Finding the spiral sinks and spiral sources of a linear system

• the7joker7
In summary: He has been told that if the eigenvalue is negative, the system will spiral in the negative direction, and if the eigenvalue is positive, the system will spiral in the positive direction. However, he is having difficulty finding information on how to find which values of A and B give complex eigenvalues.
the7joker7

Homework Statement

Basically, the problem involves a linear system dx/dt = ax + by and dy/dt = -x - y, with a and b being parameters that can take on any real value. Basically, you go through this system for several values of a and b (I did -12 to 12) to find the state at various points. That is, whether or not the point is a saddle, source, sink, perodic, or whatever else. I've found all of that. The only thing I haven't found yet is which sources and sinks are spirals. I know that it's a spiral if a complex number is involved. I've been told two different ways to find this.

The Attempt at a Solution

A: Graph the trace of the system (a + d)*x and the determinant (ad - bc) and the area inside the parabola has the spirals. I'm having a hard time doing this on the graphing tools I've found on the internet.

B: Using the quadratic x^2 - (trace*x) + determinant = 0, find what values of the trace and determinant make is such that x = sqrt(negative number). This equation simplifies to x = sqrt(trace*x - determinant). An x on both sides, so that complicates things...

The behavior of a linear system like you describe is pretty much determined by the eigenvalues of the matrix. Here's a link to an article on Equilibrium in dynamical systems, as these are often called. The section titled Two-Dimensional Space might answer some of your questions.

For the most part, I understand the relationship between behavior and eigenvalues. When you have two positive eigenvalues it's a source, two negative it's a sink, so on and so forth...

My issue is, what's the easiest way to find for which values of A and B are there complex values involved.

This is pretty hazy in my memory since it was a long time ago that I studied this stuff, but your quadratic (B) brings up some memories. It could be that this equation is the characteristic polynomial for the system, which gives you roots r1 and r2, and from which you get solutions e^(r1*t) and e^(r2 * t). If the solutions to the quadratic are complex, they will be conjugates, so you'll get r = a +/-bi. The imaginary parts will lead to solutions involving sin() and cos(), and that's why you get spiral behavior, either toward a source (if a < 0) or away to a sink (if a > 0).

Again, this is pretty fuzzy in my mind, but maybe I've given you some directions to go in.
Mark

What is a linear system?

A linear system is a set of equations that can be represented by a line or a plane in a coordinate system. It consists of linear equations with one or more variables.

What are spiral sinks and spiral sources?

Spiral sinks and spiral sources are types of equilibrium points in a linear system. A spiral sink is an equilibrium point where the solutions of the system spiral inward towards the point. A spiral source is an equilibrium point where the solutions of the system spiral outward from the point.

How do you find spiral sinks and spiral sources?

To find spiral sinks and spiral sources of a linear system, you need to determine the eigenvalues and eigenvectors of the system's matrix. The eigenvalues and eigenvectors will tell you the direction and rate of change of the solutions near the equilibrium points. If the eigenvalues are complex numbers with a negative real part, the equilibrium point is a spiral sink. If the eigenvalues are complex numbers with a positive real part, the equilibrium point is a spiral source.

Why is it important to find spiral sinks and spiral sources?

Knowing the location and nature of spiral sinks and spiral sources can help in understanding the long-term behavior of a system. These equilibrium points can act as attractors or repellors for the solutions of the system, and can give insight into the stability and dynamics of the system.

Can a linear system have more than one spiral sink or spiral source?

Yes, a linear system can have multiple spiral sinks and spiral sources depending on the values of the coefficients in the system's equations. These equilibrium points may also coexist with other types of equilibrium points, such as nodes and saddles, in the system.

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