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Vector Field on an arbitrary differentiable manifolds, and constitutes a specific choice of tangent vectors from a tangent hyperplane at each point of the manifold. A topological vector field $V : M \to T_\ast M$ of a manifold $M$, such that $V$ is a section of of the tangent bundle $T_\ast M$.
Flow
As with sufficiently smooth differential equations on $\mathbb{R}^n$, vector field on a manifold can be integrated to yield a flow. Given $V$ on $M$, the associated to $V$ is a family of diffeomorphisms $\varphi_t : M \to M$ such that:
- $\varphi_0(x) = x$ for all $x \in M$
- $\varphi_{t+\tau}(x) = \varphi_t(\varphi_\tau(x))$ for all $x \in M$ and $t, \tau \in \mathbb{R}$
- $\dot{\varphi_t}(x) = V(x)$
One thinks of $\varphi_t(x)$ as determining location of a particle starting at $x$ and moving via velocity field $V$ for $t$ units of time. For non-compact $M$ or insufficiently smooth $V$, one must worry about existence and uniqueness solutions. Smooth vector fields on compact manifolds yield smooth flows whose dynamics links topology and differential equations.
Equilibria
Primal objects of inquiry in dynamics are the equilibrium solutions which correspond to fixed points of a vector field: a vector field is said to have an equilibrium at a point $x$ if $V(x) = 0$. An isolated fixed point may have several qualitatively distinct features based on stability. We quantify stability of a fixed point $p \in X$ point by defining stable and unstable manifolds:
- Stable manifold: $W^s(p) = \{x \in X\; : \; \varphi_t(x) \to p,\; \text{as}\; t \to \infty\}$
- Unstable manifold: $W^u(p) = \{x \in X\; : \; \varphi_t(x) \to p,\; \text{as}\; t \to -\infty\}$
- Fixed point is a sink when its open neighborhood is a subset of its stable manifold.
- Fixed point is a source when its open neighborhood is a subset of its unstable manifold.
- Fixed point is a saddle if it is contained in a respective open neighborhoods of both stable and unstable manifolds.
A critical point can also be classified as degenerate if the second derivative (or Hessian) is degenerate (has a zero determinant). If there are no degenerate critical points the gradient field $-\nabla V$ is a section of a tangent bundle $T_\ast M$ transverse to the zero section. Degenerate points can complicate the analysis and lead to different topological behavior. In those cases, additional tools and techniques are needed to study the manifold's topology. Nondegenerate critical points are the basis elements of Morse theory.
Orbits
A periodic orbit of a flow is an orbit $\varphi_t(x)\}_{t \in \mathbb{R}}$ satisfying $\varphi_{t+t_0}(x) = \varphi_t(x)$ for some fixed $t_0$ and all $t \in \mathbb{R}$. Periodic orbits are submanifolds of $M$ diffeomorphic to $S^1$. One may classify periodic orbits as being stable, unstable, saddle-type, or degenerate. The existence of periodic orbits, in contrast to equilibria, is often computationally intractable. On $S^3$ it is possible to find smooth, fixed-point-free vector fields whose set of periodic orbits is all of $S^3$ or $\varnothing$.
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