Role of eigenvalues in phase portraits

[bhatiaharsh]

Hi,

In the study of dynamical systems, phase portraits play an important role. However, in almost all related text, I only see some standard examples like prey-predator, pendulum etc. I have a rather unclear thought in my head regarding the role of real/imaginary eigenvalues in the system. What role do they play with respect to the system dynamics ? Are there any physical systems that you know of, which only deal with either real or imaginary eigenvalues ? Or any practical cases where one would "want" to deal with anyone of those ?

 

[matematikawan]

Euler's formula $e^{it}=\cos{t }+ i \sin{t}$.
So if the eigenvalue is a complex number, the solution will have sinusoidal functions I think.

 

[pasmith]

Hi,

In the study of dynamical systems, phase portraits play an important role. However, in almost all related text, I only see some standard examples like prey-predator, pendulum etc. I have a rather unclear thought in my head regarding the role of real/imaginary eigenvalues in the system. What role do they play with respect to the system dynamics ?

The eigenvalues of the linearization of an ODE about a fixed point determine the stability of the fixed point. The important factor is not whether the eigenvalues are real or complex, but whether the real part of an eigenvalue is positive (unstable direction), negative (stable direction) or zero (indeterminate at linear order).

http://www.augustana.uAlberta.ca/~hackw/mat332/exhibit/fxdpt2d.html gives a classification of fixed points (other than degenerate cases) in 2D dynamical systems according to the trace and determinant of the linearization. Bear in mind that for a 2x2 matrix $M$ with eigenvalues $\lambda_1$, $\lambda_2$,
$$\mathrm{Tr}\,M = \lambda_1 + \lambda_2,\quad \det M = \lambda_1 \lambda_2.$$