Dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, fluid dynamics, the Brownian motion, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real number or or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it.

At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector space in a geometrical manifold. The evolution rule of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a given time interval only one future state follows from the current state.

However, some systems are stochastic, in that random events also affect the evolution of the state variables.

The study of dynamical systems is the focus of dynamical systems theory, which has applications to a wide variety of fields such as mathematics, physics, biology, chemistry, engineering,

economics,

Giancarlo Gandolfo (2025). 9783642135033, Springer. ISBN 9783642135033

Cliodynamics, and medicine. Dynamical systems are a fundamental part of chaos theory, logistic map dynamics, bifurcation theory, the self-assembly and self-organization processes, and the edge of chaos concept.

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Overview The concept of a dynamical system has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is an implicit relation that gives the state of the system for only a short time into the future. (The relation is either a differential equation, difference equation or other time scale.) To determine the state for all future times requires iterating the relation many times—each advancing time a small step. The iteration procedure is referred to as solving the system or integrating the system. If the system can be solved, then, given an initial point, it is possible to determine all its future positions, a collection of points known as a trajectory or orbit.

Before the advent of computers, finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system.

For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because:

• The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations. The approximations used bring into question the validity or relevance of numerical solutions. To address these questions several notions of stability have been introduced in the study of dynamical systems, such as Lyapunov stability or structural stability. The stability of the dynamical system implies that there is a class of models or initial conditions for which the trajectories would be equivalent. The operation for comparing orbits to establish their equivalence changes with the different notions of stability.
• The type of trajectory may be more important than one particular trajectory. Some trajectories may be periodic, whereas others may wander through many different states of the system. Applications often require enumerating these classes or maintaining the system within one class. Classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes. Linear dynamical systems and systems that have two numbers describing a state are examples of dynamical systems where the possible classes of orbits are understood.
• The behavior of trajectories as a function of a parameter may be what is needed for an application. As a parameter is varied, the dynamical systems may have bifurcation points where the qualitative behavior of the dynamical system changes. For example, it may go from having only periodic motions to apparently erratic behavior, as in the Turbulence.
• The trajectories of the system may appear erratic, as if random. In these cases it may be necessary to compute averages using one very long trajectory or many different trajectories. The averages are well defined for ergodic theory and a more detailed understanding has been worked out for hyperbolic systems. Understanding the probabilistic aspects of dynamical systems has helped establish the foundations of statistical mechanics and of chaos theory.

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History Many people regard French mathematician Henri Poincaré as the founder of dynamical systems.Holmes, Philip. "Poincaré, celestial mechanics, dynamical-systems theory and "chaos"." Physics Reports 193.3 (1990): 137–163. Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). These papers included the Poincaré recurrence theorem, which states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state.

Aleksandr Lyapunov developed many important approximation methods. His methods, which he developed in 1899, make it possible to define the stability of sets of ordinary differential equations. He created the modern theory of the stability of a dynamical system.

In 1913, George David Birkhoff proved Poincaré's "Last Geometric Theorem", a special case of the three-body problem, a result that made him world-famous. In 1927, he published his Dynamical Systems. Birkhoff's most durable result has been his 1931 discovery of what is now called the ergodic theorem. Combining insights from physics on the ergodic hypothesis with measure theory, this theorem solved, at least in principle, a fundamental problem of statistical mechanics. The ergodic theorem has also had repercussions for dynamics.

Stephen Smale made significant advances as well. His first contribution was the Horseshoe map that jumpstarted significant research in dynamical systems. He also outlined a research program carried out by many others.

Oleksandr Mykolaiovych Sharkovsky developed Sharkovsky's theorem on the periods of discrete dynamical systems in 1964. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period.

In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity. Palestinian mechanical engineer Ali H. Nayfeh applied nonlinear dynamics in mechanics and engineering systems.

His pioneering work in applied nonlinear dynamics has been influential in the construction and maintenance of machines and structures that are common in daily life, such as ships, cranes, bridges, buildings, skyscrapers, jet engines, rocket engines, aircraft and spacecraft.

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Formal definition In the most general sense,Giunti M. and Mazzola C. (2012), " Dynamical systems on monoids: Toward a general theory of deterministic systems and motion". In Minati G., Abram M., Pessa E. (eds.), Methods, models, simulations and approaches towards a general theory of change, pp. 173–185, Singapore: World Scientific. Mazzola C. and Giunti M. (2012), " Reversible dynamics and the directionality of time". In Minati G., Abram M., Pessa E. (eds.), Methods, models, simulations and approaches towards a general theory of change, pp. 161–171, Singapore: World Scientific. . a dynamical system is a tuple ( T, X, Φ) where T is a monoid, written additively, X is a non-empty set and Φ is a function

$\backslash Phi:\; U\; \backslash subseteq\; (T\; \backslash times\; X)\; \backslash to\; X$

with

$\backslash mathrm\{proj\}\_\{2\}(U)\; =\; X$ (where $\backslash mathrm\{proj\}\_\{2\}$ is the 2nd projection map)

and for any x in X:

$\backslash Phi(0,x)\; =\; x$

$\backslash Phi(t\_2,\backslash Phi(t\_1,x))\; =\; \backslash Phi(t\_2\; +\; t\_1,\; x),$

for $\backslash ,\; t\_1,\backslash ,\; t\_2\; +\; t\_1\; \backslash in\; I(x)$ and $\backslash \; t\_2\; \backslash in\; I(\backslash Phi(t\_1,\; x))$, where we have defined the set $I(x)\; :=\; \backslash \{\; t\; \backslash in\; T\; :\; (t,x)\; \backslash in\; U\; \backslash \}$ for any x in X.

In particular, in the case that $U\; =\; T\; \backslash times\; X$ we have for every x in X that $I(x)\; =\; T$ and thus that Φ defines a Semigroup action of T on X.

The function Φ( t, x) is called the evolution function of the dynamical system: it associates to every point x in the set X a unique image, depending on the variable t, called the evolution parameter. X is called phase space or state space, while the variable x represents an initial state of the system.

We often write

$\backslash Phi\_x(t)\; \backslash equiv\; \backslash Phi(t,x)$

$\backslash Phi^t(x)\; \backslash equiv\; \backslash Phi(t,x)$

if we take one of the variables as constant. The function

$\backslash Phi\_x:I(x)\; \backslash to\; X$

is called the flow through x and its graph is called the trajectory through x. The set

$\backslash gamma\_x\; \backslash equiv\backslash \{\backslash Phi(t,x)\; :\; t\; \backslash in\; I(x)\backslash \}$

is called the orbit through x. The orbit through x is the image of the flow through x. A subset S of the state space X is called Φ- invariant if for all x in S and all t in T

$\backslash Phi(t,x)\; \backslash in\; S.$

Thus, in particular, if S is Φ- invariant, $I(x)\; =\; T$ for all x in S. That is, the flow through x must be defined for all time for every element of S.

More commonly 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.

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Geometrical definition In the geometrical definition, a dynamical system is the tuple $\backslash langle\; \backslash mathcal\{T\},\; \backslash mathcal\{M\},\; f\backslash rangle$. $\backslash mathcal\{T\}$ is the domain for time – there are many choices, usually the reals or the integers, possibly restricted to be non-negative. $\backslash mathcal\{M\}$ is a manifold, i.e. locally a Banach space or Euclidean space, or in the discrete case a graph. f is an evolution rule t →  f  ^t (with $t\backslash in\backslash mathcal\{T\}$) such that f ^t is a diffeomorphism of the manifold to itself. So, f is a "smooth" mapping of the time-domain $\backslash mathcal\{T\}$ into the space of diffeomorphisms of the manifold to itself. In other terms, f( t) is a diffeomorphism, for every time t in the domain $\backslash mathcal\{T\}$ .

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Real dynamical system 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 R, M a manifold locally diffeomorphic to a Banach space, and Φ a continuous function. If Φ is continuously differentiable we say the system is a differentiable dynamical system. If the manifold M is locally diffeomorphic to R ⁿ, the dynamical system is finite-dimensional; if not, the dynamical system is infinite-dimensional. This does not assume a symplectic structure. When T is taken to be the reals, the dynamical system is called global or a flow; and if T is restricted to the non-negative reals, then the dynamical system is a semi-flow.

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Discrete dynamical system A discrete dynamical system, discrete-time dynamical system is a tuple ( T, M, Φ), where M is a manifold locally diffeomorphic to a Banach space, and Φ is a function. When T is taken to be the integers, it is a cascade or a map. If T is restricted to the non-negative integers we call the system a semi-cascade.

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Cellular automaton A cellular automaton is a tuple ( T, M, Φ), with T a lattice such as the or a higher-dimensional integer lattice, 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.

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Multidimensional generalization Dynamical systems are usually defined over a single independent variable, thought of as time. A more general class of systems are defined over multiple independent variables and are therefore called multidimensional systems. Such systems are useful for modeling, for example, image processing.

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Compactification of a dynamical system Given a global dynamical system ( R, X, Φ) on a locally compact and Hausdorff space topological space X, it is often useful to study the continuous extension Φ* of Φ to the one-point compactification X* 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 space and simply connected.

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Measure theoretical definition A dynamical system may be defined formally as a measure-preserving transformation of a measure space, 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 measure space, meaning that Σ is a sigma-algebra on X and μ is a finite measure on ( X, Σ). A map Φ: X → X is said to be Σ-measurable if and only if, for every σ in Σ, one has $\backslash Phi^\{-1\}\backslash sigma\; \backslash in\; \backslash Sigma$. A map Φ is said to preserve the measure if and only if, for every σ in Σ, one has $\backslash mu(\backslash Phi^\{-1\}\backslash sigma\; )\; =\; \backslash mu(\backslash sigma)$. 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 iterates $\backslash Phi^n\; =\; \backslash Phi\; \backslash circ\; \backslash Phi\; \backslash circ\; \backslash dots\; \backslash circ\; \backslash Phi$ for every integer n are studied. For continuous dynamical systems, the map Φ is understood to be a finite time evolution map and the construction is more complicated.

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Relation to geometric definition The measure theoretical definition assumes the existence of a measure-preserving transformation. Many different invariant measures can be associated to any one evolution rule. If the dynamical system is given by a system of differential equations the appropriate measure must be determined. This 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 and assumes the choice has been made. A simple construction (sometimes called the Krylov–Bogolyubov theorem) shows that for a large class of systems 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.

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 chaotic dissipative 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. A small region of phase space shrinks under time evolution.

For hyperbolic dynamical systems, the Sinai–Ruelle–Bowen measures appear to be the natural choice. They are constructed on the geometrical structure of stable manifold of the dynamical system; they behave physically under small perturbations; and they explain many of the observed statistics of hyperbolic systems.

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Construction of dynamical systems 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. But a system of ordinary differential equations must be solved before it becomes a dynamic system. For example, consider an initial value problem such as the following:

$\backslash dot\{\backslash boldsymbol\{x\}\}=\backslash boldsymbol\{v\}(t,\backslash boldsymbol\{x\})$

$\backslash boldsymbol\{x\}|\_=\backslash boldsymbol\{x\}\_0$

where

• $\backslash dot\{\backslash boldsymbol\{x\}\}$ represents the velocity of the material point x
• M is a finite dimensional manifold
• v: T × M → TM is a vector field in R ⁿ or C ⁿ and represents the change of velocity induced by the known acting on the given material point in the phase space M. The change is not a vector in the phase space  M, but is instead in the tangent space TM.

There is no need for higher order derivatives in the equation, nor for the parameter t in v( t, x), because these can be eliminated by considering systems of higher dimensions.

Depending on the properties of this vector field, the mechanical system is called

• autonomous, when v( t, x) = v( x)
• homogeneous when v( t, 0) = 0 for all t

The solution can be found using standard ODE techniques and is denoted as the evolution function already introduced above

$\backslash boldsymbol(t)=\backslash Phi(t,\backslash boldsymbol\_0)$

The dynamical system is then ( T, M, Φ).

Some formal manipulation of the system of differential equations shown above gives a more general form of equations a dynamical system must satisfy

$\backslash dot\{\backslash boldsymbol\{x\}\}-\backslash boldsymbol\{v\}(t,\backslash boldsymbol\{x\})=0\; \backslash qquad\backslash Leftrightarrow\backslash qquad\; \backslash mathfrak\backslash left(t,\backslash Phi(t,\backslash boldsymbol\_0)\backslash right)=0$

where $\backslash mathfrak\{G\}:$

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