Eigen Value Theory

Eigen values tell us the exponential part of the solution of the differential equation system
Three possible values for an eigen value :
1) Positive value: system will increase exponentially
2) Negative value: system will decay exponentially
3) Imaginary value: system will oscillate (note combinations of the above are possible)

NOTE:
Eigen values close to imaginary axis dominated on other value which is far from imaginary axis. Eigen value close to imaginary axis affect the system response.

Effect:
· If any eigenvalue has a positive real part, the system will tend to move away from the fixed point.
· If the real component of at least one eigenvalue is positive, than the system is unstable.
· If there are imaginary eigenvalue components with real positive component than the response will oscillate.
· Stability of linear dynamical systems can be determined from eigenvalues.
· Stability of nonlinear dynamical systems can be locally evaluated using eigenvalues.