System of linear ODEs

Description

The following is a generalisation of 1-dimensional first order linear equation, to a system of first-order equations. (Or a multidimensional linear ODE). Note that due to the Existence and uniqueness theorem of linearly bounded functions, the solution always exists and is defined on all of $\mathbb{R}$.

Usually, we will focus on the case, where $A$ stays constant, meaning the case of System of autonomous linear ODEs. The solution of those systems is discussed in Variation of parameters.

Definition

A differential equation $x^{\prime }(t)=A(t)x(t)+g(t)$ with $A(t)$ a matrix and $g(t)$ a vector, continuously dependent on $t$ is called a system of linear ODEs. If $g(t)$ vanished for all $t$, then the system is called homogenous. Otherwise it is called nonhomogenous.

We will assume, that $A(t)$ is not constant, otherwise it could be replaced by a matrix. There is an article System of autonomous linear ODEs on that topic.

Properties

Solution space of homogenous systems

Let $x^{\prime }=A(t)x(t)$ be a homogenous system of linear ODEs of dimension $n$. Then the solutions follow superposition, meaning given two solutions ${\lambda }_1(t),{\lambda }_2(t)$ the sum ${\lambda }_1+{\lambda }_2$ and factors $c{\lambda }_1$ for $c\in \mathbb{R}$ are solutions of the same equation as well.

Since the initial value problem has a unique solution, the solution space of the ODE is $n$-dimensional. It is generated by $n$ functions ${\mu }_1,...,{\mu }_n:\mathbb{R}\to {\mathbb{R}}^n$ called Fundamental system (ODE). Writing those $n$ functions next to another gives us a matrix called the fundamental matrix.

To find a fundamental system, we can take a system of functions and show that they are linearly independent by calculating the determinant for some $t$.

Solution space of nonhomogenous system

Let $x^{\prime }=A(t)x(t)+g(t)$ be a nonhomogenous system. Let $\mathcal{L}$ denote the solution space of solution for the homogenous system. The solution set of the nonhomogenous system is given by ${\mu }_p=\mathcal{L}$ where ${\mu }_p$ is the particular solution (the one at zero?)

Stability of non-homogenous systems

Let $x^{\prime }=A(t)x(t)+g(t)$ be a non-homogenous linear system with contunous $A,g$. Then

1. Every Attractive solution is a Asymptotically stable solution
2. A solution is unstable/stable/asymptotically stable i.f.f. $0$ in $x^{\prime }=A(t)x(t)$ is unstable/stable/asymptotically stable.

Notice how strong the last statement is. It says that every solution has the same stability properties.

To see explicit solutions go to Variation of parameters.

Stability of homogenous systems

Let the problem be as before. All solutions $\mu :]a,\infty [$ of a homogenous system are

1. stable if for all $t_0>a$, there is a $\delta >0$ such that $\Vert \Lambda (t,t_0)\Vert <\delta$ for all $t\ge t_0$.
2. asymptotically stable if ${\lim }_{t\to \infty }\Lambda (t,t_0)=0$ for all $t_0\in ]a,\infty [$.

Examples

Solutions to systems with constant $A$

See System of autonomous linear ODEs

Solutions to systems with non-constant diagonal $A(t)$

If $A$ is diagonal, then it is possible to split up the different lines and calculate them one by one.