A short introduction to stability points.

by kagiso kwenda

Mathematical: NOTES 001

On dynamical systems.

Dynamics are the stuff of legend and no one really knows how these systems even work on some levels but what we know is they run all the show- in this kinder of simulated world, the subject of dynamic in a mathematical fashion deals with the act of finding the sweet spot of a stability point but the definition of a stable point is quite contorted since the explanation is arrived at through other points which are in a relation that seems far fetched

Definitions for stable and unstable points.

A fixed point ${\bf x_\ast}$ is asymptotically stable iff there exist a neighborhood $ \mathcal{N}$ of ${\bf x_\ast}$ such that every trajectory that starts in $ \mathcal{N}$ approaches ${\bf x_\ast}$ as time tends to infinity :time will be an argument for consideration for the continuous case : $\lim_{t \rightarrow \infty} $ x(t)=${\bf x_\ast}$ for the discrete case :$\lim_{n \rightarrow \infty}$ x(n)=${\bf x_\ast}$. The neighborhood $ \mathcal{N}$ represents the idea that the stability is local.While the trajectories on the space $ \mathcal{N}$ may not tend to ${\bf x_\ast}$, there is region around $ \mathcal{N}$ i.e in which the trajectories do tend to ${\bf x_\ast}$.

Where as a fixed point ${\bf x_\ast}$ is unstable in a neighborhood $ \mathcal{N}$ of ${\bf x_\ast} $ iff there exists a sequence of ${\bf x_i}$ of points , with $\lim_{i \rightarrow \infty}$ x(i)= ${\bf x_\ast}$ such as that every trajectory, with an ${\bf x_i }$ as starting point,contains points that are outside $\mathcal{N}$.

What's the important traits.

We see clearly that a path or trajectory is one component which is important ,we also see that a trajectory is avoidable to reach a stable point.Thus making it a trajectory that's is sought out.So from this we also learn that a trajectory of arriving at a stable point is not unique.

On marginal stability.

A fixed point is marginally stable iff it is neither stable or unstable. This might be harder to see but there will be points were they do not lead to stability or unstability of the system.