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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.

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2 Answers 2

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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.

  1. If $\lambda_1<0<\lambda_2$, then one component, or one part of the solution is getting pulled toward the origin, while the other part is repelled. This leads to the saddle.
  2. If $\lambda_1< \lambda_2<0$, then both parts of the solution are pulled toward the origin, and we get a sink.
  3. If $0<\lambda_1< \lambda_2$, then both parts are pushed away from the origin, and so we get a source.

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.

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It all has to do with the general form $$ y= C_1 e^{\lambda _1 t}+C_2 e^{\lambda _1 t}$$

Not that for a positive $\lambda$ and a nonzero coefficient $C$, the branch $Ce^{\lambda t}$ approaches $\pm \infty$ depending on the sign of $C$ as $t$ approaches infinity. On the other hand for a negative $\lambda$, $Ce^{\lambda t} $ tends to zero as $t$ approaches infinity.

You can analyse case by case to see what happens if $t$ approaches $\infty$ or $-\infty$

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