How to Determine Stability of Linear Difference Equations

1 First‑order linear difference equation with constant coefficients

For a first‑order linear difference equation with constant coefficients, let the equation be \(x_{k+1}=a x_k + b\) where \(a\) is a constant. The equilibrium point \(x^*\) is obtained by solving \(x^*=a x^* + b\). If \(|a|<1\), the equilibrium point \(x^*\) is stable; otherwise it is unstable.

Generally, the stability of an equilibrium point of a nonlinear first‑order difference equation can be reduced to the stability of the corresponding linearized equation. By linearizing around \(x^*\) we obtain a linear equation with coefficient \(a\); the necessary and sufficient condition for stability is \(|a|<1\).

For a system of \(n\) first‑order linear difference equations written in vector form \(\mathbf{x}_{k+1}=A\mathbf{x}_k + \mathbf{b}\) with constant matrix \(A\), the equilibrium point is stable if and only if all eigenvalues of \(A\) satisfy \(|\lambda_i|<1\).

2 Second‑order linear difference equation with constant coefficients

Consider a second‑order linear difference equation \(x_{k+2}=a_1 x_{k+1}+a_2 x_k + b\) where \(a_1, a_2\) are constants. The equilibrium point is stable exactly when the roots of the characteristic polynomial \(r^2 - a_1 r - a_2 = 0\) lie inside the unit circle (i.e., have magnitude less than one). The same reasoning applies to general equilibrium points and can be extended to higher‑order linear difference equations.

3 First‑order nonlinear difference equation

For a first‑order nonlinear difference equation \(x_{k+1}=f(x_k)\) with known function \(f\), the equilibrium point \(x^*\) satisfies \(f(x^*)=x^*\). To study its stability, linearize the equation around \(x^*\): \(x_{k+1}\approx f'(x^*) (x_k - x^*) + x^*\). The linearized coefficient is \(a = f'(x^*)\). When \(|a|<1\), the equilibrium is stable; when \(|a|>1\), it is unstable.

References

Python Mathematical Experiments and Modeling, Si Shougui & Sun Xijing, Science Press.