For an autonomous system \[ \dot x=f(x), \] uniqueness turns solutions into a local action of time on state space. Phase portraits describe orbits geometrically, independent of their time parametrization.
Local and global flows
Definition. Let \(\phi_t(x)\) denote the solution at time \(t\) starting from \(x\), wherever defined. It is a local flow if \[ \phi_0(x)=x,\qquad \phi_{t+s}(x)=\phi_t(\phi_s(x)) \] whenever both sides are defined.
Proof of the flow law. Both \(t\mapsto\phi_{t+s}(x)\) and \(t\mapsto\phi_t(\phi_s(x))\) solve the same autonomous initial-value problem at \(t=0\) with value \(\phi_s(x)\). Uniqueness makes them equal.
If every solution exists for all \(t\in\mathbb R\), the vector field is complete and \(\phi_t\) is a one-parameter group of homeomorphisms; for \(f\in C^k\), these are \(C^k\) diffeomorphisms.
Variational equation and flow Jacobian
Theorem. If \(f\in C^1\), the derivative \[ J(t,x)=D_x\phi_t(x) \] solves \[ \dot J(t,x)=Df(\phi_t(x))J(t,x),\qquad J(0,x)=I. \]
Proof. Differentiate the integral equation \[ \phi_t(x)=x+\int_0^tf(\phi_s(x))\,\mathrm{d}s \] with respect to \(x\), justified by differentiable dependence on initial data.
Corollary (Liouville formula). \[ \det J(t,x) =\exp\left( \int_0^t\operatorname{div}f(\phi_s(x))\,\mathrm{d}s \right). \]
This follows from Jacobi's determinant formula. Divergence-free flows preserve volume; negative divergence contracts it.
Orbits, equilibria, and periodic trajectories
An equilibrium satisfies \(f(x^*)=0\). A nonstationary orbit is \(\{\phi_t(x):t\in I_x\}\). In the plane, uniqueness prevents distinct orbits from crossing. A periodic orbit has \(\phi_T(x)=x\) for some least \(T>0\). Its stability is studied by the derivative of a Poincaré return map, which removes the neutral direction along the orbit.
Omega-limit sets
For a forward precompact orbit, \[ \omega(x)= \bigcap_{T>0}\overline{\{\phi_t(x):t\ge T\}} \] is nonempty, compact, connected, and invariant. It records asymptotic states. A strict Lyapunov function forces omega-limit sets into the set where its derivative vanishes.
Poincaré--Bendixson in the plane
For a \(C^1\) planar flow, a nonempty compact omega-limit set containing no equilibrium is a periodic orbit. This strong classification is special to two dimensions; in dimension three, recurrent sets and chaotic dynamics are possible.