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.