# Stablility of Fixed Point of a Dynamical System

** Published:**

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):

\[\dot x = f(x)\]A **fixed point** of this system is given by:

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 \(\dot x = f(x)\), a stable fixed point will force this particle to move towards itself. On the other hand, an unstable fixed point will force a particle away from itself.

### Mathematical Intuition:

For a dynamical system, \(\dot x = f(x)\), a fixed point is \(f(x) = 0\).

If \(f^{\prime}(x) \gt 0\), we have magnitude of \(f(x)\) increasing at x. This can be represented as \(f(x - \delta) \lt 0 \lt f(x + \delta)\) for a sufficiently small value of \(\delta \gt 0\).

Thus, if we start from \(x+\delta\) which is “close” \(x\), the ODE from equation (1) will keep on increasing the value of $x$. And it we start from \(x-\delta\) which is “close” \(x\), equation (1) will move the particle way from \(x\) by decreasing the value of \(x\).

Therefore, if \(f^{\prime} (x) \gt 0\), we have an **unstable fixed point** and vice versa.

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

## Intuitive Example:

For the differential equation **\(x^{\prime} = sin(x)\)**:

Using linear stability analysis, fixed points occurs when \(f(x)=sin(x)=0\) or \(x=kπ\) where \(k\) is integer.

\(f^{\prime}(x)=cos(kπ)=1\) if \(k\) is even. and \(f^{\prime}(x)=cos(kπ)=−1\) if \(k\) is odd.

Therefore, \(x\) is **unstable** when \(k\) is *even*, and **stable** if \(k\) 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.

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