Let II be an interval, A:I→R n×nA:I\rightarrow\mathbb{R}^{\,n\times n} be a matrix-valued function with continuous entries and g‾:I→Rn\underline{g}:I\rightarrow\mathbb{R}^n be continuous. A linear system of first-order equations is a system of the form

y‾′=A(t) y‾+g‾(t),\underline{y}'=A(t)\,\underline{y}+\underline{g}(t),

where the independent variable is written tt and derivatives of vector- and matrix-valued functions are taken entrywise. The associated homogeneous system is the system with g‾=0‾.\underline{g}=\underline{0}.

y‾′=A(t) y‾\underline{y}'=A(t)\,\underline{y}

Linear systems have the following properties:

  • every initial value problem y‾(t0)=y‾0\underline{y}\left(t_0\right)=\underline{y}_0 has exactly one solution on all of I;I;
  • the superposition principle holds: the solutions of the homogeneous system form a vector space, and the general solution of the inhomogeneous system is the sum of a particular solution and the general solution of the homogeneous system;
  • the solution space of the homogeneous system has dimension n,n, since y‾↦y‾(t0)\underline{y}\mapsto\underline{y}\left(t_0\right) is an isomorphism onto Rn.\mathbb{R}^n.

Let y(n)+an−1y(n−1)+⋯+a0y=gy^{(n)}+a_{n-1}y^{(n-1)}+\cdots+a_0y=g be a linear equation of order n.n. The companion matrix of the equation is the matrix with ones on its superdiagonal, −a0,…,−an−1-a_0,\ldots,-a_{n-1} in its last row and zeros elsewhere. A linear equation of order nn is the linear system whose matrix is its companion matrix.

Let y‾′=A(t) y‾\underline{y}'=A(t)\,\underline{y} be a homogeneous linear system on I.I. A fundamental matrix of the system is a matrix-valued function Φ:I→R n×n\Phi:I\rightarrow\mathbb{R}^{\,n\times n} whose columns are a basis of the solution space. A fundamental matrix satisfies Φ′=A Φ\Phi'=A\,\Phi and Φ(t)\Phi(t) is invertible for every t.t.

Let y‾′=A(t) y‾\underline{y}'=A(t)\,\underline{y} be a homogeneous linear system with fundamental matrix Φ.\Phi. The Wronskian of the system is the determinant W=det⁡Φ.W=\det\Phi. Liouville’s formula states that the Wronskian satisfies

W(t)=W(t0)exp⁡(∫t0ttr A(s) ds),W(t)=W\left(t_0\right)\exp\left(\int_{t_0}^{t}\mathrm{tr}\,A(s)\,ds\right),

so nn solutions form a basis if and only if their Wronskian is non-zero at one, and hence every, point.

Let y‾′=A(t) y‾+g‾(t)\underline{y}'=A(t)\,\underline{y}+\underline{g}(t) be a linear system with fundamental matrix Φ\Phi of the homogeneous system. Solutions of linear systems have the following properties:

  • the general solution of the homogeneous system is Φ(t) c‾\Phi(t)\,\underline{c} for c‾∈Rn;\underline{c}\in\mathbb{R}^n;
  • the solution of the homogeneous initial value problem is Φ(t) Φ(t0)−1y‾0;\Phi(t)\,\Phi\left(t_0\right)^{-1}\underline{y}_0;
  • variation of parameters gives the solution of the inhomogeneous initial value problem.
y‾(t)=Φ(t) Φ(t0)−1y‾0+Φ(t)∫t0tΦ(s)−1g‾(s) ds\underline{y}(t)=\Phi(t)\,\Phi\left(t_0\right)^{-1}\underline{y}_0+\Phi(t)\int_{t_0}^{t}\Phi(s)^{-1}\underline{g}(s)\,ds

Let A∈R n×nA\in\mathbb{R}^{\,n\times n} be a constant matrix. The matrix exponential of AA is the sum of the exponential series with AA in place of the scalar.

eA:=∑k=0∞Akk!e^{A}:=\sum_{k=0}^{\infty}\frac{A^k}{k!}

The series converges absolutely in the Frobenius norm, since ∥Ak∥F≤∥A∥Fk\lVert A^k\rVert_F\leq\lVert A\rVert_F^k and ∑k=0∞∥A∥Fkk!=e∥A∥F,\sum_{k=0}^{\infty}\frac{\lVert A\rVert_F^k}{k!}=e^{\lVert A\rVert_F}, and R n×n\mathbb{R}^{\,n\times n} with this norm is complete. The matrix exponential has the following properties for A,B∈R n×n,A,B\in\mathbb{R}^{\,n\times n}, invertible PP and s,t∈R:s,t\in\mathbb{R}:

e0=In,e(s+t)A=esAetA,(etA)−1=e−tA,ddtetA=A etA=etAAe^{0}=I_n,\qquad e^{\left(s+t\right)A}=e^{sA}e^{tA},\qquad\left(e^{tA}\right)^{-1}=e^{-tA},\qquad\frac{d}{dt}e^{tA}=A\,e^{tA}=e^{tA}A eA+B=eAeB  if  AB=BA,ePAP−1=P eAP−1,det⁡eA=etr Ae^{A+B}=e^{A}e^{B}\,\,\mathrm{if}\,\,AB=BA,\qquad e^{PAP^{-1}}=P\,e^{A}P^{-1},\qquad\det e^{A}=e^{\mathrm{tr}\,A}

The exponential law can fail for non-commuting matrices.

Let A∈R n×nA\in\mathbb{R}^{\,n\times n} be a constant matrix. A linear system with constant coefficients is the system y‾′=A y‾.\underline{y}'=A\,\underline{y}. Since ddtetA=A etA,\frac{d}{dt}e^{tA}=A\,e^{tA}, the matrix etAe^{tA} is a fundamental matrix of the system and the solution of its initial value problem is

y‾(t)=e(t−t0)A y‾0.\underline{y}(t)=e^{\left(t-t_0\right)A}\,\underline{y}_0.

Linear systems with constant coefficients have the following properties:

  • if AA is diagonalisable as A=PΛP−1A=P\Lambda P^{-1} then etA=P etΛP−1e^{tA}=P\,e^{t\Lambda}P^{-1} where etΛe^{t\Lambda} is diagonal with entries eλit;e^{\lambda_it};
  • if v‾\underline{v} is an eigenvector of AA with eigenvalue λ\lambda then y‾=eλtv‾\underline{y}=e^{\lambda t}\underline{v} is a solution, and an eigen-basis gives a fundamental set of solutions;
  • a pair of complex conjugate eigenvalues α±iβ\alpha\pm i\beta with eigenvectors u‾±iw‾\underline{u}\pm i\underline{w} gives, by Euler’s formula, the real solutions eαt(u‾cos⁡βt−w‾sin⁡βt)e^{\alpha t}\left(\underline{u}\cos\beta t-\underline{w}\sin\beta t\right) and eαt(u‾sin⁡βt+w‾cos⁡βt).e^{\alpha t}\left(\underline{u}\sin\beta t+\underline{w}\cos\beta t\right).

Let A∈R n×nA\in\mathbb{R}^{\,n\times n} be a square matrix, λ\lambda be an eigenvalue of AA and k≥1.k\geq 1. A generalised eigenvector of rank kk for λ\lambda is a vector w‾\underline{w} with (A−λIn)kw‾=0‾\left(A-\lambda I_n\right)^k\underline{w}=\underline{0} and (A−λIn)k−1w‾≠0‾.\left(A-\lambda I_n\right)^{k-1}\underline{w}\neq\underline{0}. Generalised eigenvectors have the following properties:

  • every matrix has a basis of generalised eigenvectors;
  • since λIn\lambda I_n and A−λInA-\lambda I_n commute, etAw‾=eλtet(A−λIn)w‾e^{tA}\underline{w}=e^{\lambda t}e^{t\left(A-\lambda I_n\right)}\underline{w} and the second series terminates, which gives the solution
y‾(t)=eλt∑j=0k−1tjj!(A−λIn)jw‾;\underline{y}(t)=e^{\lambda t}\sum_{j=0}^{k-1}\frac{t^j}{j!}\left(A-\lambda I_n\right)^j\underline{w};
  • these solutions, with polynomial factors of degree less than the algebraic multiplicity, always give a fundamental set, including when AA is defective.

For example, the companion matrix of y′′−2y′+y=0y''-2y'+y=0 has the repeated eigenvalue 11 with a single eigenvector, and the generalised eigenvector recovers the solution t et.t\,e^t.

Let U⊆RnU\subseteq\mathbb{R}^n be open and F‾:U→Rn\underline{F}:U\rightarrow\mathbb{R}^n be C1.C^1. An autonomous system is a system of the form y‾′=F‾(y‾),\underline{y}'=\underline{F}\left(\underline{y}\right), whose right-hand side is a vector field on U.U. The set UU is the phase space of the system.

Let y‾′=F‾(y‾)\underline{y}'=\underline{F}\left(\underline{y}\right) be an autonomous system on U.U. A trajectory or orbit of the system is the image in UU of a solution. The phase portrait of the system is the collection of its trajectories. An equilibrium point of the system is a zero y‾∗\underline{y}^* of F‾,\underline{F}, whose trajectory is the single point. Trajectories have the following properties:

  • a trajectory is tangent to the vector field at every point;
  • by uniqueness, distinct trajectories do not intersect;
  • a trajectory that returns to a point is a closed curve corresponding to a periodic solution.

Let A∈R 2×2A\in\mathbb{R}^{\,2\times 2} be invertible with eigenvalues λ1,λ2,\lambda_1,\lambda_2, so that 0‾\underline{0} is the only equilibrium point of y‾′=A y‾.\underline{y}'=A\,\underline{y}. The eigenvalues are determined by the trace τ=λ1+λ2\tau=\lambda_1+\lambda_2 and the determinant Δ=λ1λ2\Delta=\lambda_1\lambda_2 through λ2−τλ+Δ=0,\lambda^2-\tau\lambda+\Delta=0, and the phase portrait is classified as follows:

  • if Δ<0\Delta<0 the eigenvalues are real with opposite signs and the equilibrium point is a saddle: trajectories approach along the eigenvector of the negative eigenvalue and leave along the eigenvector of the positive one;
  • if Δ>0\Delta>0 and τ2≥4Δ\tau^2\geq 4\Delta the eigenvalues are real with the same sign and the equilibrium point is a node, towards which every trajectory moves if τ<0\tau<0 and away from which every trajectory moves if τ>0;\tau>0;
  • if Δ>0\Delta>0 and τ2<4Δ\tau^2<4\Delta the eigenvalues are complex with real part τ2\frac{\tau}{2} and the equilibrium point is a spiral, inwards if τ<0\tau<0 and outwards if τ>0,\tau>0, or a centre surrounded by closed trajectories if τ=0.\tau=0.

Let y‾′=F‾(y‾)\underline{y}'=\underline{F}\left(\underline{y}\right) be an autonomous system and y‾∗\underline{y}^* be an equilibrium point. The equilibrium point y‾∗\underline{y}^* is stable if and only if for every ε>0\varepsilon>0 there exists δ>0\delta>0 such that every solution with ∥y‾(t0)−y‾∗∥<δ\left\lVert\underline{y}\left(t_0\right)-\underline{y}^*\right\rVert<\delta exists for all t≥t0t\geq t_0 and satisfies ∥y‾(t)−y‾∗∥<ε.\left\lVert\underline{y}(t)-\underline{y}^*\right\rVert<\varepsilon. The equilibrium point y‾∗\underline{y}^* is asymptotically stable if and only if it is stable and every solution that starts sufficiently close to it converges to it as t→∞.t\rightarrow\infty. The equilibrium point y‾∗\underline{y}^* is unstable if and only if it is not stable.

Let y‾′=A y‾\underline{y}'=A\,\underline{y} be a linear system with constant coefficients. Stability of the origin has the following properties:

  • the origin is asymptotically stable if and only if every eigenvalue of AA has negative real part, in which case ∥etA∥F→0\left\lVert e^{tA}\right\rVert_F\rightarrow 0 exponentially;
  • the origin is unstable if some eigenvalue of AA has positive real part;
  • a centre is stable but not asymptotically stable.

Let y‾′=F‾(y‾)\underline{y}'=\underline{F}\left(\underline{y}\right) be an autonomous system and y‾∗\underline{y}^* be an equilibrium point. The linearisation of the system at y‾∗\underline{y}^* is the linear system u‾′=JF‾(y‾∗)u‾\underline{u}'=J_{\underline{F}}\left(\underline{y}^*\right)\underline{u} for the deviation u‾=y‾−y‾∗,\underline{u}=\underline{y}-\underline{y}^*, obtained by discarding the remainder of the first-order Taylor expansion of F‾.\underline{F}.

Let y‾′=F‾(y‾)\underline{y}'=\underline{F}\left(\underline{y}\right) be an autonomous system and y‾∗\underline{y}^* be an equilibrium point. The equilibrium point y‾∗\underline{y}^* is a hyperbolic equilibrium point if and only if no eigenvalue of JF‾(y‾∗)J_{\underline{F}}\left(\underline{y}^*\right) has zero real part.

Let y‾′=F‾(y‾)\underline{y}'=\underline{F}\left(\underline{y}\right) be an autonomous system and y‾∗\underline{y}^* be a hyperbolic equilibrium point. The linearisation theorem states that y‾∗\underline{y}^* is asymptotically stable if every eigenvalue of JF‾(y‾∗)J_{\underline{F}}\left(\underline{y}^*\right) has negative real part and unstable if some eigenvalue has positive real part. The Hartman–Grobman theorem states that near y‾∗\underline{y}^* the phase portrait of the system is the image of the phase portrait of the linearisation under a homeomorphism.

Linearisation is inconclusive at non-hyperbolic equilibrium points. The pendulum equation θ′′+sin⁡θ=0,\theta''+\sin\theta=0, written as the system θ′=v,\theta'=v, v′=−sin⁡θ,v'=-\sin\theta, has a centre of the linearisation at (0,0),\left(0,0\right), which is stable since the energy 12v2−cos⁡θ\frac{1}{2}v^2-\cos\theta is constant along trajectories, and a saddle at (π,0).\left(\pi,0\right).

Let y‾′=F‾(y‾)\underline{y}'=\underline{F}\left(\underline{y}\right) be an autonomous system on U,U, y‾∗\underline{y}^* be an equilibrium point and V:U→RV:U\rightarrow\mathbb{R} be C1C^1 with V(y‾∗)=0V\left(\underline{y}^*\right)=0 and V>0V>0 elsewhere near y‾∗.\underline{y}^*. Along a solution the chain rule gives ddtV(y‾(t))=∇V(y‾)⋅F‾(y‾).\frac{d}{dt}V\left(\underline{y}(t)\right)=\nabla V\left(\underline{y}\right)\cdot\underline{F}\left(\underline{y}\right). The function VV is a Lyapunov function for y‾∗\underline{y}^* if and only if ∇V⋅F‾≤0\nabla V\cdot\underline{F}\leq 0 near y‾∗.\underline{y}^*.

Let y‾′=F‾(y‾)\underline{y}'=\underline{F}\left(\underline{y}\right) be an autonomous system and y‾∗\underline{y}^* be an equilibrium point with a Lyapunov function V.V. Lyapunov’s stability theorem states that y‾∗\underline{y}^* is stable, and asymptotically stable if ∇V⋅F‾<0\nabla V\cdot\underline{F}<0 away from y‾∗.\underline{y}^*. The energy of the pendulum, shifted by its minimum, is a Lyapunov function at (0,0).\left(0,0\right).