Stability Analysis of Nonlinear DEs

Consider a nonlinear first order ODE system:

$$\frac{d x_i}{dt}=\dot{x}_i =f_i(x_1,\cdots,x_n),\;\;\;\;\;(i=1,\cdots,n)$$

If we define

$${\bf x}=\left[\begin{array}{c}x_1 \vdots x_n\end{array}\... ...1,\cdots,x_n) \vdots f_n(x_1,\cdots,x_n)\end{array}\right]$$

then the system can be written in matrx form as

$$\dot{\bf x}={\bf f}({\bf x})$$

which can be further expressed in Taylor series around a specific point ${\bf x}={\bf x}^*$ containing the constant (zero order) term, first order term, and all the higher order terms:

$$\dot{\bf x}={\bf f}({\bf x}^*)+{\bf J}({\bf x}^*)({\bf x}-{\bf x}^*)+\cdots ={\bf J}^*\delta{\bf x}+\cdots$$

where $\delta{\bf x}={\bf x}-{\bf x}^*$ and ${\bf J}^*$ is the Jacobian matrix of the vector function ${\bf f}({\bf x})$ evaluated at ${\bf x}={\bf x}^*$:

$${\bf J}^*={\bf J}({\bf x}^*)=\left[\begin{array}{ccc}\frac{\... ...tial f_n}{\partial x_n} \end{array}\right]_{{\bf x}={\bf x}^*}$$

We also note that

$$\dot{\delta{\bf x}}=\frac{d}{dt}({\bf x}-{\bf x}^*)=\frac{d}{dt}{\bf x} =\dot{\bf x}={\bf f}({\bf x})$$

and the equation above can be written as

$$\dot{\delta{\bf x}}=\dot{\bf x}={\bf f}({\bf x}) ={\bf J}^*\delta{\bf x}+\cdots$$

We assume $\delta{\bf x}$ is small, then we can neglect all higher order terms than first order in the Taylor expansion and get

$$\dot{\delta{\bf x}}\approx {\bf f}({\bf x}^*)+{\bf J}^*\delta{\bf x}$$

If we further let ${\bf x}^*$ be the equilibrium point of the system, i.e., ${\bf f}({\bf x}^*)={\bf0}$, then the equation above becomes a first order homogeneous LCCODE system:

$$\dot{\delta{\bf x}} = {\bf J}^*\;\delta{\bf x}$$

The solution of this system is in the form of $e^{\lambda t}{\bf v}$, where $\lambda$ is an eigenvalue of ${\bf J}^*$ and ${\bf v}$ is the corresponding eigenvector, satisfying ${\bf J}^*{\bf v}=\lambda{\bf v}$. In general, such an eigenvalue is complex and can be written in terms of its real and imaginary parts, $\lambda=\sigma+j\omega$, and the general solution can be written as

$$e^{\lambda t}{\bf v}=e^{(\sigma+j\omega)t} =e^{\sigma t} e^{j\omega t}=e^{\sigma t}[\cos(\omega t)+j \sin(\omega t)]$$

If $\sigma=Re(\lambda)<0$, the magnitude of the solution $e^{\sigma t}$ will attenuate to zero, i.e., the system is stable; however, if $Re(\lambda)>0$, the magnitude of the solution $e^{\sigma t}$ will grow without a bound, i.e., the system is unstable.

As we only keep a locally linear approximation around the equilibrium point, the corresponding analysis of the system is a linear stability analysis, which describes the behavior of the system close to the equilibrium point.