Stability of Fixed Point of a Dynamical System

Stability theory is used to address the stability of solutions of differential equations. A dynamical system can be represented by a differential equation. The stability of the trajectories of this system under perturbations of its initial conditions can also be addressed using the stability theory.

Fixed Point

Consider a dynamical system given by the following ordinary differential equation (ODE):

\[\dot x = f(x) \tag{1}\]

A fixed point of this system is given by:

\[\dot x = 0 \tag{2}\]

Therefore, \(f(x) = 0\) or roots of the function \(f(x)\) form the fixed points of the dynamical system.

Stable and Unstable Fixed Points

In layman’s terms, you can say the following about stable and unstable fixed points.

Stable Fixed Point: Put a system to an initial value that is “close” to its fixed point. The trajectory of the solution of the differential equation \(\dot x = f(x)\) will stay close to this fixed point.

Unstable Fixed Point: Again, start the system with initial value “close” to its fixed point. If the fixed point is unstable, there exists a solution that starts at this initial value but the trajectory of the solution will move away from this fixed point.

In other words, one can also think of a stable fixed point as the attractor and unstable fixed point as the repeller. A particle governed by \(\dot x = f(x)\) is drawn towards a stable fixed point and pushed away from an unstable one.

Mathematical Intuition:

For the dynamical system in equation (1), write \(x^{*}\) for a fixed point, so that \(f(x^{*}) = 0\). The state of the particle is \(x\), and it starts a small distance \(\delta \gt 0\) away from \(x^{*}\).

Suppose \(f^{\prime}(x^{*}) \gt 0\). Then \(f\) is increasing as it crosses zero at \(x^{*}\), which means \(f(x^{*} - \delta) \lt 0 \lt f(x^{*} + \delta)\) for sufficiently small \(\delta\).

Start the particle at \(x = x^{*} + \delta\). Equation (1) gives \(\dot x = f(x) \gt 0\), so \(x\) grows and the particle moves further to the right of \(x^{*}\). Start it instead at \(x = x^{*} - \delta\). Now \(\dot x \lt 0\), so \(x\) shrinks and the particle moves further to the left. In both cases the particle leaves the neighborhood of \(x^{*}\).

Therefore, \(f^{\prime} (x^{*}) \gt 0\) gives an unstable fixed point. \(f^{\prime} (x^{*}) \lt 0\) gives a stable fixed point.

Note: The conditions \(f^{\prime} (x^{*}) \lt 0\) and \(f^{\prime} (x^{*}) \gt 0\) are sufficient to guarantee stability and instability respectively. They are not necessary, i.e., it is possible to have stable and unstable fixed points where \(f^{\prime} (x^{*}) = 0\).

Intuitive Example:

For the differential equation \(\dot x = \sin(x)\):

Using linear stability analysis, fixed points occur when \(f(x)=\sin(x)=0\), that is at \(x^{*}=k \pi\) where \(k\) is an integer.

\(f^{\prime}(x^{*})=\cos(k \pi)=1\) if \(k\) is even and \(f^{\prime}(x^{*})=\cos(k \pi)= - 1\) if \(k\) is odd.

Therefore, \(x^{*}\) is unstable when \(k\) is even, and stable when \(k\) is odd.

High Dimensional Dynamical Systems

This post discussed the definition of a fixed point of a dynamical system. A simple one-dimensional dynamical system is used as an illustration to explain the concept. A more detailed discussion on general nonlinear, continuous-time, multi-dimensional dynamical systems and their fixed points is provided in my next post.

Reference and Further Readings:

• Strogatz, S. H. (2018). Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. CRC press. [book]
• Deshpande, A. M. (2021). Stability of Fixed Points of High Dimensional Dynamical Systems.[web]