"What is a dynamical system?"

When we talk about systems in the most general sense, we are talking about anything that takes in a certain number of inputs and produces a certain number of outputs based on those inputs.

Figure 1: Generalized System

Generalized System

In the figure above, the utut inputs could be the jets on a satellite and the ytyt outputs could be the gyros describing the "bearing" of the satellite.

There are two basic divisions of systems: static and dynamic. In a static system, the current outputs are based solely on the instantaneous values of the current inputs. An example of a static system is a resistor hooked up to a current source:

Figure 2: Resistor connected to a current source

Resistor connected to a current source

At any given moment, the voltage across the resistor (the output) depends only on the value of the current running through it (the input). The current at any time tt is simply multiplied by the constant value describing the resistance RR to give the voltage VV. Now, let's see what happens if we replace the resistor with a capacitor.

Figure 3: Simple capacitor connected to a current source

Simple capacitor connected to a current source

Solving for the voltage in the current voltage relationship above, we have:

So in the case of the capacitor, the output voltage depends on the history of the current flowing through it. In a sense, this system has memory. When a system depends on the present and past input, it is said to be a dynamical system.

"Describing dynamical systems"

As seen in voltage-current relationship of a capacitor, differential equations have memory and can thus be used to describe dynamical systems. Take the following RLC circuit as an example:

Figure 4: RLC circuit: 2nd order

RLC circuit: 2nd order

In circuits (as well as in other applications), memory elements can be thought of as energy storage elements. In this circuit diagram, there are two energy-storing components: the capacitor and the inductor. Since there are two memory elements, it makes sense that the differential equation describing this system is second order.

In the most general case of describing a system with differential equations, higher order derivatives of output variables can be described as functions of lower order derivatives of the output variables and some derivatives of the input variables. Note that by saying "function" we make no assumptions about linearity or time-invariance.

By simply rearranging the equation for the RLC circuit above, we can show that that system is in fact covered by this general relationship.

Of course, dynamical systems are not limited to electrical circuits. Any system whose output depends on current and past inputs is a valid dynamical system. Take for example, the following scenario of relating a satellite's position to its inputs thrusters.

"Planar Orbit Satellite"

Example 1

Using a simple model of a satellite, we can say that its position is controlled by a radial thruster urur, which contributes to its vertical motion, and a tangential thruster u θ u θ which contributes to its motion tangential to its orbit. To simplify the analysis, let's assume that the satellite circles the earth in a planar orbit, and that its position is described by the distance r from the satellite to the center of the Earth and the angle θ as shown in the figure.

Figure 5: Simple planar orbit satellite example

Simple planar orbit satellite example

Using the laws of motion, the following set of differential equations can be deduced:

d2dt2rt-ddtrtθ2=ur-kr2 t2 r t t1 r t θ 2 ur k r 2 (5)

2ddtrtddtθt+rddtθt=uθ 2 t1 rt t1 θt r t1 θt uθ (6)