Dynamical Systems Described by Differential Equations and Equilibrium Points 📂Dynamics

Dynamical Systems Described by Differential Equations and Equilibrium Points

Definition ¹

Given a space $V$ and function $f : V \to V$, assume the following vector field is given as a differential equation: $$ \dot{v} = f(v) $$

If variable $t$ is included in the differential equation and $t$ is not explicitly shown, it is referred to as an Autonomous Differential Equation.
If a constant function $f_{0} (v)$ is a solution to the autonomous differential equation $\dot{v} = f(v)$, then $f_{0}$ is called an Equilibrium Point.

Description

Autonomous Systems

Dynamical systems expressed by autonomous differential equations are referred to as Autonomous Systems. Geometrically, since most are expressed as vector fields, they are simply called vector fields in the appropriate context. Usually, variable $t$ represents time, and the fact that the equation includes variable $t$ without explicitly showing it, for example, refers to an equation like the following: $$ \dot{y} = y $$ A non-trivial solution to the above differential equation is $y = e^t$. The reason for the term autonomous becomes clear when considering non-autonomous differential equations, which are, as the name suggests, differential equations where variable $t$ is explicitly shown. An example would be as follows, with the addition of term $\sin t$: $$ \dot{y} = y + \sin t $$ Systems represented by such differential equations can be seen as being influenced by some external interference over time $t$, rather than being $y$ itself. In this sense, calling equations that are not non-autonomous, autonomous differential equations seems appropriate.

Fixed Points

Equilibrium Points have a physical sense to them, and in mathematics, the term Fixed Point is preferred. In a system, a fixed point, as the name suggests, does not move. A point not moving means that the derivative indicating the change of position is entirely $0$, and since it’s a fixed point, it’s a constant function. Strictly speaking, it’s an element of the function space $C^{1} (X)$ constituted by the solutions to the differential equation rather than the domain $X$ defined for the function, i.e., it’s a fixed point as a function. However, depending on the textbook, it may also be loosely referred to as an element of $X$ that is a fixed point.

Notation in Differential Equations

In differential geometry, the notation of differentiation with respect to $s$ and $t$: $$ {{ df } \over { ds }} = f^{\prime} \quad \text{and} \quad {{ df } \over { dt }} = \dot{f} $$ Whether it’s dot $\dot{}$ or prime $'$, differentiation is differentiation, but in the context of differential geometry, symbols can be differentiated as shown above. Usually, $s$ represents the parameter of a unit speed curve, and $t = t(s)$ represents the parameter of a curve reparametrized by the length of the curve.

Although not strictly necessary, in dynamics, the notation using dot $\dot{y}$ is often used in place of prime $y '$, as dynamics generally deals with vector fields in terms of changes over time $t$.

Examples

Consider the Lorenz attractor as an example: $$ \begin{cases} \dot{x} = - \sigma x + \sigma y \\ \dot{y} = - xz + \rho x - y \\ \dot{z} = xy - \beta z \end{cases} $$ Fixed points can be obtained by substituting $0$ for all left sides, since they describe points that do not move on domain $\mathbb{R}^3$. $$ \begin{cases} \displaystyle 0 = - \sigma x + \sigma y \\ \displaystyle 0 = - xz + \rho x - y \\ \displaystyle 0 = xy - \beta z \end{cases} $$ Through simple calculations, the following three fixed points $F_{i}$ can be found: $$ F_{1} = F_{1}(t) = (0,0,0) \\ F_{2} = F_{2}(t) = \left( \sqrt{\beta (\rho - 1)},\sqrt{\beta (\rho - 1)}, (\rho-1) \right) \\ F_{3} = F_{3}(t) = \left( -\sqrt{\beta (\rho - 1)},-\sqrt{\beta (\rho - 1)}, (\rho-1) \right) $$ Note that it is expressed as a function like $F_{i} = F_{i} (t)$. At first glance, $F_{i}$ seems like a point in $\mathbb{R}^{3}$, but according to its definition, it’s obtained as a constant function whose value does not change over time $t$ and as a solution to the Lorenz differential equation. Conceptually, there’s no difference from a point in three-dimensional space.