Consider a linear system with two nonzero, real, distinct eigenvalues λ1 and λ2.
If λ1 < 0 <λ2, then the origin is a saddle. There are two lines in the phase portrait that correspond to straight-line solutions. Solutions along one line tend toward (0, 0) as t increases, and solutions on the other line tend away from (0, 0). All other solutions come from and go to infinity.
If λ1 <λ2 < 0, then the origin is a sink. All solutions tend to (0, 0) as t approaches infinity, and most tend to (0, 0) in the direction of the λ2-eigenvectors.
If 0 <λ2 <λ1, then the origin is a source. All solutions except the equilibrium solution go to infinity as t approaches infinity, and most solution curves leave the origin in the direction of the λ2-eigenvectors.
I understand how we get this kind of structures when I work out specific problems (with specific numbers), but what is the intuitive explanation why this is always a case?
For example, if λ1 < 0 <λ2, how do we know that no matter what the matrix of the system is, we will get a specific phase portrait and the equilibrium will be a saddle?
I will be grateful for an intuitive explanation.
Best Answer
Your differential system looks like this: $$\dot x=Ax\qquad x(0)=x_0$$ where $A$ is a $2\times 2$ matrix. Suppose that $x_0$ is an eigenvector of $A$. Then, when $t=0$, $\dot x=\lambda x_0$, so the derivative of the solution curve points in the direction of $x_0$. If $\lambda<0$, the curve is moving towards the origin. If $\lambda>0$, the curve is moving away from the origin.
These eigendirections give us a "skeleton" around which to build the geometry of the set of solutions to the differential system. If we have two different eigenvectors $v_1$ and $v_2$ with eigenvalues $\lambda_1$ and $\lambda_2$, (which your question implicitly assumes), then we can write any $x(t)$ as $$x(t)=a(t)v_1+b(t) v_2$$ for some scalar functions $a$ and $b$. Then $$\dot x=\dot a(t)v_1+\dot b(t) v_2=Ax=a(t)Av_1+b(t)Av_2=\lambda_1 a(t)v_1+\lambda_2 b(t) v_2$$ which, by the uniqueness of the decompositions in the base $(v_1,v_2)$, implies that $$\dot a(t)=\lambda_1a(t)\qquad \dot b(t)=\lambda_2b(t)$$ So the behavior of the solutions of the system is really dependent on the behavior along each eigenvector.
As for why we can tell all of this without knowing the specifics of the matrix, the answer is that if a matrix has two distinct eigenvalues, we know everything about the behavior of that matrix just from its eigenvalues and eigenvectors.