V \in C^1(\mathbb{R}^n, \mathbb{R}_{\ge 0})$; $V(x) = 0$ then $x = 0$;
$\dot x = f(x)$, $(\nabla V, f) \le 0$.
Lyapunov function is analogous to the energy of a dissipative physical system,
but is not unique and thus easier to find.
A fixed point is Lyapunov stable if and only if the dynamical system has a Lyapunov function.
A fixed point is locally asymptotically stable if and only if
the dynamical system has a Lyapunov function whose time derivative is locally negative-definite.
A fixed point is globally asymptotically stable
if the dynamical system has a radially unbounded global Lyapunov function
whose time derivative is globally negative-definite.

**Global invariant set** for a dynamical system
is the subset $S$ of its state space whose orbits stay within itself:
$\bigcup_{x \in S} \phi_x(G) \subset S$.
Every fixed point or peoriodic orbit is a global invariant set.
A dynamical system can be restricted to any of its global invariant sets.
**Local invariant set** for a dynamical system
is the subset of its state space whose local orbits stay within itself:
$\exists t > 0$: $\bigcup_{x \in S} \phi_x([-t, t]) \subset S$.
**Local stable set** $W^s_{\text{loc}}(x^∗)$ of a critical point
is a subset of a neighborhood $U$ of the point
where forward trajectories stay within and tend toward the point:
$W^s_{\text{loc}}(x^∗) = \{x \in U : \phi_x([0, \infty)) \subset U,
\lim_{t \to \infty} \phi_x(t) = x^∗\}$.
**Local unstable set** is defined analogously but backwards:
$W^u_{\text{loc}}(x^∗) = \{x \in U : \phi_x((-\infty, 0]) \subset U,
\lim_{t \to -\infty} \phi_x(t) = x^∗\}$. [@Guckenheimer1983, p. 13]

**Invariant subspace** for a linear dynamical system
is an invariant set such that it is a vector subspace $V$ of the state space.
**Stable**, **unstable**, and **center** invariant subspaces $V^s, V^u, V^c$
of a linear dynamical system are the spans of the eigenspaces associated with
eigenvalues that have negative, positive, and zero real parts, respectively.

**Invariant manifold** for a dynamical system
is an invariant set that is a topological manifold as a topological subspace of the state space.
**Stable manifold** $W^s$, or **contracting manifold** [@Arnold1973],
is an invariant manifold where forward trajectories are bounded.
**Unstable manifold** $W^u$, or **dialating manifold**, is analogously defined but backwards.
*Hadamard-Perron Theorem* or *stable-unstable manifold theorem* for hyperbolic fixed point:
For a hyperbolic fixed point of an autonomous dynamical system,
its local stable and unstable invariant sets are submanifolds as smooth as the vector field,
and their tangent spaces coincide with the stable and unstable invariant subspaces
of the linearized dynamical system at the point:
$T_x W^s_\text{loc}(x) = V^s(x)$; $f \in C^k$ then $W^s_\text{loc}(x) \in C^k$.
Local stable manifold can be extended to a global stable manifold by flowing backwards,
and analogously for local unstable manifold but forward:
$W^s(x) = \bigcup_{t \le 0} \phi_t (W^s_\text{loc}(x))$.
Global stable and unstable manifolds need not be submanifolds.
**Center manifold** $W^c$ of a fixed point of a dynamical system
is a local invariant manifold whose tangent space at the point
is the center invariant subspace of the linearized dynamical system at the point.
*Center manifold theorem* [@Guckenheimer and Holmes, 1983]:
Every non-hyperbolic fixed point of a smooth dynamical system has a center manifold.
Center manifold at a fixed point need not be unique.

**ω-limit set** $\omega_x$ of a trajectory of a dynamical system
is the set of limit points of all convergent subsequences of the forward trajectory:
$\omega_x = \{\lim \phi_x(t_n) : \lim t_n = +\infty\}$;
or equivalently, $\omega_x = \bigcap_{t \in \mathbb{R}} \overline{\phi_x([t, \infty))}$.
**α-limit set** $\alpha_x$ is defined analogously but backwards.
**Recurrent point** of a dynamical system is a point that belongs to its own ω-limit set:
$\forall U(x)$, $\forall T > 0$, $\exists t > T$: $\phi(t, x) \in U$.
Chain recurrent point...

**Non-wandering point** or **nostalgic point** of a dynamical system is a point such that:
$\forall U(x)$, $\forall T > 0$, $\exists t > T$, $\exists y \in U$: $\phi(t, y) \in U$.
**Non-wandering set** or **Ω-set** $\Omega(\phi)$ of a dynamical system
is the set of all its non-wandering points.
The non-wandering set of a dynamical system is a close invariant set,
and it is non-empty if the phase space is compact.
Topological conjugacies and equivalences preserve non-wandering sets.

**Attracting set** of a dynamical system is a closed invariant set
that has a neighborhood where forward trajectories stay within and tend toward the invariant set:
$\exists U(A)$: $\forall x \in U$, $\gamma_x^+ \subset U$, $\lim_{t\to\infty} d(\phi(t, x), A) = 0$.
**Repelling set** is defined analogously but backwards.

**Indecomposable** closed invariant set is one where any point can be reached from any point
by a finite number of forward transformations with arbitrarily small connecting distance:
$\forall x, y \in A$, $\forall \varepsilon > 0$,
$\exists (x_i)_{i=0}^n \subset X$, $\exists (t_i)_{i=0}^{n-1} \subset G_+$:
$x_0 = x$, $d(\phi(t_i, x_i), x_{i+1}) < \varepsilon$, $x_n = y$.
**Attractor** of a dynamical system is an indecomposable, closed, forward-invariant set
whose every ε-neighborhood has a nonzero-volume subset
whose forward trajectories are within the neighborhood
and the ω-limit sets are within the invariant set [@Guckenheimer1983, Sec 1.6, 5.4]:
$\forall \varepsilon > 0$, $\exists S \subset U_\varepsilon(A)$, $v(S) > 0$,
$\bigcup_{x \in S} \gamma_x^+ \subset U_\varepsilon(A)$, $\bigcup_{x \in S} \omega_x \subset A$.
**Repellor** is defined analogously but backwards.
Attractor can be thought of as an attracting set that contains a dense orbit.
**Strange attractor** is an attractor that includes a transverse homoclinic orbit.

Types of attractors:

- stable fixed point;
- periodic points: discrete periodic sequence;
- sink: attractive periodic trajectory;
- limit torus: periodic components with incommensurate frequencies;
- strange attractor: contains a transversal homoclinic orbit; with a fractal dimension;

periodic attractor. expanding attractor.

Nonlinear dynamical systems are typically chaotic. Notes on chaos

Space of dynamical systems...
**C^s norm** $\|\cdot\|_s$ of a $C^s$ vector field on a subset of a normed space
is the maximum of the sup norms of the vector field and its partial derivatives up to order s:
$v \in \mathfrak{X}^s(X)$,
$\|v\|_s = \max_{|I| \le s} \left\|\frac{\partial v}{\partial x^I}\right\|_\infty$.
**C^s topology** on a space of $C^s$ vector fields is the topology induced by the $C^s$ norm.
**Perturbation** (摄动) of a $C^s$ vector field on a subset of a normed space
is an element in a neighborhood of the vector field in the $C^s$-normed space of vector fields:
$v' \in \mathfrak{X}^s(X)$, $\|v' -v\|_s < \varepsilon$.

*Perturbation Theorem* for fixed point:
For every neighborhood of a regular singular point in a $C^1$ vector field,
there is a neighborhood of the vector field
where all vector fields in has a unique singular point in the neighborhood of the point:
$v \in \mathfrak{X}^1(X)$, $v(x) = 0$, $d_x v \in \text{GL}(T_x X)$,
$\forall U(x) \subset X$, $\exists U(v) \subset \mathfrak{X}^1(X)$:
$\forall v' \in U(v)$, $\exists x' \in U(x)$, $v'(x') = 0$.

Perturbation method...

Averaging method...

**Structurally stable** dynamical system is a dynamical system
that is in the interior of its topological conjugacy/equivalence class:
$f \in (C^r(X), \mathcal{T}_r)$, $\exists U(f)$: $U \subset [f]$.

**Morse-Smale system** is a dynamical system whose non-wandering set consists of
a finite number of hyperbolic fixed points and hyperbolic peoriodic orbits,
and it has no heteroclinic or homoclinic orbit.
*Theorem* [@Peixoto1962]:
A dynamical system on a compact orientable 2-manifold is $C^1$ structurally stable
if and only if it is Morse-Smale; Morse-Smale systems are open and dense in $\mathfrak{X}^r(X)$.
*Theorem* [@Palis and Smale, 1970];
Every Morse-Smale system on a compact manifold is $C^1$ structurally stable.

**Anosov system**...
*Theorem* [@Anosov1962; @Anosov1967]:
Every Anosov system on a compact manifold is $C^1$ structurally stable.

**AS system** is a dynamical system that satisfies Axiom A and the strong transversality condition.
*Theorem* [@Robinson1974; @Robinson1976]:
Every AS system is $C^1$ structurally stable.

*C^0 density of C^1 structurally stable systems* [@Smale1971; @Shub1972]:
Every $C^r$ diffeomorphism on a compact manifold, $r \in \mathbb{N}_+$,
is $C^r$ isotopic to a $C^1$ structurally stable system by an arbitrarily $C^0$-small isotopy.

Other definitions of stability of dynamical systems have also been introduced,
the most interesting of which is Ω-stability.
**Ω-stable** dynamical system is a dynamical system
that is in the interior of its Ω-conjugacy/equivalence class:
$f \in (C^r(X), \mathcal{T}_r)$, $\exists U(f)$: $U \subset [f|_\Omega]$.
An Axiom A system on a compact manifold is Ω-stable if it has the no-cycle property.

**Bifurcation theory** (分岔; branching) is the study of
structural changes in a parametrized family of dynamical systems.
**Catastrophe theory** is the application of bifurcation theory to the study of
discontinuous or qualitative changes in long-term behavior of natural and social systems
under varying external conditions.

Parametrized first-order ODE: $\dot x = f(x, \varepsilon)$

Local bifurcation... Global bifurcation...

Codimension-one bifurcation... Saddle-node bifurcation... Hopf bifurcation [@Hopf1943]... The periodic attractor shrinks until it merges with the unstable focus to form a stable focus.

Index of fixed point...
*Poincare-Hopf Theorem*:
The index sum of a flow on a compact manifold equals the Euler characteristic of the manifold.

Slow variable... Slow motion... Slow manifold...

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- P Grindrod, Patterns and Waves (1991).
- J. Smoller, Shock Waves and Reaction Diffusion Equations (1994).