2 · ordinary differential equations

phase space, vector fields and flows

An autonomous system of ordinary differential equations on an open set $U\subset\mathbb{R}^n$, $$ \dot{x}=v(x),\qquad x\in U, $$ is equivalently described by the phase-velocity vector field $v$. The solution curves are the integral curves of $v$.

definition (autonomous vector field). a map $v:U\to\mathbb{R}^n$ of class $C^1$ (or at least locally Lipschitz) is a vector field on $U$. a curve $x:I\to U$ is an integral curve of $v$ if $\dot{x}(t)=v(x(t))$ for all $t\in I$.

definition (local flow). a local flow of $v$ is a map $(t,x)\mapsto\varphi_t(x)$, defined on an open set of $\mathbb{R}\times U$ containing $\{0\}\times U$, such that $t\mapsto\varphi_t(x_0)$ is the unique integral curve of $v$ with $\varphi_0(x_0)=x_0$. write $$ \varphi_t(x_0)=x(t;x_0) $$ whenever the solution is defined.

Under standard regularity assumptions the local flow exists for small $|t|$ and satisfies the group property $$ \varphi_{t+s}=\varphi_t\circ\varphi_s $$ whenever both sides are defined. Thus $\{\varphi_t\}$ is a local one-parameter group of diffeomorphisms of the phase space. The mathematical motivation for this language is that every intrinsic geometric notion (orbits, invariant sets, conjugacies) can be expressed solely in terms of the action of the flow, independently of any particular coordinate representation of $v$.

theorem (group property of the local flow). let $v$ be locally Lipschitz on open $U\subset\mathbb{R}^n$. if $\varphi_s(x_0)$ is defined and both sides of the following formula make sense, then $$ \varphi_{t+s}(x_0)=\varphi_t\bigl(\varphi_s(x_0)\bigr). $$

proof. fix $s$ and set $y_0=\varphi_s(x_0)$. the curve $t\mapsto\varphi_{t+s}(x_0)$ solves $\dot x=v(x)$ with value $y_0$ at $t=0$. the curve $t\mapsto\varphi_t(y_0)$ solves the same ODE with the same initial condition. by uniqueness of solutions (proved below via Picard), the two curves coincide on their common interval of definition. renaming the free variable yields the group property. $\square$

action of diffeomorphisms and coordinate-free character

If $\Phi$ is a diffeomorphism of the ambient space, it acts on vector fields by push-forward, $$ (\Phi_*v)(y)=T\Phi\cdot v(\Phi^{-1}(y)). $$ Direction fields (unoriented line fields) transform in the same way. Two vector fields that are related by such a push-forward generate flows that are conjugate.

definition (push-forward of a vector field). for a $C^1$ diffeomorphism $\Phi:U\to\Phi(U)$ and a vector field $v$ on $U$, $$ (\Phi_*v)(y) =D\Phi(\Phi^{-1}(y))\cdot v(\Phi^{-1}(y)). $$

theorem (conjugacy of flows). if $\psi_t$ is the local flow of $\Phi_*v$ and $\varphi_t$ is the local flow of $v$, then on a common domain of definition $$ \Phi\circ\varphi_t=\psi_t\circ\Phi. $$

proof. fix $x_0\in U$ and set $\gamma(t)=\Phi\bigl(\varphi_t(x_0)\bigr)$. then $\gamma(0)=\Phi(x_0)$ and $$ \dot\gamma(t) =D\Phi(\varphi_t(x_0))\cdot v(\varphi_t(x_0)) =(\Phi_*v)\bigl(\gamma(t)\bigr). $$ thus $\gamma$ is the integral curve of $\Phi_*v$ starting at $\Phi(x_0)$, which is by definition $t\mapsto\psi_t(\Phi(x_0))$. uniqueness gives $\Phi\circ\varphi_t=\psi_t\circ\Phi$. $\square$

remark. consequently the qualitative dynamics is an invariant of the equivalence class of the vector field under diffeomorphisms. this coordinate-free viewpoint is the natural setting in which to formulate structural stability, topological classification, and bifurcation theory.

local rectification, existence and uniqueness, periodic orbits

The rectification theorem asserts that near any non-singular point the field can be straightened to a constant field. Global (or at least local) existence and uniqueness of solutions follow from the contraction-mapping principle applied to the integral equation of Picard.

theorem (local existence and uniqueness, Picard). let $v$ be Lipschitz with constant $K$ on a closed ball $\overline{B_r(x_0)}\subset U$, with $\|v\|\le M$ on that ball. then there exists $T>0$ such that the initial-value problem $$ \dot x=v(x),\qquad x(0)=x_0 $$ admits a unique $C^1$ solution on $[-T,T]$ with image remaining in $B_r(x_0)$.

proof. a continuous curve $x:[-T,T]\to\mathbb{R}^n$ solves the ODE with initial value $x_0$ if and only if it solves the integral equation $$ x(t)=x_0+\int_0^t v\bigl(x(s)\bigr)\,ds. $$ let $X$ be the Banach space of continuous maps $[-T,T]\to\mathbb{R}^n$ with the supremum norm, and let $$ \mathcal{C} =\bigl\{x\in X:\ \|x(t)-x_0\|\le r\ \forall\,t\in[-T,T]\bigr\}, $$ a closed subset of $X$. define the Picard operator $$ (Px)(t)=x_0+\int_0^t v\bigl(x(s)\bigr)\,ds. $$ if $T\le r/M$ and $x\in\mathcal{C}$, then $\|(Px)(t)-x_0\|\le MT\le r$, so $P:\mathcal{C}\to\mathcal{C}$. for $x,y\in\mathcal{C}$, $$ \|(Px)(t)-(Py)(t)\| \le\int_0^{|t|}K\|x(s)-y(s)\|\,ds \le KT\|x-y\|_\infty, $$ so if $KT<1$ the operator $P$ is a contraction of the complete metric space $\mathcal{C}$. the Banach fixed-point theorem yields a unique fixed point, which is the unique continuous solution of the integral equation, hence the unique $C^1$ solution of the ODE on $[-T,T]$. $\square$

remark (maximal interval). iterating the local existence theorem along a solution produces a maximal open interval of existence. if that interval is finite in one direction, the solution must leave every compact subset of $U$ (escape time theorem).

theorem (rectification / flow box). let $v$ be $C^1$ near $x_0$ with $v(x_0)\neq 0$. then there exist open neighborhoods $W\ni x_0$ and $V\subset\mathbb{R}^n$, and a $C^1$ diffeomorphism $\Psi:V\to W$, such that $$ \Psi_*\Bigl(\frac{\partial}{\partial u^1}\Bigr)=v\Big|_W. $$ in coordinates $(u^1,\dots,u^n)$ the field is the constant field $\partial/\partial u^1$.

proof. without loss of generality take $x_0=0$ and $v(0)=e_1=(1,0,\dots,0)$ after an invertible linear change of coordinates (possible because $v(0)\neq 0$). for $y=(0,y')\in\{0\}\times\mathbb{R}^{n-1}$ near $0$ and for small $t$, set $$ \Psi(t,y'):=\varphi_t(0,y'), $$ where $\varphi$ is the local flow of $v$. then $\Psi(0,0)=0$ and $$ D\Psi(0,0)\,e_1 =\frac{\partial}{\partial t}\Big|_{t=0}\varphi_t(0) =v(0)=e_1, $$ while the partial derivatives of $\Psi$ with respect to the $y'$ directions at $(0,0)$ span a complement of $\mathrm{span}\{e_1\}$ because at $t=0$ one has $\Psi(0,y')=(0,y')$. thus $D\Psi(0,0)$ is invertible. the inverse function theorem yields a local $C^1$ diffeomorphism $\Psi$. by construction, $$ \frac{\partial}{\partial t}\Psi(t,y') =v\bigl(\Psi(t,y')\bigr), $$ which is exactly $\Psi_*(\partial/\partial t)=v$ in flow-box coordinates $(u^1,u')=(t,y')$. $\square$

definition (periodic orbit). a non-constant integral curve $\gamma$ is periodic if there exists $T>0$ with $\gamma(t+T)=\gamma(t)$ for all $t$ on the maximal interval. the image is a closed orbit in phase space.

remark. existence of a periodic orbit is a global topological question that cannot be decided by local rectification alone: near a non-singular point the rectified field has only straight open orbits.

linearization at singular points

At an equilibrium $x_*$ one has $v(x_*)=0$. The linearization is the linear vector field $$ \dot{\xi}=A\xi,\qquad A=Dv(x_*). $$

definition (equilibrium / singular point). a point $x_*\in U$ with $v(x_*)=0$ is an equilibrium of $v$. the matrix $A=Dv(x_*)$ is the linearization (Jacobian) of $v$ at $x_*$.

definition (matrix exponential). for $A\in\mathrm{M}_n(\mathbb{R})$ or $\mathrm{M}_n(\mathbb{C})$, $$ e^{tA}=\sum_{k=0}^\infty\frac{(tA)^k}{k!}. $$ the series converges absolutely in the operator norm for every $t$ and every $A$.

theorem (fundamental solution of linear systems). the unique solution of $\dot\xi=A\xi$, $\xi(0)=\xi_0$ is $\xi(t)=e^{tA}\xi_0$. moreover $$ \frac{d}{dt}e^{tA}=A\,e^{tA}=e^{tA}A. $$

proof. termwise differentiation of the power series (justified by absolute convergence on compact time intervals) yields $$ \frac{d}{dt}e^{tA} =\sum_{k=1}^\infty\frac{k t^{k-1}A^k}{k!} =\sum_{k=0}^\infty\frac{t^k A^{k+1}}{k!} =A\,e^{tA}. $$ thus $t\mapsto e^{tA}\xi_0$ solves the linear ODE with value $\xi_0$ at $t=0$. uniqueness for linear systems with continuous coefficients is a special case of Picard uniqueness. $\square$

Complexification of the real space $\mathbb{R}^n$ allows the use of the Jordan canonical form over $\mathbb{C}$. The real invariant subspaces corresponding to eigenvalues with negative, zero, and positive real parts yield the stable, center, and unstable subspaces.

definition (hyperbolic equilibrium). an equilibrium $x_*$ is hyperbolic if no eigenvalue of $A=Dv(x_*)$ has zero real part.

theorem (hartman-grobman, statement). if $x_*$ is a hyperbolic equilibrium of a $C^1$ vector field $v$, then there exist neighborhoods of $x_*$ and of $0$ and a homeomorphism $h$ between them conjugating the nonlinear flow of $v$ to the linear flow of $A=Dv(x_*)$.

remark. the homeomorphism $h$ need not be differentiable. a complete proof uses a discrete Hartman map for a time-$t_0$ diffeomorphism near $x_*$ and a fixed-point argument in a suitable function space; the statement is recorded here for the classification program, while the hyperbolicity hypothesis is the one that geometric users must check in examples.

classification, lyapunov functions and topological equivalence

In the plane the linear classification produces nodes, saddles, foci, and centers according to the eigenvalues of $A$. The same local phase portraits persist for the nonlinear system when the equilibrium is hyperbolic (Hartman-Grobman). Lyapunov's direct method supplies a coordinate-free criterion for stability without solving the ODE.

definition (lyapunov function). a $C^1$ function $V$ defined near an equilibrium $x_*$ is a Lyapunov function if $V(x_*)=0$, $V(x)>0$ for $x\neq x_*$ in a punctured neighborhood, and the orbital derivative $$ \dot V(x):=DV(x)\cdot v(x) $$ satisfies $\dot V(x)\le 0$ on that neighborhood.

theorem (lyapunov stability). if a Lyapunov function $V$ exists near $x_*$, then $x_*$ is (Lyapunov) stable. if moreover $\dot V(x)<0$ for all $x\neq x_*$ in a neighborhood, then $x_*$ is asymptotically stable.

proof. stability: fix $\varepsilon>0$ small enough that the closed ball $\overline{B_\varepsilon(x_*)}$ lies in the domain of $V$. let $$ m:=\min\bigl\{V(x):\ \|x-x_*\|=\varepsilon\bigr\}>0 $$ (continuous function on a compact set, positive because $V$ vanishes only at $x_*$). by continuity of $V$ there is $\delta>0$ such that $\|x-x_*\|<\delta$ implies $V(x)<m$. now let $x(t)$ be a solution with $\|x(0)-x_*\|<\delta$. if it first hits the sphere of radius $\varepsilon$ at time $t_1$, then $V(x(t_1))\ge m$, but $V(x(t))$ is nonincreasing along solutions because $\dot V\le 0$, so $V(x(t_1))\le V(x(0))<m$, a contradiction. hence the solution remains in $B_\varepsilon$ for all forward time of existence, and for short times of backward existence in the standard formulation of stability one argues on a maximal interval on which the orbit stays in a compact set. thus $x_*$ is stable.

asymptotic stability under $\dot V<0$: stability is already secured. let $x(t)$ start near $x_*$. the function $t\mapsto V(x(t))$ is strictly decreasing and bounded below by $0$, so it converges to some $\ell\ge 0$. if the $\omega$-limit set contains a point $y\neq x_*$ then continuous dependence and the strict decrease of $V$ produce a contradiction (along the orbit through $y$, $V$ drops below $\ell$). for planar and finite-dimensional settings it follows that $x(t)\to x_*$ (standard $\omega$-limit argument for continuous flows on compact neighborhoods trapped by the Lyapunov sublevel sets). $\square$

remark (on the asymptotic half). the strict inequality $\dot V<0$ is the classical sufficient criterion. refinements (LaSalle) allow $\dot V\le 0$ when the largest invariant set in $\{\dot V=0\}$ is reduced to $\{x_*\}$; those refinements are deferred to the dynamical systems chapter.

definition (topological equivalence). two vector fields are topologically equivalent if there exists a homeomorphism mapping oriented orbits of one onto oriented orbits of the other. hyperbolic equilibria of the same index (dimension of the unstable subspace) are topologically equivalent by Hartman-Grobman; centers and weak foci require finer invariants.

extension to differentiable manifolds

On a smooth manifold $M$ a vector field is a section of the tangent bundle $TM$. The notions of flow, linearization, and Lyapunov function carry over verbatim: charts reduce local questions to $\mathbb{R}^n$, while global invariants do not.

definition (index of an isolated singular point, in $\mathbb{R}^n$). if $x_*$ is an isolated zero of a continuous vector field $v$ and $S_\varepsilon$ is a small sphere about $x_*$ on which $v\neq 0$, the index of $x_*$ is the degree of the normalized map $$ \frac{v}{|v|}:S_\varepsilon\to S^{n-1}. $$

theorem (poincare-hopf, statement). if $M$ is a compact smooth manifold and $v$ is a continuous vector field with finitely many isolated zeros, then the sum of the indices of the zeros equals the Euler characteristic $\chi(M)$.

remark. thus the global topology of $M$ constrains the possible singular sets of any continuous vector field. the full proof uses either obstruction theory or triangulation and local-to-global additivity of degree; the theorem is the precise sense in which "every continuous vector field on $S^2$ has a singular point" (hairy ball) is a special case $\chi(S^2)=2$.

theorem (no nowhere-vanishing continuous vector field on even spheres, sketch from poincare-hopf). if $n$ is even then $\chi(S^n)=2\neq 0$, so any continuous vector field on $S^n$ has at least one zero.

proof. the Euler characteristic of $S^n$ is $2$ for even $n$ and $0$ for odd $n$. if a continuous field were nowhere zero, the zero set would be empty and the index sum would be $0$, contradicting Poincare-Hopf when $\chi(S^n)\neq 0$. $\square$

conservative systems with one degree of freedom

A second-order equation $\ddot q=f(q)$ is equivalent to the planar system $$ \dot q=p,\qquad\dot p=f(q). $$ When $f=-V'$ the total energy $$ E=\frac12 p^2+V(q) $$ is a first integral. The phase portrait is completely determined by the level sets of $E$: closed level curves correspond to periodic (librational) motions, while unbounded level curves correspond to unbounded trajectories. Critical points of $V$ produce equilibria whose stability is decided by the second-derivative test; the global arrangement of separatrices is read off from the critical values of the potential.

theorem (energy integral for $\ddot q=-V'(q)$). along every $C^2$ solution of $\ddot q=-V'(q)$ the function $E(q,\dot q)=\frac12\dot q^2+V(q)$ is constant.

proof. $$ \frac{dE}{dt} =\dot q\,\ddot q+V'(q)\dot q =\dot q\bigl(\ddot q+V'(q)\bigr) =0 $$ by the equation of motion. $\square$

theorem (phase portrait from the potential). assume $V$ is $C^2$ and $q_*$ is a nondegenerate critical point. if $V''(q_*)>0$ then the equilibrium $(q_*,0)$ of the planar system is a center for the linearized system and is a Lyapunov-stable equilibrium of the nonlinear system (with Lyapunov function $E-E(q_*,0)$). if $V''(q_*)<0$ then $(q_*,0)$ is a saddle of both the linearization and the nonlinear flow in a neighborhood.

proof. linearization of $(\dot q,\dot p)=(p,-V'(q))$ at $(q_*,0)$ is $$ A=\begin{pmatrix}0&1\\-V''(q_*)&0\end{pmatrix}. $$ eigenvalues satisfy $\lambda^2+V''(q_*)=0$. if $V''(q_*)>0$ then $\lambda=\pm i\omega$ (purely imaginary), which is the linear center. the function $L(q,p)=E(q,p)-V(q_*)$ is positive definite near $(q_*,0)$ because $V(q)-V(q_*)$ has a local minimum of quadratic order and $\frac12 p^2$ is positive definite in $p$; moreover $\dot L=0$ by energy conservation, so Lyapunov's theorem gives stability. if $V''(q_*)<0$ then $\lambda=\pm\sqrt{|V''(q_*)|}$ are real of opposite sign, so the linearization is a saddle. hyperbolicity plus Hartman-Grobman (or an elementary topological argument for planar saddles) yields a nonlinear saddle. $\square$

remark. this elementary picture already exhibits the fundamental dichotomy between bounded and unbounded motions that persists, in far more subtle form, in higher-dimensional Hamiltonian systems (to be met again under symplectic methods).

summary

The geometric theory of ordinary differential equations replaces the classical search for closed-form solutions by the study of the phase flow as a dynamical system on a manifold. Local analytic normal forms, topological indices, and Lyapunov functions together provide a coherent set of tools for classifying equilibria, detecting periodic orbits, and relating the differential equation to the topology of the underlying space.

exercises

exercise 1 (picard contraction). for $\dot x=-x$ on $\mathbb{R}$, write the Picard iteration starting from $x^{(0)}(t)\equiv 1$ with $x(0)=1$, and show that it converges to $e^{-t}$ on any fixed interval $[-T,T]$. compute the Lipschitz constant of $v(x)=-x$ and verify that the proof above applies on every ball.

exercise 2 (linear classification in the plane). draw the phase portrait of $\dot\xi=A\xi$ for $$ A=\begin{pmatrix}-1&0\\0&-2\end{pmatrix},\qquad A=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad A=\begin{pmatrix}0&-1\\1&0\end{pmatrix}. $$ identify node, saddle, and center. which of these equilibria are hyperbolic?

exercise 3 (energy portrait). take $V(q)=\frac14 q^4-\frac12 q^2$. locate the critical points, classify the equilibria of the planar system $(\dot q,\dot p)=(p,-V'(q))$, and sketch the level sets of $E$ that produce a figure-eight separatrix (homoclinic pair).

exercise 4 (lyapunov for a focus). for $$ \dot x=-x-y,\qquad \dot y=x-y $$ show that $V=x^2+y^2$ is a strict Lyapunov function at the origin, and conclude asymptotic stability. compute the eigenvalues of the linearization and compare with the classification table of planar linear systems.

exercise 5 (hartman-grobman necessity of hyperbolicity). give a planar example of a $C^\infty$ vector field with a nonhyperbolic equilibrium that is topologically inequivalent to its linearization. (a standard choice is a weak focus or a nilpotent node; sketch phase portraits and eigenvalues.) explain why hyperbolicity is required in the Hartman-Grobman theorem.