📂Dynamics

Non-Smooth Systems in Dynamics

Terms

A Nonsmooth Dynamical System is defined as a dynamical system expressed through $S_{k} \subset \mathbb{R}^{n}$ in terms of $f_{k} : S_{k} \to \mathbb{R}^{n}$ defined by a piecewise smooth system $$\dot{x} = f_{k} (x) \qquad , k = 1, \cdots, s$$ or heteroclinic mapping $F : \mathbb{R}^{n} \rightrightarrows \mathbb{R}^{n}$ with respect to a differential inclusion $$\dot{x} \in F(x)$$.

Description

Many definitions and theorems regarding dynamics, especially those dynamical systems represented by differential equations, assume that the given system $\dot{x} = f(x)$ has a smooth $f$. Consequently, every point $x$ in the system is uniquely directed by a corresponding vector $f(x)$, determined by the vector field. However, actual systems in the real world might exhibit variations in $f(x)$ due to the inclusion of switches or sudden external controls being applied, indicating that $f(x)$ can change from one moment to the next.

While nonsmooth systems are undoubtedly challenging to handle, they also hold significant potential for applications and research value.