Linear_ODE_limit_behaviour.md

Limit behaviour of solutions of linear ODEs

General theory (reminder)

Consider $\dot x=Ax$.

• If $Av=λv$, then $x(t)=e^{λt}v$ is a solution of $\dot x=Ax$.
• If $A$ is diagonalizable, then any solution is a linear combination of $e^{λ_kt}v_k$: $x(t)=\sum_{k=1}^nc_ke^{λ_kt}v_k$.

Limit behaviour

What happens to a generic solution $x(t)$ of $\dot x=Ax$ as $t→∞$? Here “generic” means that we can ignore finitely many lines on the plane (or hyperplanes in the space).

Diagonalizable systems

If $A$ is diagonalizable, then $x(t)=c₁e^{λ₁t}v₁+c₂e^{λ₂t}v₂$. The limit behaviour depends on the order of $λ₁$, $λ₂$, and $0$ on the real line. We assume $λ₁≤λ₂$, otherwise we swap the indices $1$ and $2$. We also exclude the cases corresponding to degenerate matrices (i.e., we assume that $λ₁λ₂≠0$).

Name	Inequalities	Limit behaviour of a generic orbit	Genericity assumption	Otherwise
Saddle point	$λ₁<0<λ₂$	$x(t)≈c₂e^{λ₂t}v₂→∞$	$c₂≠0$	$x(t)=c_1e^{λ_1t}v_1→0$
Stable node	$λ₁<λ₂<0$	$x(t)≈c₂e^{λ₂t}v₂→0$	$c₂≠0$	$x(t)=c_1e^{λ_1t}v_1→0$
Unstable node	$0<λ₁<λ₂$	$x(t)≈c₂e^{λ₂t}v₂→∞$	$c₂≠0$	$x(t)=c_1e^{λ_1t}v_1→∞$ unless $x(t)=0$
Stable dicritical node	$λ₁=λ₂<0$	$x(t)=e^{λ₁t}x(0)→0$	-	-
Unstable dicritical node	$0<λ₁=λ₂$	$x(t)=e^{λ₁t}x(0)→∞$	$x(0)≠0$	$x(t)=0$

Complex eigenvalues

If $A$ has complex eigenvalues $λ=a+bi$ and $\bar λ=a-bi$, then the solutions are given by $x(t)=e^{at}\left[(c₁\cos(bt)+c₂\sin(bt))v+(c₂\cos(bt)-c₁\sin(bt))w\right]$, where $v+iw$ is an eigenvector corresponding to $λ=a+bi$. The limit behaviour of a generic orbit depends on the sign of $a$.

Name	(In)equalities	Limit behaviour of a generic orbit	Genericity assumption	Otherwise
Stable focus	$a<0$	$x(t)→0$, $C_1e^{at}\le|x(t)|\le C_2e^{at}$	-	-
Unstable focus	$a>0$	$x(t)→∞$, $C_1e^{at}\le|x(t)|\le C_2e^{at}$	$x(0)≠0$	$x(t)=0$
Center	$a=0$	$x(t)$ is periodic	-	-