Dynamical system

In mathematics, physics, engineering and system theory, a dynamical system is the description of how a system evolves in time.

"We express our observables as numbers and record them over time"chaos book: https://chaosbook.org/version17/chapters/ChaosBook.pdf ch 2.1
     

For example, an astronomer can experimentally record the positions of how the planets move in the sky, and this can be considered a complete enough description of a dynamical system. In the case of planets there is also enough knowledge to codify this information as a set of differential equations with initial conditions, or as a map from the present state to a future state in a predefined state space with a time parameter t , or as an orbit in remark 2.1

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,

Erwin Kreyszig (2026). 9780470646137, Wiley. ISBN 9780470646137

economics,

Giancarlo Gandolfo (2026). 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

chaos book: https://chaosbook.org/version17/chapters/ChaosBook.pdf Appendix 1.1]]
     

Part 1, Geometry of chaos The relation from one state and another is either explicit such as a function in the parameter t predicting position and velocity of a particle or implicit such as a differential equation, difference equation or other time scale. Some times it may not be possible to define such a description, there may not even be a differential equation predicting /ref>

Important properties are existence and uniqueness of solutions, integrability (i.e. the existence of conserved quantities), the possibility to solve the system and be able to compute the state at any point in time. Other properties are whether the system is discrete, continuous, differentiable, smooth function, deterministic,ergodic, stochastic or Chaos theory.Here is an example of /ref>Yet another /ref>

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.One of the first to get the intuition of numerical computations for weather forecasting is Richardson, he imagined a set of human people doing computations 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. Appendix 1.1 - Chaos is born
• 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.
  (2026). 9783319332741 ISBN 9783319332741
• 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 /ref>

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Examples Simple 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.

File:Three-body Problem Animation with COM.gif|Three body problem: Approximate trajectories of three identical bodies located at the vertices of a scalene triangle and having zero initial velocities. File:Arnoldcatmap.svg|Arnold cat map: picture showing how the linear map stretches the unit square and how its pieces are rearranged when the modulo operation is performed. The lines with the arrows show the direction of the contracting and expanding File:Ising-tartan.png|Baker's map: Example of a measure that is invariant under the action of the (unrotated) baker's map: an invariant measure. Applying the baker's map to this image always results in exactly the same image. File:Stadium billiard.gif|alt=|Billiards: A particle moving inside the Bunimovich stadium, a well-known chaotic billiard. File:OuterBilliardsDefinition.png|Outer billiards: defined relative to a pentagon File:Bouncing_ball_strobe_edit.jpg|Bouncing ball dynamics: The motion is not quite parabola due to air resistance. File:Circle map bifurcation.jpeg|Bifurcation diagram for a Circle map. Black regions correspond to Arnold tongues. File:Miimcr.png|The recursive application of a Complex quadratic polynomial as a complex plane map gives a Dynamical system. Here there is a Dynamical plane with a Julia set and critical orbit. File:Double pendulum simulation.gif|right|Motion of the double compound pendulum (from numerical integration of the equations of motion) File:Dyadic trans.gif|right|Dyadic transformation xy plot where x =  x₀ ∈ 0, 1 is Rational number and y =  x _n for all  n File:Lorenz attractor yb.svg|The Lorenz attractor arises in the study of the Lorenz system, a dynamical system. File:Heard Island Karman vortex street.jpg|An example of a Kármán vortex street, an emergent phenomenon from Fluid dynamics. File:Kicked Rotor Phase Portrait.png| The Kicked Rotor, a famous chaotic system File:PIA17173 Titan resonances in Saturn's C ring.jpg|Orbital resonance in Saturn's rings. File:Hyperion true.jpg|The chaotic rotation of Hyperion. The Solar System as a whole is full of examples of dynamical systems from Celestial mechanics

Other classical examples include:

• Hénon map
• Irrational rotation
• Kaplan–Yorke map
• Lorenz attractor
• Quadratic map simulation system
• Rössler map
• Swinging Atwood's machine
• Tent map

Any mathematical map can be treated as the definition of a dynamical system for example:

• List of chaotic maps

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History Many people regard French mathematician Henri Poincaré as the founder of dynamical systems. 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. Appendix 1.1.1

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. Appendix 1.2 - Chaos grows up

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. Appendix 1.4 - Periodic orbit theory

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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Generalizations The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory, physics such as Phase space, quantum state and thermodynamic state, engineering such as system theory, control theory and even information theory.

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Mathematical intuition From a mathematics perspective in the most general case the state space X is treated as a generic set of abstract algebra. This space X has a semi-group structure on it (i.e. where only associativity is required) and there is most often a natural choice for an Identity element, which is typically attached to the origin of the chosen reference frame. This semi-group can be intuitively interpreted as the time coordinate t Classes of Dynamical systems pp 6 Time in fact has an addition operation and an origin, the identity, like a group. The action of the semi-group on X is a set of maps from X to itself parametric in the time t, and this is intuitively the time evolution. A FANCY DEFINITION pp 6

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Generalizing the state space It is possible to allow different choices of the state space such as a function space (e.g. the pressure, temperature and velocity of a gas in a rocket are a function in the space of solutions of some fluid dynamics PDEs and they may vary over time),

(2026). 9783540309918 . ISBN 9783540309918

a Quantum state space (e.g. the state of an atom can be described by a set of functions in an hilbert space and a set of probabilities for these),

(2026). 9780198504009 ISBN 9780198504009

or a manifold (e.g. the state of a black hole can be described by a metric tensor on a Riemann manifold and it's position will be a vector in the same manifold).

(2026). 9783642419911 ISBN 9783642419911

Other choices can be a Phase space, a Configuration space or even a discrete space (e.g. the set of prime numbers or a /ref>

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Time as a multidimensional manifold

Time can be generalized too as a generic set of continuous parameters, for example the control theory parameters of a holonomic system can be a manifold. There is no need that time has a direction, that is smooth or even that it has whatsoever meaning similar to the intuition of time, in fact it can be generalized to even more general algebraic objects Here a few examples where time is generalized into real numbers, complex numbers or discrete n-dimensional /ref>

A general class of systems are defined over multiple independent variables and are therefore called multidimensional systems. Such systems are useful for modeling, and for example in image processing.

Time typically is considered often an external parameter as in classical and quantum mechanics, and this is typically called time domain representation, and it goes hand in hand with the Hamiltonian mechanics formulation. This is not necessarily always the case: general relativity for example is reference frame independent,

(2026). 9783642419911 ISBN 9783642419911

and gravity has an influence over time too, and in quantum electrodynamics the use of the Lagrangian mechanics formulation is more common where time and space are on same footing. In both cases the literature still talks about dynamical systems.

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Discrete dynamical system

Time can also be a discrete parameter. When time is generalized to the multi-dimensional case, i.e. as a general set of control or external parameters, this space can be interpreted as a Lattice, i.e. as the discrete points of a manifold or the Tick size of a stock price. Discrete Time events therefore can be counted by integers, for example like the measurements of the position of the planets in the sky, but this can be very different than the intuition of time as a clock that has equispaced time events. One of the tasks is typically to extract some mathematical model from the data.

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Not deterministic

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.

(1995). 9780521341875, Cambridge University Press. . ISBN 9780521341875

However, some systems are not deterministic they may allow multiple future states (i.e. the maps are generalized into multivalued functions and not uniquely defined everywhere) and the system can be subject to a bifurcation.

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Stochastic

Some systems are also stochastic, either in the input parameters such as an oscillator with a random force, or in the initial conditions, or in the predicted variables as in a Stochastic differential equation. In that random events also affect the evolution of the state variables, and this includes stochastic which are not continuous, a prototype example of a stochastic dynamical system are .

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Chaotic and Quantum systems

Last but not least there are chaotic systems (i.e. typically deterministic but not predictable) such as:

• complex dynamics
• Hyperbolic dynamics
• /ref>
• non-deterministic chaos"Non-deterministic chaos is a new dynamical paradigm where a non-deterministic system is influenced by random perturbations" /ref>

And quantum systems (i.e. deterministic until they are measured), or /ref>

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Formal definition

Assume that X is a non empty Set with elements called States. Assume a general transformation:

:$T\; :\; X\; \backslash to\; X$

/ref> Adding different structures on T and on X allows to model different properties of the dynamical system.

It is possible to model time evolution: $\backslash hat\{T\}$ can be a semigroup with one parameter $t$ called time that will also belong to a semi-group such as $N\; (t>0)$ in the discrete time case, $R^\{+\}\; (t>0)$ in the continuous time case.

A semigroup structure introduces associativity

:$\backslash hat\{T\_1\}(\backslash hat\{T\_2\}\backslash hat\{T\_3\})=(\backslash hat\{T\_1\}\backslash hat\{T\_2\})\backslash hat\{T\_3\}$

/ref>

:$\backslash hat\{T\}(t\_1+t\_2)=\backslash hat\{T\}(t\_1)\backslash hat\{T\}(t\_2)$

this is also ultimately a homomorphism

It is possible to define an origin of time $t=0$ adding an identity to the semi-group

:$\backslash hat\{T\}(0)=\backslash mathbf\{1\}$

/ref>

:$\backslash exists!\; \backslash hat\{T\}^\{-1\}:\; \backslash hat\{T\}^\{-1\}\; =\; \backslash hat\{T\}(-t),\; \backslash hat\{T\}(-t)\backslash hat\{T\}(t)\; =\; \backslash mathbf\{1\}$

More commonly there are multiple classes of definitions for a dynamical system: a first one is motivated by ordinary differential equations and is geometrical in flavor, there is an additional diffeomorphism structure; a second one is motivated by ergodic theory and is measure theoretical in flavor, there is an additional topological structure and a last one which is motivated by Category theory and is more abstract algebra in flavour.

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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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Algebraic dynamical systems

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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 the system is called a differentiable dynamical system /ref> If the manifold M is locally diffeomorphic to R ⁿ, the dynamical system is finite-dimensional; if not, the dynamical system is infinite-dimensionalThe Korteweg–De Vries equation is an example with infinite degrees of freedom and infinite integrals of motion. 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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Classical definition The modern geometrical definition assumes a map that provides an explicit description of the dynamical system, this is motivated by ergodic theory, by partial differential equations and by mathematical techniques that go beyond differential equations. An explicit description is often not available, the classical geometrical definition is implicit, rooted in classical mechanics, and based on a standard set of ordinary differential equations and a finite set of degrees of freedom: This definition implies the existence and uniqueness of solutions of such equations.

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Lagrangian Dynamical system It is also possible to cast the geometrical definition in terms of a variational principle

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Hamiltonian Dynamical system Dually to the Lagrangian it is possible to use a Hamiltonian formulation which includes a Symplectic or Poisson manifold structure on the phase spaceArnold Mathematical methods of Classical Mechanics(1989), sec 19, pp 83, example end of page

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Non integrable systems To be complete there are also systems that are typically not integrable systems such as dissipative systems, nonholonomic systems and systems that have a contact manifold structure for example systems that have a no slip boundary condition (i.e some constraint on the velocity on the boundary)

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Discrete dynamical system A 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. T can be taken to be the integers or the non negative integers. The manifold itself can be a graph or made discrete for example with a discrete topologyA graph and a general discrete space can be a Hausdorff space, can have a measure, and at least if it is finite is compact. It is not strictly a good example of a Banach space because Cauchy sequences may not make sense. This boundary between finite and infinite is interesting in the field of arithmetic geometry.

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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 /ref>

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 \circ \dots \circ \Phi_{t_0} |date=February 2026}}. 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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Topological dynamical system A topological dynamical system is a global dynamical system ( T, X, Φ) on a locally compact and Hausdorff space topological space X. T is a topological isomorphism and therefore a homeomorphism Terry tao lecture 1

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Compactification It is often useful to study the continuous extension Φ* of Φ to the one-point compactification X* of X. Even after losing the differential structure of the original system, there are compactness arguments to analyze the new system ( R, X*, Φ*). This is similar in spirit to Projective geometry where all limit points to infinity are the same point.

Another more general technique is to use Stone–Čech compactificationTerry tao lecture 3 which is similar in spirit to affine geometry where all limit points at infinity are considered different.

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Relevance In compact dynamical systems the limit set of any orbit is non-empty, compact space and simply connected.

As an example in a topological dynamical system the limit orbit of an attractor is contained within the manifold itself. This is a non trivial statement for multiple reasons: limit orbits may never be reached; limit orbits may have Lebesgue measure zero; attaching a probability to a limit orbit would be non trivial; an attractor may also have multiple limit orbits and the distinction between different compactifications may be relevant.

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Definition with Category theory

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Categories vs semi-groups

  "A category X of mathematical objects has a semigroup G of homomorphisms acting on it (topological spaces have continuous maps, sets have arbitrary maps, groups, rings fields or algebras have homomorphisms, measure spaces have measurable maps). We can view each of these categories as a dynamical system. One can even include the category of dynamical systems with suitable homomorphisms. But this viewpoint is not a very useful in itself"A final link:  https://people.math.harvard.edu/~knill/teaching/math118/118_dynamicalsystems.pdf
     

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Definition with monoids In the context of category theory, categories are always defined together with an Identity map, therefore these definitions are based on monoids instead of semi-groups.Note that it is always possible to add an Identity on the semi-group, define a monoid and pull back the monoid structure onto the original semi-group

A dynamical system 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. . 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.

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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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