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Dynamical system is a set of n first-order ordinary differential equations of the form

(8.1)

where x_{a} = x_{a}(t) are unknown functions of time, a = 1, 2.......... n and f_{a} are arbitrary differentiable functions of variables x_{a} and possibly of time t. If functions f_{a} do not depend on time explicitly, dynamical system is called autonomous, otherwise it is called non-autonomous. Using the index notation, dynamical system (8.1) can be written briefly in the form

(8.2)

where x stands for the n tuple of variables x_{a}. Autonomous system is then

In this notation we suppress time-dependence of x_{a} on time because this dependence is assumed implicitly. Motivated by Hamilton's formalism, we intend to interpret the solution x_{a} = x_{a}(t) as the motion in the phase space. Phase space is an abstract space

with coordinates x_{a}. Arbitrary point x âˆˆ M represents the state of physical system described by equations (8.1). Solution of dynamical system is not unique unless we specify the initial conditions, i.e. values of coordinates x_{a} at some given initial time t_{0},

Usually we set t_{0} = 0. The nâˆ’tuple of initial coordinates x_{a0} will be denoted simply by x_{0} âˆˆ M.

Suppose we choose a point x_{0}âˆˆ M at time t_{0} = 0 as in figure 8.1. A mathematical theorem guarantees that there exists unique solution x = x(t) satisfying (8.1) such that x(0) = x_{0}. The solution x = x(t) is also called the phase trajectory. Equations (8.1) essentially state that vector f(x(t)) evaluated at arbitrary point of the trajectory is in fact tangent to the trajectory, see figure 8.1. Hence, we can interpret vector field f(x) as a velocity. Although it can be very difficult or even impossible to solve the equations of motion (8.1), the velocity gives us a good idea about the behaviour of the system.

**Fig. 8.1. Two-dimensional dynamical system of the form . Initial position is at x _{0} = x(0). The "velocity" vector at x_{0} is f (x_{0} ) and determines the tra jectory of the system in the in nitesimal neighbourhood of the initial point.**

**Example**

Let us see an illustrating example. We are already familiar with the equation of harmonic oscillator

This is a second order equation but we can bring into into the rs-order form by setting

Then we have

and

Hence, instead of single equation of second order we now have two equations of first order

(8.3)

Clearly, this is a dynamical system (8.1) if we set f_{1} = x_{2} and f_{2} = x_{1}. Thus, the velocity eld is

f (x_{1}; x_{2}) = (x_{2}; x_{1}) (8.4)

This vector eld can be visualised in Mathematica by function VectorPlot:

**Resulting gure is**

This picture agrees with our previous analysis when we used the conservation of energy to show that the phase tra jectories of harmonic oscillator are circles (or ellipses when using SI units). Another possibility is to use function StreamPlot with the same arguments which yields

**Implementation in Mathematica**

In this section we show how to implement a dynamical system in Mathematica in a convenient way.

This code deserves a brief explanation. Arguments of the function DynDys are

- pure function f { this is a vector function representing the right hand side of dynamical system (8.1);
- initial conditions IC { list of the initial values of variables x
_{a}at time t = 0; - tmax { upper bound of interval t 2 (0; t
_{max}).

Hence, function DynSys can be called, e.g. with the arguments

In this example, pure function f is

{#2, -#1} &

which is equivalent to

f (x_{1}; x_{2}) = (x_{2}; x_{1}):

Clearly, this corresponds to harmonic oscillator (8.4).

Initial conditions IC are set to

x_{1}(0) = 1; x_{2}(0) = 1

and we want to nd the solution in time interval (0; 10).

Now suppose that we called function DynSys with the arguments above and let us explain how this function works. Thus, we assume that the arguments are

f = {#2, -#1}&

IC = {1, 1}

tmax = 10.

The rst command

vars = Table[ x[a][t], {a, 1, Length[IC]}];

creates a list of variables xa_{(}t) in the form

vars = { x[1][t], x[2][t] }.

The left hand side of equations is generated simply by calling

lhs = D[vars, t]

which yields

lhs = { x[1]'[t], x[2]'[t] }.

Now we form the right hand side of equations. Recall that vars is the list of variables. We want to evaluate functions fa at point x_{a}, i.e. we need the expression

However, we cannot write simply f[vars] because this would mean

f[{x[1][t], x[2][t]}]

while what we need is

f[ x[1][t], x[2][t] ].

Hence, we must turn the list vars into the sequence of arguments by replacing its head. Command

rhs = f[Sequence @@ vars].

leads to correct application of function f to arguments x_{a}:

Having de ned the left hand side and the right hand side of dynamical system separately, we join them in a usual way,

eqs = Equal @@@ Transpose[ {lhs, rhs} ]

Next we de ne the initial conditions with values speci ed in argument IC={1,1}. We need to produce the list

{ x[1][0] == 1, x[2][0] == 1 }

The left hand side of initial conditions consists of the elements of vars evaluated at time 0,

vars /. t->0,

the right hand side consists of the elements of IC. We con join them together by

inConds = Equal @@@ Transpose[{vars /. t -> 0, IC}];

Finally we solve the list of equations of motion and initial conditions by NDSolve:

NDSolve[Join[eqs, inConds], vars, {t, 0, tmax}].

Now, the reader should be familiar with functionality of function DynSys. We use this function in the following examples.

To nalize this section we show how to use function DynSys to solve the motion of harmonic oscillator.

**Chaotic pendulum **

In the previous chapters we introduced the mathematical pendulum as a simple example of physical system which has only one degree of freedom (angle of de

ection Î¸) and is described by the Lagrange equation

This equations is non-linear because of the presence of the sine. We have seen that this equation cannot be solved in terms of elementary functions but we were able to nd the numerical solution. Moreover, using the Hamiltonian formalism we were able to plot the phase tra jectories without actually solving the equation of motion.

We can generalize the model of mathematical pendulum in several ways. First, any realistic system is dissipative, i.e. there are resisting forces acting against the motion of the pendulum. As an approximation, resisting force is proportional to velocity and has opposite direction. In the case of pendulum, velocity is proportional to Î¸^{.} and hence equation of pendulum with resisting force has the form

where b is the constant characterizing the strength of resisting force. For example, it can be related to the viscosity of the medium in which the pendulum moves.

Pendulum with resisting force is called damped pendulum.

Next we can assume that in addition to restoring gravitational force there is an external force acting on the pendulum. Such force is called driving force. In the presence of driving force, even if the initial velocity of the pendulum is zero (and the pendulum is at equilibrium position), driving force will make the pendulum to move.

Resulting motion of the pendulum will be a "mixture" of two motions: periodic motion due to self-oscillations of the pendulum, and motion due to driving force.

Pendulum with the driving force, driven pendulum with the friction is described by equation

(8.5)

where we assume that driving force is harmonic with angular frequency Î© and amplitude F_{0}. In the subsequent analysis we will show that this kind of pendulum exhibits chaotic behaviour and hence we also call it chaotic pendulum.

We start with rewriting equation (8.5) in the form of dynamical system. This is straightforward since we can de ne

We will freely pass from notation (Î¸, p; Ï†) to equivalent notation (x_{1}, x_{2}, x_{3}) according to the context. By de nition, variable Ï† satis es equation

while variable p (which is clearly related to the momentum of the pendulum) was de ned by

which can be consequently regarded as an equation for Î¸. The only true dynamical equation is an equation for p which follows from (8.5):

Thus, variables (x_{1}, x_{2}, x_{3}) = (Î¸, p; Ï†) are determined by dynamical system

(8.6)

supplemented with initial conditions, i.e. values of x_{a} at time t = 0.

Dynamical system (8.6) can be solved in Mathematica using function DynSys de ned in the previous section. We choose initial conditions

and investigate how values b, Î© and F_{0} affect the behaviour of the pendulum. First we set

b = 0, F0 = 0, Î© = 0.

So, we reduce driven pendulum to the case of ordinary mathematical pendulum without friction and driving force (damping coeffcient b = 0 and the amplitude of the force is F_{0 }= 0).

Here we have chosen t_{max} = 10 but the reader should adjust this parameter in order to reproduce all gures below. Now we can plot the phase tra jectory in a usual way.

Let us add the friction now and set

b = 0.1,

see gure 8.2 for the result. We can see that the phase tra jectory is a spiral which, in the limit ends at the origin of the phase plane. This means that the oscillations are damped until the pendulum stops. Slightly more "fancy" picture can be obtained by

which results in gure 8.3.

Fig. 8.2. Parameters b = 0.1, F_{0} = 0. Non-zero friction leads to damped oscillations of the pendulum.

Fig. 8.3. Phase tra jectory of damped pendulum together with the time dependence of deection Î¸ = Î¸(t).

Let us see how the driving force affects the motion. For this purpose we set

b = 0, F_{0 }= 1, Î© = 2, and choose the initial conditions to

In other words, the pendulum is initially at its equilibrium position and, hence, without driving force it would stay at rest. However, the presence of driving force leads to solution plotted in gure 8.4.

Fig. 8.4. Motion of the pendulum without friction under the external driving force with amplitude F_{0 }= 1 and angular frequency Î© = 2. Initial position of the pendulum is Î¸ (0) = p(0) = 0.

In the following example we choose

b = 1, F0 = 1, Î© = 1,

see gure 8.5. An interesting feature of this solution is the presence of short transient stage during which the phase tra jectory follows outgoing spiral but then settles at circular periodic orbit. Such behaviour is called limit cycle.

To experiment with values of parameters b, F_{0} and Î© and with initial values Î¸(0), p(0) and Ï†(0). We can see that resulting motion of the pendulum is a consequence of complicated and delicate interplay between three motions:

Fig. 8.5. Motion of the pendulum with friction (b = 1) under the external driving force with amplitude F_{0} = 1 and angular frequency Î© = 1. Initial position of the pendulum is Î¸(0) = p(0) = 0.

- self-oscillations of the pendulum;
- resisting force (friction);
- external driving force.

While for some combinations of parameters the motion is perfectly understandable (like in the absence of the friction and the driving force or in the absence of driving force but in the presence of friction), for general values the motion is unpredictable, chaotic.

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