# Properties of Nonlinear System (Related to Definition)

Definition 1.1 (Linearity of The Function)
The function f(x) is linear with respect to independent variable x if and only if it satisfies two conditions:
1. Additivity : $f\left({x}_{1},{x}_{2} \right)=f\left({x}_{1} \right)+f\left({x}_{2} \right)$, V${x}_{1},{x}_{2}$ in domain of the function f,

2. Homogenity: $f\left(\alpha x \right)=\alpha f\left(x \right)$, Vx in domain of function f and all scalar $\alpha$

Definition 1.2 (Locally Lipschitz Function)

The function f is said to be locally Lipschitz for the variable x(t) if near the point $x\left(0 \right)={x}_{0}$ it satisfies the Lipschitz criterion:

$\left|\left|f\left(x \right)-f\left(y \right) \right| \right|\leq \left|\left|x-y \right| \right|$

for all x and y in the vicinity of ${x}_{0}$ where k is a positive constant and the norm is Euclidian.

Definition 1.3 (Forced and Unforced System)

A continuous system is said to be forced if an input signal is present:

$\dot{x}\left(t \right)=f\left[t,x\left(t \right),u\left(t \right) \right]$; V$t\geq 0$, $u\left(t \right)\neq 0$

A continuous system is said to be unforced if there is no input (excitation) signal:

$\dot{x}\left(t \right)=f\left[t,x\left(t \right)\right]$; V$t\geq 0$, $u\left(t \right)= 0$

Definition 1.4 (Time-Varying and Time-Invariant System)

A system is time-invariant if the function f does not explicitly depend on time, i.e. the system can be described with:

$\dot{x}\left(t \right)=f\left[x\left(t \right),u\left(t \right) \right]$; V$t\geq 0$

A system is time-varying if the function f explicitly depend on the time, i.e. the system can be described with:

$\dot{x}\left(t \right)=f\left[t,x\left(t \right),u\left(t \right) \right]$; V$t\geq 0$

Definition 1.5 (Equilibrium State)

An equilibrium state can be defined as a state which the system retain if no external signal $u\left(t \right)$ is present. Mathematically, the equilibrium state is expressed by the vector ${x}_{e}\in {R}^{n}$. The system remains in equilibrium state if the initial moment the state is the equilibrium state, i.e. at $t={t}_{0}-x\left({t}_{0} \right)={x}_{e}$. In the other words, if the system begins from the equilibrium state, it remains in this state $x\left(t \right)={x}_{e}$,V$t\geq {t}_{0}$ under presumption that no external signal act, i.e. $u\left(t \right)=0$

Definition 1.6 (System’s Trajectory-Solution)

An unforced system described by the vector differential equation:

$\dot{x}\left(t \right)=f\left[t,x\left(t \right) \right]$;V$t\geq 0$

where $x\in {R}^{n}$, $t\in {R}_{+}$and continuous function $f:{R}_{+}\times {R}^{n}\rightarrow {R}^{n}$

Definition 1.7 (Attractive Equilibrium State-Attractor)

Equilibrium state ${x}_{e}$is attractive if for every  ${t}_{0}\in {R}_{+}$there exist a number $\eta \left({t}_{0} \right)>0$such that

$\left|\left|{x}_{0} \right| \right|<\eta \left({t}_{0} \right)\Rightarrow s\left({t}_{0}+t,{t}_{0},{x}_{0} \right)\rightarrow {x}_{e}$as $t\rightarrow \sim$

Definition 1.8 (Uniformly Attractive Equilibrium State)

Equilibrium state ${x}_{e}$is uniformly attractive if for every ${t}_{0}\in {R}_{+}$, there exist a number $\eta \left({t}_{0} \right)>0$so that the following condition is true:

$\left|\left|{x}_{0} \right| \right|<\eta$,${t}_{0}\geq 0\Rightarrow s\left({t}_{0}+t,{t}_{0},{x}_{0} \right)\rightarrow {x}_{e}$ as $t\rightarrow \sim$ uniformly in ${x}_{0}$and${t}_{0}$.

Source:

Vukic, Zoran, Kuljaca, Ljubomir, Donlagic, Dali, Tesnjak, Sejid, Nonlinear Control Systems, Control Engineering Series, Marcel Dekker, New York USA, 2003.