Discrete-Time Dynamical Systems Suppose we measure changes in a system over a period of time, and notice patterns in the data. If possible, weâd like to quantify these patterns of change into a dynamical rule - a rule that speciï¬es how the system will change over a period of time. In doing so, we will be able to predict future. Dynamical systems are about the evolution of some quantities over time. This evolution can occur smoothly over time or in discrete time steps. Here, we introduce dynamical systems where the state of the system evolves in discrete time steps, i.e., discrete dynamical systems. About this book. Discrete dynamical systems are essentially iterated functions. Given the ease with which computers can do iteration, it is now possible for anyone with access to a personal computer to generate beautiful images whose roots lie in discrete dynamical systems. Images of Mandelbrot and Julia sets abound in. Lectures on Dynamical Systems Anatoly Neishtadt. Time can be either discrete, whose set of values is the set of integer numbers Z, or continuous, whose set of values is the set of real numbers R. Law of evolution is the rule which allows us, if we know the state of the. 0n form a discrete-time dynam-ical system. We will use the term dynamical system to refer to either discrete-time or continuous-time dynamical systems. Most concepts and results in dy-namical systems have both discrete-time and continuous-time versions. The continuous-time version can often be deduced from the discrete-time ver.
Description
This book covers important topics like stability, hyperbolicity, bifurcation theory and chaos, topics which are essential in order to understand the fascinating behavior of nonlinear discrete dynamical systems. The theory is illuminated by several examples and exercises, many of them taken from population dynamical studies. Solution methods of linear systems as well as solution methods of discrete optimization (control) problems are also included. In an Appendix it is explained how to estimate parameters in nonlinear discrete models.
Content
The dynamical system concept is a mathematicalformalization for any fixed 'rule' which describes the time dependence of a point's position in its ambient space. Download iphone ringtone for android. The concept unifies very different types of such 'rules' in mathematics: the different choices made for how time is measured and the special properties of the ambient space may give an idea of the vastness of the class of objects described by this concept. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the ambient space may be simply a set, without the need of a smooth space-time structure defined on it.
Formal definition[edit]
There are two classes of definitions for a dynamical system: one is motivated by ordinary differential equations and is geometrical in flavor; and the other is motivated by ergodic theory and is measure theoretical in flavor. The measure theoretical definitions assume the existence of a measure-preserving transformation. This appears to exclude dissipative systems, as in a dissipative system a small region of phase space shrinks under time evolution. A simple construction (sometimes called the KrylovâBogolyubov theorem) shows that it is always possible to construct a measure so as to make the evolution rule of the dynamical system a measure-preserving transformation. In the construction a given measure of the state space is summed for all future points of a trajectory, assuring the invariance.
The difficulty in constructing the natural measure for a dynamical system makes it difficult to develop ergodic theory starting from differential equations, so it becomes convenient to have a dynamical systems-motivated definition within ergodic theory that side-steps the choice of measure.
General definition[edit]
In the most general sense,[1][2]a dynamical system is a tuple (T, M, Φ) where T is a monoid, written additively, M is a non-empty set and Φ is a function
with
The function Φ(t,x) is called the evolution function of the dynamical system: it associates to every point in the set M a unique image, depending on the variable t, called the evolution parameter. M is called phase space or state space, while the variable x represents an initial state of the system.
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We often write
if we take one of the variables as constant.
is called flow through x and its graphtrajectory through x. The set
is called orbit through x.Note that the orbit through x is the image of the flow through x.A subset S of the state space M is called Φ-invariant if for all x in S and all t in T
Thus, in particular, if S is Φ-invariant,I(x)=T{displaystyle I(x)=T} for all x in S. That is, the flow through x must be defined for all time for every element of S.
Geometrical cases[edit]
In the following cases, M is a manifold (or its extreme case a graph). Dynamical systems are defined as tuples of which one element is a manifold.
Real dynamical system[edit]
A real dynamical system, real-time dynamical system, continuous time dynamical system, or flow is a tuple (T, M, Φ) with T an open interval in the real numbers R, M a manifold locally diffeomorphic to a Banach space, and Φ a continuous function. If T=R we call the system global, if T is restricted to the non-negative reals we call the system a semi-flow. If Φ is continuously differentiable we say the system is a differentiable dynamical system. If the manifold M is locally diffeomorphic to Rn, the dynamical system is finite-dimensional; if not, the dynamical system is infinite-dimensional. Note that this does not assume a symplectic structure.
Discrete dynamical system[edit]
A discrete dynamical system, discrete-time dynamical system, map or cascade is a tuple (T, M, Φ) where T is the set of integers, M is a manifold locally diffeomorphic to a Banach space, and Φ is a function. If T is restricted to the non-negative integers we call the system a semi-cascade.[3]
Cellular automaton[edit]
A cellular automaton is a tuple (T, M, Φ), with T a lattice such as the integers or a higher-dimensional integer grid, M is a set of functions from an integer lattice (again, with one or more dimensions) to a finite set, and Φ a (locally defined) evolution function. As such cellular automata are dynamical systems. The lattice in M represents the 'space' lattice, while the one in T represents the 'time' lattice.
Measure theoretical definition[edit]
A dynamical system may be defined formally, as a measure-preserving transformation of a sigma-algebra, the triplet (T, (X, Σ, μ), Φ) Here, T is a monoid (usually the non-negative integers), X is a set, and (X, Σ, μ) is a probability space. A map Φ: X â X is said to be Σ-measurable if and only if, for every Ï in Σ, one has Φâ1(Ï) â Σ. A map Φ is said to preserve the measure if and only if, for every Ï in Σ, one has μ(Φâ1(Ï)) = μ(Ï). Combining the above, a map Φ is said to be a measure-preserving transformation of X, if it is a map from X to itself, it is Σ-measurable, and is measure-preserving. The triplet (T, (X, Σ, μ), Φ), for such a Φ, is then defined to be a dynamical system.
The map Φ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iteratesÏn=ÏâÏââ¦âÏ{displaystyle scriptstyle phi ^{n}=phi circ phi circ ldots circ phi } for every integer n are studied. For continuous dynamical systems, the map Φ is understood to be finite time evolution map and the construction is more complicated.
Relation to geometric definition[edit]
Many different invariant measures can be associated to any one evolution rule. In ergodic theory the choice is assumed made, but if the dynamical system is given by a system of differential equations the appropriate measure must be determined. Some systems have a natural measure, such as the Liouville measure in Hamiltonian systems, chosen over other invariant measures, such as the measures supported on periodic orbits of the Hamiltonian system. For many dissipative chaotic systems the choice of invariant measure is technically more challenging. The measure needs to be supported on the attractor, but attractors have zero Lebesgue measure and the invariant measures must be singular with respect to the Lebesgue measure.
For hyperbolic dynamical systems, the SinaiâRuelleâBowen measures appear to be the natural choice. They are constructed on the geometrical structure of stable and unstable manifolds of the dynamical system; they behave physically under small perturbations; and they explain many of the observed statistics of hyperbolic systems.
Construction of dynamical systems[edit]
The concept of evolution in time is central to the theory of dynamical systems as seen in the previous sections: the basic reason for this fact is that the starting motivation of the theory was the study of time behavior of classical mechanical systems, that is the study of the initial value problems for their describing systems of ordinary differential equations.
where
The solution is the evolution function already introduced in above
Some formal manipulation of the system of differential equations shown above gives a more general form of equations a dynamical system must satisfy
where G:(TÃM)MâC{displaystyle {mathfrak {G}}:{{(Ttimes M)}^{M}}to mathbf {C} } is a functional from the set of evolution functions to the field of the complex numbers.
![]() Discrete Dynamical Systems Galor PdfCompactification of a dynamical system[edit]
Given a global dynamical system (R, X, Φ) on a locally compact and Hausdorfftopological spaceX, it is often useful to study the continuous extension Φ* of Φ to the one-point compactificationX* of X. Although we lose the differential structure of the original system we can now use compactness arguments to analyze the new system (R, X*, Φ*).
In compact dynamical systems the limit set of any orbit is non-empty, compact and simply connected.
References[edit]
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