[METHODS] Stability of systems

Find discrete modes from a characteristic equation

For $a\ne0$ and

a x_{n+1}+b x_n+c x_{n-1}=0,

try a non-trivial solution $x_n=K r^n$ with $K\ne0$ and $r\ne0$ . Substitution gives

K r^{n-1}(ar^2+br+c)=0.

For this nonzero mode, the prefactor is nonzero, so its characteristic equation is

ar^2+br+c=0.

The zero-root case, if it occurs, is not covered by this $r^{n-1}$ factorisation and must be checked directly in the recurrence. When the non-zero roots $r_1,r_2$ are distinct, the general solution is

x_n=K_1r_1^n+K_2r_2^n.

If there is a repeated non-zero root $r$ , the general solution is instead

x_n=(K_1+K_2n)r^n.

The expression $ar^2+br+c$ is a complex polynomial. Its non-zero roots are exactly the numbers for which the proposed exponential mode satisfies the substitution. The factor $n$ in the repeated-root solution matters for stability: even when $|r|=1$ , the term $nr^n$ is unbounded.

If $c=0$ , then zero is a characteristic root and the recurrence reduces to $a x_{n+1}+b x_n=0$ for $n\ge1$ . Here $x_0$ and $x_1$ are independent initial values. For $b\ne0$ , the later terms are $x_n=x_1(-b/a)^{n-1}$ for $n\ge1$ ; for $b=0$ , all terms from $x_2$ onward are zero. This handles the zero-root case without using $0^0$ .

Find continuous modes from a characteristic equation

For distinct roots $r_1,r_2$ of $ar^2+br+c=0$ , the solution has the form

x(t)=K_1e^{r_1t}+K_2e^{r_2t}.

For a repeated root $r$ , the solution has the form

x(t)=(K_1+K_2t)e^{rt}.

If $r=\alpha+i\beta$ , then

e^{rt}=e^{\alpha t}e^{i\beta t}.

The factor $e^{\alpha t}$ controls growth or decay, while $e^{i\beta t}$ supplies oscillation. When the coefficients and required solution are real, a conjugate pair $\alpha\pm i\beta$ (with $\beta\ne0$ ) combines into

e^{\alpha t}\bigl(A\cos(\beta t)+B\sin(\beta t)\bigr), \qquad A,B\in\mathbb R.

The constant-coefficient ODE method in Calculus is the same calculation, with repeated roots treated in more detail. The damped oscillator below shows each case.

[1]:Continuous-time system
[2]:Complex polynomial
[3]:Continuous stability from real parts

Methods from Stability of systems

Find discrete modes from a characteristic equation

Find continuous modes from a characteristic equation

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