Welcome to the fifth module of the MOOC Algorithmic
Information Dynamics.

As probably, you know by now my name is Narsis
and today I will talk to you about dynamical

systems.

At the end of the 17th century, Leibniz (1646-1716)
and Newton (1643-1727), independently one

from the other, invented a brilliant mathematical
tool: infinitesimal calculus or differential

and integral calculus.

This is an incredibly efficient crystal ball
to predict the future, provided the system

in question is governed by a differential
equation.

Using it , Poincaré's work on celestial mechanics
(Poincaré 1899), and specifically in a 270-page,

prize-winning, and initially flawed paper
(Poincaré 1890) start the qualitative theory

of dynamical systems.

The methods developed therein laid the basis
for the local and global analysis of nonlinear

differential equations, including the use
of first-return (Poincaré) maps, stability

theory for fixed points and periodic orbits,
stable and unstable manifolds, and the Poincaré

recurrence theorem.

In this module, the idea is to make you a
very generic introduction to dynamical systems

. we talk about how systems typically only
occupy a small subset of the overall space

as they cycle through some set of states.

We will be talking about attractors and the
fundamental role they play within dynamics

of a system.

We will discuss chaotic and complex regime
consisting of multiple attractors and equilibria

very briefly.

The concept of Boolean network as a discrete
dynamical system will be introduced and we

learn how we can analyse them.

Ok let see what we mean with dynamical system.

Within science and mathematics, dynamics is
the study of how things change with respect

to time, as opposed to describing things simply
in terms of their static properties.

The patterns we observe all around us in how
the state of things change overtime is an

alternative ways through which we can describe
the phenomena we see in our world.

A dynamical system is a set of possible states,
together with a rule that determines the present

state in terms of past states.

As examples for dynamical systems, you can
think of any system that is evolving in time.

For example, the pendulum, or whether evolution,
or the evolution of population of bacteria

or any kind of season that evolves through
time.

A dynamical system have two parts State space
and function, and we describe such a system

by them.

So lets see what they are.

Dynamical system is study of the thing, which
are changing overtime!

Those things are states and a state space
is a model used within dynamic systems to

capture this change in a system’s state
overtime.

Formally, State space is the set of all possible
states of a dynamical system.

Each state of the system corresponds to a
unique point in the state space.

For example, the state of an idealized pendulum
is uniquely defined by its angle and angular

velocity, so the state space is the set of
all possible pairs "(angle, velocity)", which

form the cylinder.

In general, any abstract set could be a state
space of some dynamical system.

It could be finite, consisting of just a few
points or consisting of an infinite number

of points forming a smooth manifold, as usually
the case in ordinary differential equations

and mappings.

Such a state space is often called a phase
space.

A state space could be infinite-dimensional,
as in partial differential equations and delay

differential equations.

In symbolic dynamics, it is a Cantor set,
which is zero-dimensional.

Moreover, the second part of dynamical system,
Function tells us, given the current state,

what the state of the system will be in the
next instant of time.

For investigating dynamical systems, it is
necessary to specify some characteristics

that provide a subdivision into special classes
of dynamical systems.

Specific methods are available for some of
these classes, thus such a classification

can help to simplify the analysis.

A dynamical system is deterministic If the
present state can be determined uniquely from

the past states (no randomness is allowed).

Stochastic models possess some inherent randomness.

Chaotic model is a deterministic model with
a behaviour that cannot be entirely predicted.

They are predictable in the very short term,
but appears random for longer periods.

An important characteristic of a dynamical
system is whether it is continuous or discrete.

In a discrete system, the state variables
change only at a countable number of points

in time.

These points in time are the ones at which
the event occurs/change in state.

In a continuous dynamical system, the state
variables change in a continuous way, and

not abruptly from one state to another (infinite
number of states).

When the reals are acting, the system is called
a continuous dynamical system, and when the

integers are acting, the system is called
a discrete dynamical system.

Continuous systems are given by differential
equations whereas discrete dynamical systems

(often called maps) are specified by difference
equations.

Let’s start by discrete, We denote time
by k or n, and the system can be solved by

iteration calculation called iterative maps.

Iterative maps give us less information but
are much simpler and better suited to dealing

with very many entities, where feedback is
important.

Typical example is annual progress of a bank
account.

If the initial deposit is 100000 euros and
annual interest is 3%, then we can describe

the system by:

In a continuous system, the 
time interval between our measurements is

negligibly small making it appear as one long
continuum and this is done through the language

of calculus and using differential equation
or a set of them.

For example, vertical throw is described by
initial conditions h(0), v(0) and equations:

where h is height and v is velocity of a body.

Calculus and differential equations have formed
a key part of the language of modern science

since the days of Newton and Leibniz.

Even though an analytical treatment of dynamical
systems is usually very complicated, obtaining

a numerical solution is (often) straightforward.

Solving differential equations numerically
can be done by a number of schemes.

The techniques for solving differential equations
based on numerical approximations were developed

before programmable computers existed.

During WorldWarII, it was common to find rooms
of people (usually women) working on mechanical

calculators to numerically solve systems of
differential equations for military calculations.

The easiest way is by the 1st order Euler’s
Method which uses the idea of local linearity

or linear approximation, where we use small
tangent lines over a short distance to approximate

the solution to an initial-value problem.

If we zoom in small enough, every curve looks
like a straight line, and therefore, the tangent

Line is a great way for us to calculate what
is happening over a period.

Today we have many numerical methods to solve
differential equations.

And there is no "best way" or "best method",
since the method to be chosen heavily depends

on the problem (stiff or non stiff, equation
or system, smooth or non smooth right hand

side, etc.).

For a "general" or "standard", a 4th order
Runge-Kutta is good, it's easy to add an error

estimator to it with no or little additional
cost.

A predictor-corrector Adams-method also does
the job, they are also quite popular.

Differential equations are great for few elements
they give us lots of information but they

also become very complicated very quickly.

Where as differential equations are central
to modern science iterative maps are central

to the study of nonlinear systems and their
dynamics as they allow us to take the output

to the previous state of the system and feed
it back into the next iteration, thus making

them well designed to capture the feedback
characteristic of nonlinear systems.

Another important classification of dynamical
system is based on linearity.

Let start by defining a Nonlinear System.

Nonlinear System is a set of (one or more)nonlinear
equations.

Nonlinear equations are equations where the
unknown quantity that we want to solve for

appears in a nonlinear fashion.

For example, if the quantity in question is
a function y(t),then terms such as y^2or sin

y would be nonlinear.

More precisely, a nonlinear equation is one
where a linear combination of solutions is

not a new solution.

In a linear system, function that is describing
the system behaviour must satisfy two basic

properties

additivity homogeneity .

Which one is linear?

g(x) = 3x; g(y) = 3y or f(x) = x2; f(y) = y2?

additivity g(x+y) = 3x + 3y = g(x) + g(y)

homogeneity 5 * g(x) = 5* 3x = 15x = g(5x)

If a system of ordinary differential equations
do not depend on the independent variable,

it called an Autonomous system.

If the independent variable is time, we call
it time-invariant system.

In an autonomous system If the input signal
x(t) produces an output y(t) then any time

shifted input, x(t + δ), results in a time-shifted
output y(t + δ).

Consider these two systems,

System A:
System B:

Are they Autonomous?

We examine system A and u can do the same
for system B. lets Start with a delay of the

input

Now we delay the output by δ

Therefore the system

is not time-invariant or is nonautonomous.