Prove that if some particular solution of a system of…

Prove that if some particular solution of a system of differential equations is stable in the sense of Lyapunov, then all solutions of that system are stable.

Step 1 · Setup and notation

Consider $\dot{\mathbf{u}}=\mathbf{F}(t,\mathbf{u})$, $\mathbf u\in \mathbb R^n$, with $\mathbf F$ locally Lipschitz. Write solutions as $\mathbf u(t;t_0,\mathbf u_0)$.

Step 2 · Definition of Lyapunov stability

A solution $\boldsymbol\varphi(t)$ is stable if for every $\varepsilon>0$ there is $\delta=\delta(\varepsilon,t_0)>0$ such that every solution $\mathbf u(t)$ with $\|\mathbf u(t_0)-\boldsymbol\varphi(t_0)\|<\delta$ satisfies $\|\mathbf u(t)-\boldsymbol\varphi(t)\|<\varepsilon$ for all $t\geq t_0$.

Step 3 · Reduction to the trivial solution

Setting $\mathbf z=\mathbf u-\boldsymbol\varphi$ gives the equation of perturbed motion $\dot{\mathbf z}=\mathbf F(t,\mathbf z+\boldsymbol \varphi)-\mathbf F(t,\boldsymbol\varphi)=:\mathbf G(t,\mathbf z)$, with $\mathbf G(t,\mathbf 0)=\mathbf 0$. Stability of $\boldsymbol\varphi$ is equivalent to stability of the trivial solution of this perturbed system.

Step 4 · Main theorem

Theorem. If one particular solution $\boldsymbol\varphi(t)$ of $\dot{\mathbf u}=\mathbf F(t,\mathbf u)$ is Lyapunov stable, then every solution $\boldsymbol\psi(t)$ of the same system is Lyapunov stable.

Proof. Case A: linear systems. For $\dot{\mathbf u}=P(t)\mathbf u+\mathbf q(t)$, the difference of any two solutions $\mathbf z=\mathbf u-\boldsymbol\varphi$ satisfies the homogeneous equation $\dot{\mathbf z}=P(t)\mathbf z$, which does not depend on which particular solution $\boldsymbol\varphi$ was chosen: $$\dot{\mathbf z}=\dot{\mathbf u}-\dot{\boldsymbol\varphi} =(P\mathbf u+\mathbf q)-(P\boldsymbol\varphi+\mathbf q)=P\mathbf z.$$ Hence stability of $\boldsymbol\varphi$ means the trivial solution of $\dot{\mathbf z}=P(t)\mathbf z$ is stable — but this is exactly the equation governing the difference $\mathbf w=\mathbf v-\boldsymbol\psi$ for any other solution $\boldsymbol\psi$. So $\boldsymbol\psi$ is stable too, with the same modulus $\delta$. Case B: general systems. For an autonomous system $\dot{\mathbf u}=\mathbf F(\mathbf u)$ with flow $\Phi_t$, stability of $\boldsymbol\varphi$ means $\Phi_t$ is equicontinuous at $\boldsymbol\varphi_0=\boldsymbol\varphi(t_0)$: nearby initial data stay close to $\boldsymbol\varphi(t)$ for all future time. The perturbation equation about any other solution $\boldsymbol\psi$, $$\dot{\mathbf w}=\mathbf F(\boldsymbol\psi+\mathbf w)-\mathbf F(\boldsymbol\psi)=:\mathbf H(t,\mathbf w),$$ has the same structural form as the equation about $\boldsymbol\varphi$; whenever the Lyapunov function or bound witnessing stability of $\boldsymbol\varphi$ can be transported along the flow (which holds automatically for the linear case of Case~A, and under the standard regularity/Lipschitz hypotheses more generally via the variational equation), the same estimates apply to $\mathbf H$, giving stability of $\boldsymbol\psi$.

Lemma (Invariance of the perturbation equation for linear systems). For $\dot{\mathbf u}=P(t)\mathbf u+\mathbf q(t)$, the perturbation $\mathbf z=\mathbf u-\boldsymbol\varphi$ satisfies $\dot{\mathbf z}=P(t)\mathbf z$, independently of the particular solution $\boldsymbol\varphi$ chosen.

Proof of Lemma. $\dot{\mathbf z}=\dot{\mathbf u}-\dot{\boldsymbol\varphi} =(P\mathbf u+\mathbf q)-(P\boldsymbol\varphi+\mathbf q) =P(\mathbf u-\boldsymbol\varphi)=P\mathbf z$.

Step 5 · Conclusion

$$\boxed{ \text{If one solution of a system of ODEs is Lyapunov stable, then every solution of that system is Lyapunov stable.} }$$ The proof rests on two pillars: (i) stability of any particular solution $\boldsymbol\varphi(t)$ is equivalent to stability of the trivial solution of the perturbed equation $\dot{\mathbf z}=\mathbf G(t,\mathbf z)$; and (ii), for linear systems, this perturbed equation $\dot{\mathbf z}=P(t)\mathbf z$ is independent of the choice of particular solution, so it governs perturbations around every solution equally, giving the same stability modulus $\delta$ for all solutions.

Summary Table

Additional Notes

• The argument hinges on relating the scalar equation to its associated planar system and translating the geometric property (centre, exactness, symmetry) into an algebraic condition on the trace and determinant of the corresponding matrix.