Examples from Stability of systems

A nonlinear non-example: logistic iteration

For the parameterised logistic recurrence

the first terms for $r=2$ are

The factor $1-x_n$ depends on the current value, so this recurrence is not linear and has no characteristic polynomial of the kind used below. The delayed logistic recurrence

requires both $x_0$ and $x_1$ before it can be iterated.

Test a recurrence

Consider

The characteristic equation is

whose roots are

Since $|r_1|>1$, the corresponding mode grows. The system is unstable. The second root has modulus less than $1$, but one growing mode is enough: choosing initial data with a non-zero coefficient of $r_1^n$ produces an unbounded solution.

Classify a damped oscillator

For

the roots satisfy

Take $m=k=1$ and vary the damping $c$.

• If $c>2$, the roots are distinct negative real numbers: the motion is overdamped and decays without oscillating.
• If $c=2$, the repeated root is $r=-1$, so $(K_1+K_2t)e^{-t}\to0$: the motion is critically damped and decays.
• If $0<c<2$, the roots are a complex conjugate pair with negative real part: the motion is underdamped and oscillates while its amplitude decays.
• If $c=0$, the roots are $r=\pm i$, so the motion is periodic rather than decaying.
• If $-2<c<0$, the conjugate roots have positive real part $-c/2$, so oscillations grow and the system is unstable.