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In mathematics, a **dynamical system** is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

- Overview
- History
- Basic definitions
- Examples
- Further examples
- Linear dynamical systems
- Flows
- Maps
- Local dynamics
- Rectification
- Near periodic orbits
- Conjugation results
- Bifurcation theory
- Ergodic systems
- Nonlinear dynamical systems and chaos
- Geometrical definition
- Measure theoretical definition
- Multidimensional generalization
- See also
- References
- Further reading
- External links

At any given time, a dynamical system has a state given by a tuple of real numbers (a vector) that can be represented by a point in an appropriate state space (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.^{ [1] }^{ [2] } However, some systems are stochastic, in that random events also affect the evolution of the state variables.

In physics, a **dynamical system** is described as a "particle or ensemble of particles whose state varies over time and thus obeys differential equations involving time derivatives".^{ [3] } In order to make a prediction about the system's future behavior, an analytical solution of such equations or their integration over time through computer simulation is realized.

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,^{ [4] }^{ [5] } biology,^{ [6] } chemistry, engineering,^{ [7] } economics,^{ [8] } history, 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.

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, 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 transition to turbulence of a fluid.
- 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 systems 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.

Many people regard French mathematician Henri Poincaré as the founder of dynamical systems.^{ [9] } 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 Smale horseshoe 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, Palestinian mechanical engineer Ali H. Nayfeh applied nonlinear dynamics in mechanical and engineering systems.^{ [10] } 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.^{ [11] }

A dynamical system is a manifold *M* called the phase (or state) space endowed with a family of smooth evolution functions Φ^{t} that for any element *t* ∈ *T*, the time, map a point of the phase space back into the phase space. The notion of smoothness changes with applications and the type of manifold. There are several choices for the set *T*. When *T* is taken to be the reals, the dynamical system is called a * flow *; and if *T* is restricted to the non-negative reals, then the dynamical system is a *semi-flow*. When *T* is taken to be the integers, it is a *cascade* or a *map*; and the restriction to the non-negative integers is a *semi-cascade*.

Note: There is a further technical condition that Φ^{t} is an action of *T* on *M*. That includes the facts that Φ^{0} is the identity function and that Φ^{s+t} is the composition of Φ^{s} and Φ^{t}. This is a monoid action, which doesn't require the existence of negative values for *t*, and doesn't require the functions Φ^{t} to be invertible.

The evolution function Φ^{ t} is often the solution of a *differential equation of motion*

The equation gives the time derivative, represented by the dot, of a trajectory *x*(*t*) on the phase space starting at some point *x*_{0}. The vector field *v*(*x*) is a smooth function that at every point of the phase space *M* provides the velocity vector of the dynamical system at that point. (These vectors are not vectors in the phase space *M*, but in the tangent space *T _{x}M* of the point

There is no need for higher order derivatives in the equation, nor for time dependence in *v*(*x*) because these can be eliminated by considering systems of higher dimensions. Other types of differential equations can be used to define the evolution rule:

is an example of an equation that arises from the modeling of mechanical systems with complicated constraints.

The differential equations determining the evolution function Φ^{ t} are often ordinary differential equations; in this case the phase space *M* is a finite dimensional manifold. Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds—those that are locally Banach spaces —in which case the differential equations are partial differential equations. In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity.

- Arnold's cat map
- Baker's map is an example of a chaotic piecewise linear map
- Billiards and outer billiards
- Bouncing ball dynamics
- Circle map
- Complex quadratic polynomial
- Double pendulum
- Dyadic transformation
- Hénon map
- Irrational rotation
- Kaplan–Yorke map
- List of chaotic maps
- Lorenz system
- Quadratic map simulation system
- Rössler map
- Swinging Atwood's machine
- Tent map

Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. In a linear system the phase space is the *N*-dimensional Euclidean space, so any point in phase space can be represented by a vector with *N* numbers. The analysis of linear systems is possible because they satisfy a superposition principle: if *u*(*t*) and *w*(*t*) satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will *u*(*t*) + *w*(*t*).

For a flow, the vector field v(*x*) is an affine function of the position in the phase space, that is,

with *A* a matrix, *b* a vector of numbers and *x* the position vector. The solution to this system can be found by using the superposition principle (linearity). The case *b* ≠ 0 with *A* = 0 is just a straight line in the direction of *b*:

When *b* is zero and *A* ≠ 0 the origin is an equilibrium (or singular) point of the flow, that is, if *x*_{0} = 0, then the orbit remains there. For other initial conditions, the equation of motion is given by the exponential of a matrix: for an initial point *x*_{0},

When *b* = 0, the eigenvalues of *A* determine the structure of the phase space. From the eigenvalues and the eigenvectors of *A* it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin.

The distance between two different initial conditions in the case *A* ≠ 0 will change exponentially in most cases, either converging exponentially fast towards a point, or diverging exponentially fast. Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the (necessary but not sufficient) conditions for chaotic behavior.

A discrete-time, affine dynamical system has the form of a matrix difference equation:

with *A* a matrix and *b* a vector. As in the continuous case, the change of coordinates *x* → *x* + (1 − *A*)^{ –1}*b* removes the term *b* from the equation. In the new coordinate system, the origin is a fixed point of the map and the solutions are of the linear system *A*^{ n}*x*_{0}. The solutions for the map are no longer curves, but points that hop in the phase space. The orbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map.

As in the continuous case, the eigenvalues and eigenvectors of *A* determine the structure of phase space. For example, if *u*_{1} is an eigenvector of *A*, with a real eigenvalue smaller than one, then the straight lines given by the points along *α* *u*_{1}, with *α* ∈ **R**, is an invariant curve of the map. Points in this straight line run into the fixed point.

There are also many other discrete dynamical systems.

The qualitative properties of dynamical systems do not change under a smooth change of coordinates (this is sometimes taken as a definition of qualitative): a *singular point* of the vector field (a point where *v*(*x*) = 0) will remain a singular point under smooth transformations; a *periodic orbit* is a loop in phase space and smooth deformations of the phase space cannot alter it being a loop. It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood. In the qualitative study of dynamical systems, the approach is to show that there is a change of coordinates (usually unspecified, but computable) that makes the dynamical system as simple as possible.

A flow in most small patches of the phase space can be made very simple. If *y* is a point where the vector field *v*(*y*) ≠ 0, then there is a change of coordinates for a region around *y* where the vector field becomes a series of parallel vectors of the same magnitude. This is known as the rectification theorem.

The *rectification theorem* says that away from singular points the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space *M* the dynamical system is *integrable*. In most cases the patch cannot be extended to the entire phase space. There may be singular points in the vector field (where *v*(*x*) = 0); or the patches may become smaller and smaller as some point is approached. The more subtle reason is a global constraint, where the trajectory starts out in a patch, and after visiting a series of other patches comes back to the original one. If the next time the orbit loops around phase space in a different way, then it is impossible to rectify the vector field in the whole series of patches.

In general, in the neighborhood of a periodic orbit the rectification theorem cannot be used. Poincaré developed an approach that transforms the analysis near a periodic orbit to the analysis of a map. Pick a point *x*_{0} in the orbit γ and consider the points in phase space in that neighborhood that are perpendicular to *v*(*x*_{0}). These points are a Poincaré section *S*(*γ*, *x*_{0}), of the orbit. The flow now defines a map, the Poincaré map *F* : *S* → *S*, for points starting in *S* and returning to *S*. Not all these points will take the same amount of time to come back, but the times will be close to the time it takes *x*_{0}.

The intersection of the periodic orbit with the Poincaré section is a fixed point of the Poincaré map *F*. By a translation, the point can be assumed to be at *x* = 0. The Taylor series of the map is *F*(*x*) = *J* · *x* + O(*x*^{2}), so a change of coordinates *h* can only be expected to simplify *F* to its linear part

This is known as the conjugation equation. Finding conditions for this equation to hold has been one of the major tasks of research in dynamical systems. Poincaré first approached it assuming all functions to be analytic and in the process discovered the non-resonant condition. If *λ*_{1}, ..., *λ*_{ν} are the eigenvalues of *J* they will be resonant if one eigenvalue is an integer linear combination of two or more of the others. As terms of the form *λ*_{i}– ∑ (multiples of other eigenvalues) occurs in the denominator of the terms for the function *h*, the non-resonant condition is also known as the small divisor problem.

The results on the existence of a solution to the conjugation equation depend on the eigenvalues of *J* and the degree of smoothness required from *h*. As *J* does not need to have any special symmetries, its eigenvalues will typically be complex numbers. When the eigenvalues of *J* are not in the unit circle, the dynamics near the fixed point *x*_{0} of *F* is called * hyperbolic * and when the eigenvalues are on the unit circle and complex, the dynamics is called *elliptic*.

In the hyperbolic case, the Hartman–Grobman theorem gives the conditions for the existence of a continuous function that maps the neighborhood of the fixed point of the map to the linear map *J* · *x*. The hyperbolic case is also *structurally stable*. Small changes in the vector field will only produce small changes in the Poincaré map and these small changes will reflect in small changes in the position of the eigenvalues of *J* in the complex plane, implying that the map is still hyperbolic.

The Kolmogorov–Arnold–Moser (KAM) theorem gives the behavior near an elliptic point.

When the evolution map Φ^{t} (or the vector field it is derived from) depends on a parameter μ, the structure of the phase space will also depend on this parameter. Small changes may produce no qualitative changes in the phase space until a special value *μ*_{0} is reached. At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation.

Bifurcation theory considers a structure in phase space (typically a fixed point, a periodic orbit, or an invariant torus) and studies its behavior as a function of the parameter *μ*. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures. By using Taylor series approximations of the maps and an understanding of the differences that may be eliminated by a change of coordinates, it is possible to catalog the bifurcations of dynamical systems.

The bifurcations of a hyperbolic fixed point *x*_{0} of a system family *F _{μ}* can be characterized by the eigenvalues of the first derivative of the system

Some bifurcations can lead to very complicated structures in phase space. For example, the Ruelle–Takens scenario describes how a periodic orbit bifurcates into a torus and the torus into a strange attractor. In another example, Feigenbaum period-doubling describes how a stable periodic orbit goes through a series of period-doubling bifurcations.

In many dynamical systems, it is possible to choose the coordinates of the system so that the volume (really a ν-dimensional volume) in phase space is invariant. This happens for mechanical systems derived from Newton's laws as long as the coordinates are the position and the momentum and the volume is measured in units of (position) × (momentum). The flow takes points of a subset *A* into the points Φ^{ t}(*A*) and invariance of the phase space means that

In the Hamiltonian formalism, given a coordinate it is possible to derive the appropriate (generalized) momentum such that the associated volume is preserved by the flow. The volume is said to be computed by the Liouville measure.

In a Hamiltonian system, not all possible configurations of position and momentum can be reached from an initial condition. Because of energy conservation, only the states with the same energy as the initial condition are accessible. The states with the same energy form an energy shell Ω, a sub-manifold of the phase space. The volume of the energy shell, computed using the Liouville measure, is preserved under evolution.

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let *F* be a phase space volume-preserving map and *A* a subset of the phase space. Then almost every point of *A* returns to *A* infinitely often. The Poincaré recurrence theorem was used by Zermelo to object to Boltzmann's derivation of the increase in entropy in a dynamical system of colliding atoms.

One of the questions raised by Boltzmann's work was the possible equality between time averages and space averages, what he called the ergodic hypothesis. The hypothesis states that the length of time a typical trajectory spends in a region *A* is vol(*A*)/vol(Ω).

The ergodic hypothesis turned out not to be the essential property needed for the development of statistical mechanics and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems. Koopman approached the study of ergodic systems by the use of functional analysis. An observable *a* is a function that to each point of the phase space associates a number (say instantaneous pressure, or average height). The value of an observable can be computed at another time by using the evolution function φ^{ t}. This introduces an operator *U*^{ t}, the transfer operator,

By studying the spectral properties of the linear operator *U* it becomes possible to classify the ergodic properties of Φ^{ t}. In using the Koopman approach of considering the action of the flow on an observable function, the finite-dimensional nonlinear problem involving Φ^{ t} gets mapped into an infinite-dimensional linear problem involving *U*.

The Liouville measure restricted to the energy surface Ω is the basis for the averages computed in equilibrium statistical mechanics. An average in time along a trajectory is equivalent to an average in space computed with the Boltzmann factor exp(−β*H*). This idea has been generalized by Sinai, Bowen, and Ruelle (SRB) to a larger class of dynamical systems that includes dissipative systems. SRB measures replace the Boltzmann factor and they are defined on attractors of chaotic systems.

Simple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random, despite the fact that they are fundamentally deterministic. This seemingly unpredictable behavior has been called * chaos *. Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent space perpendicular to a trajectory can be well separated into two parts: one with the points that converge towards the orbit (the *stable manifold*) and another of the points that diverge from the orbit (the *unstable manifold*).

This branch of mathematics deals with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible attractors?" or "Does the long-term behavior of the system depend on its initial condition?"

Note that the chaotic behavior of complex systems is not the issue. Meteorology has been known for years to involve complex—even chaotic—behavior. Chaos theory has been so surprising because chaos can be found within almost trivial systems. The logistic map is only a second-degree polynomial; the horseshoe map is piecewise linear.

A dynamical system is the tuple , with a manifold (locally a Banach space or Euclidean space), the domain for time (non-negative reals, the integers, ...) and *f* an evolution rule *t* → *f*^{ t} (with ) such that *f ^{ t}* is a diffeomorphism of the manifold to itself. So, f is a mapping of the time-domain into the space of diffeomorphisms of the manifold to itself. In other terms,

A dynamical system may be defined formally, as a measure-preserving transformation of a sigma-algebra, the quadruplet (*X*, Σ, μ, τ). Here, *X* is a set, and Σ is a sigma-algebra on *X*, so that the pair (*X*, Σ) is a measurable space. μ is a finite measure on the sigma-algebra, so that the triplet (*X*, Σ, μ) is a probability space. A map τ: *X* → *X* is said to be Σ-measurable if and only if, for every σ ∈ Σ, one has . A map τ is said to **preserve the measure** if and only if, for every σ ∈ Σ, one has . Combining the above, a map τ is said to be a **measure-preserving transformation of X**, if it is a map from

The map τ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates for integer *n* are studied. For continuous dynamical systems, the map τ is understood to be a finite time evolution map and the construction is more complicated.

Dynamical systems are defined over a single independent variable, usually 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.

- Behavioral modeling
- Cognitive modeling
- Complex dynamics
- Dynamic approach to second language development
- Feedback passivation
- Infinite compositions of analytic functions
- List of dynamical system topics
- Oscillation
- People in systems and control
- Sharkovskii's theorem
- System dynamics
- Systems theory
- Principle of maximum caliber

**Chaos theory** is a branch of mathematics focusing on the study of *chaos* — dynamical systems whose apparently random states of disorder and irregularities are actually governed by underlying patterns and deterministic laws that are highly sensitive to initial conditions. Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state. A metaphor for this behavior is that a butterfly flapping its wings in Texas can cause a hurricane in China.

In mathematics, the **Lyapunov exponent** or **Lyapunov characteristic exponent** of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation vector diverge at a rate given by

In the mathematical field of dynamical systems, an **attractor** is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed.

**Ergodic theory** is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc. Thus, the statistics with which we are concerned are properties of the dynamics.

In mathematics, in the study of dynamical systems with two-dimensional phase space, a **limit cycle** is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of a great many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912).

**Bifurcation theory** is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a **bifurcation** occurs when a small smooth change made to the parameter values of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations occur in both continuous systems and discrete systems. The name "bifurcation" was first introduced by Henri Poincaré in 1885 in the first paper in mathematics showing such a behavior. Henri Poincaré also later named various types of stationary points and classified them with motif.

In mathematics and physics, the **Poincaré recurrence theorem** states that certain dynamical systems will, after a sufficiently long but finite time, return to a state arbitrarily close to, or exactly the same as, their initial state.

In those branches of mathematics called dynamical systems and ergodic theory, the concept of a **wandering set** formalizes a certain idea of movement and mixing in such systems. When a dynamical system has a wandering set of non-zero measure, then the system is a dissipative system. This is very much the opposite of a conservative system, for which the ideas of the Poincaré recurrence theorem apply. Intuitively, the connection between wandering sets and dissipation is easily understood: if a portion of the phase space "wanders away" during normal time-evolution of the system, and is never visited again, then the system is dissipative. The language of wandering sets can be used to give a precise, mathematical definition to the concept of a dissipative system. The notion of wandering sets in phase space was introduced by Birkhoff in 1927.

In the mathematical theory of bifurcations, a **Hopf****bifurcation** is a critical point where a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues—of the linearization around the fixed point—crosses the complex plane imaginary axis. Under reasonably generic assumptions about the dynamical system, a small-amplitude limit cycle branches from the fixed point.

In the mathematics of evolving systems, the concept of a **center manifold** was originally developed to determine stability of degenerate equilibria. Subsequently, the concept of center manifolds was realised to be fundamental to mathematical modelling.

In mathematics, a **homoclinic orbit** is a trajectory of a flow of a dynamical system which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the stable manifold and the unstable manifold of an equilibrium.

In mathematics, **stability theory** addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using L^{p} norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.

The **Lorenz system** is a system of ordinary differential equations first studied by Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the **Lorenz attractor** is a set of chaotic solutions of the Lorenz system. In popular media the "butterfly effect" stems from the real-world implications of the Lorenz attractor, i.e. that in any physical system, in the absence of perfect knowledge of the initial conditions, our ability to predict its future course will always fail. This underscores that physical systems can be completely deterministic and yet still be inherently unpredictable even in the absence of quantum effects. The shape of the Lorenz attractor itself, when plotted graphically, may also be seen to resemble a butterfly.

The **dynamical system** concept is a mathematical formalization for any fixed "rule" that describes the time dependence of a point's position in its ambient space. 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.

In mathematics, an **invariant measure** is a measure that is preserved by some function. Ergodic theory is the study of invariant measures in dynamical systems. The Krylov–Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and space under consideration.

In mathematics, a **conservative system** is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink over time. Precisely speaking, they are those dynamical systems that have a null wandering set: under time evolution, no portion of the phase space ever "wanders away", never to be returned to or revisited. Alternately, conservative systems are those to which the Poincaré recurrence theorem applies. An important special case of conservative systems are the measure-preserving dynamical systems.

**Numerical continuation** is a method of computing approximate solutions of a system of parameterized nonlinear equations,

**Synchronization of chaos** is a phenomenon that may occur when two, or more, dissipative chaotic systems are coupled.

**Supersymmetric theory of stochastic dynamics** or **stochastics** (**STS**) is an exact theory of stochastic (partial) differential equations (SDEs), the class of mathematical models with the widest applicability covering, in particular, all continuous time dynamical systems, with and without noise. The main utility of the theory from the physical point of view is a rigorous theoretical explanation of the ubiquitous spontaneous long-range dynamical behavior that manifests itself across disciplines via such phenomena as 1/f, flicker, and crackling noises and the power-law statistics, or Zipf's law, of instantonic processes like earthquakes and neuroavalanches. From the mathematical point of view, STS is interesting because it bridges the two major parts of mathematical physics – the dynamical systems theory and topological field theories. Besides these and related disciplines such as algebraic topology and supersymmetric field theories, STS is also connected with the traditional theory of stochastic differential equations and the theory of pseudo-Hermitian operators.

- ↑ Strogatz, S. H. (2001).
*Nonlinear Dynamics and Chaos: with Applications to Physics, Biology and Chemistry*. Perseus. - ↑ Katok, A.; Hasselblatt, B. (1995).
*Introduction to the Modern Theory of Dynamical Systems*. Cambridge: Cambridge University Press. ISBN 978-0-521-34187-5. - ↑ "Nature". Springer Nature. Retrieved 17 February 2017.
- ↑ Melby, P.; et al. (2005). "Dynamics of Self-Adjusting Systems With Noise".
*Chaos: An Interdisciplinary Journal of Nonlinear Science*.**15**(3): 033902. Bibcode:2005Chaos..15c3902M. doi:10.1063/1.1953147. PMID 16252993. - ↑ Gintautas, V.; et al. (2008). "Resonant forcing of select degrees of freedom of multidimensional chaotic map dynamics".
*J. Stat. Phys*.**130**. arXiv: 0705.0311 . Bibcode:2008JSP...130..617G. doi:10.1007/s10955-007-9444-4. S2CID 8677631. - ↑ Jackson, T.; Radunskaya, A. (2015).
*Applications of Dynamical Systems in Biology and Medicine*. Springer. - ↑ Kreyszig, Erwin (2011).
*Advanced Engineering Mathematics*. Hoboken: Wiley. ISBN 978-0-470-64613-7. - ↑ Gandolfo, Giancarlo (2009) [1971].
*Economic Dynamics: Methods and Models*(Fourth ed.). Berlin: Springer. ISBN 978-3-642-13503-3. - ↑ Holmes, Philip. "Poincaré, celestial mechanics, dynamical-systems theory and "chaos"."
*Physics Reports*193.3 (1990): 137-163. - ↑ Rega, Giuseppe (2019). "Tribute to Ali H. Nayfeh (1933-2017)".
*IUTAM Symposium on Exploiting Nonlinear Dynamics for Engineering Systems*. Springer. pp. 1–2. ISBN 9783030236922. - ↑ "Ali Hasan Nayfeh".
*Franklin Institute Awards*. The Franklin Institute. 4 February 2014. Retrieved 25 August 2019.

Works providing a broad coverage:

- Ralph Abraham and Jerrold E. Marsden (1978).
*Foundations of mechanics*. Benjamin–Cummings. ISBN 978-0-8053-0102-1. (available as a reprint: ISBN 0-201-40840-6) *Encyclopaedia of Mathematical Sciences*( ISSN 0938-0396) has a sub-series on dynamical systems with reviews of current research.- Christian Bonatti; Lorenzo J. Díaz; Marcelo Viana (2005).
*Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective*. Springer. ISBN 978-3-540-22066-4. - Stephen Smale (1967). "Differentiable dynamical systems".
*Bulletin of the American Mathematical Society*.**73**(6): 747–817. doi: 10.1090/S0002-9904-1967-11798-1 .

Introductory texts with a unique perspective:

- V. I. Arnold (1982).
*Mathematical methods of classical mechanics*. Springer-Verlag. ISBN 978-0-387-96890-2. - Jacob Palis and Welington de Melo (1982).
*Geometric theory of dynamical systems: an introduction*. Springer-Verlag. ISBN 978-0-387-90668-3. - David Ruelle (1989).
*Elements of Differentiable Dynamics and Bifurcation Theory*. Academic Press. ISBN 978-0-12-601710-6. - Tim Bedford, Michael Keane and Caroline Series,
*eds.*(1991).*Ergodic theory, symbolic dynamics and hyperbolic spaces*. Oxford University Press. ISBN 978-0-19-853390-0.CS1 maint: multiple names: authors list (link) - Ralph H. Abraham and Christopher D. Shaw (1992).
*Dynamics—the geometry of behavior, 2nd edition*. Addison-Wesley. ISBN 978-0-201-56716-8.

Textbooks

- Kathleen T. Alligood, Tim D. Sauer and James A. Yorke (2000).
*Chaos. An introduction to dynamical systems*. Springer Verlag. ISBN 978-0-387-94677-1. - Oded Galor (2011).
*Discrete Dynamical Systems**. Springer. ISBN 978-3-642-07185-0.* - Morris W. Hirsch, Stephen Smale and Robert L. Devaney (2003).
*Differential Equations, dynamical systems, and an introduction to chaos*. Academic Press. ISBN 978-0-12-349703-1. - Anatole Katok; Boris Hasselblatt (1996).
*Introduction to the modern theory of dynamical systems*. Cambridge. ISBN 978-0-521-57557-7. - Stephen Lynch (2010).
*Dynamical Systems with Applications using Maple 2nd Ed*. Springer. ISBN 978-0-8176-4389-8. - Stephen Lynch (2014).
*Dynamical Systems with Applications using MATLAB 2nd Edition*. Springer International Publishing. ISBN 978-3319068190. - Stephen Lynch (2017).
*Dynamical Systems with Applications using Mathematica 2nd Ed*. Springer. ISBN 978-3-319-61485-4. - Stephen Lynch (2018).
*Dynamical Systems with Applications using Python*. Springer International Publishing. ISBN 978-3-319-78145-7. - James Meiss (2007).
*Differential Dynamical Systems*. SIAM. ISBN 978-0-89871-635-1. - David D. Nolte (2015).
*Introduction to Modern Dynamics: Chaos, Networks, Space and Time*. Oxford University Press. ISBN 978-0199657032. - Julien Clinton Sprott (2003).
*Chaos and time-series analysis**. Oxford University Press. ISBN 978-0-19-850839-7.* - Steven H. Strogatz (1994).
*Nonlinear dynamics and chaos: with applications to physics, biology chemistry and engineering*. Addison Wesley. ISBN 978-0-201-54344-5. - Teschl, Gerald (2012).
*Ordinary Differential Equations and Dynamical Systems*. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0. - Stephen Wiggins (2003).
*Introduction to Applied Dynamical Systems and Chaos*. Springer. ISBN 978-0-387-00177-7.

Popularizations:

- Florin Diacu and Philip Holmes (1996).
*Celestial Encounters*. Princeton. ISBN 978-0-691-02743-2. - James Gleick (1988).
*Chaos: Making a New Science*. Penguin. ISBN 978-0-14-009250-9. - Ivar Ekeland (1990).
*Mathematics and the Unexpected (Paperback)*. University Of Chicago Press. ISBN 978-0-226-19990-0. - Ian Stewart (1997).
*Does God Play Dice? The New Mathematics of Chaos*. Penguin. ISBN 978-0-14-025602-4.

Wikimedia Commons has media related to Dynamical systems . |

- Arxiv preprint server has daily submissions of (non-refereed) manuscripts in dynamical systems.
- Encyclopedia of dynamical systems A part of Scholarpedia — peer reviewed and written by invited experts.
- Nonlinear Dynamics. Models of bifurcation and chaos by Elmer G. Wiens
- Sci.Nonlinear FAQ 2.0 (Sept 2003) provides definitions, explanations and resources related to nonlinear science

- Online books or lecture notes

- Geometrical theory of dynamical systems. Nils Berglund's lecture notes for a course at ETH at the advanced undergraduate level.
- Dynamical systems. George D. Birkhoff's 1927 book already takes a modern approach to dynamical systems.
- Chaos: classical and quantum. An introduction to dynamical systems from the periodic orbit point of view.
- Learning Dynamical Systems. Tutorial on learning dynamical systems.
- Ordinary Differential Equations and Dynamical Systems. Lecture notes by Gerald Teschl

- Research groups

- Dynamical Systems Group Groningen, IWI, University of Groningen.
- Chaos @ UMD. Concentrates on the applications of dynamical systems.
- , SUNY Stony Brook. Lists of conferences, researchers, and some open problems.
- Center for Dynamics and Geometry, Penn State.
- Control and Dynamical Systems, Caltech.
- Laboratory of Nonlinear Systems, Ecole Polytechnique Fédérale de Lausanne (EPFL).
- Center for Dynamical Systems, University of Bremen
- Systems Analysis, Modelling and Prediction Group, University of Oxford
- Non-Linear Dynamics Group, Instituto Superior Técnico, Technical University of Lisbon
- Dynamical Systems, IMPA, Instituto Nacional de Matemática Pura e Applicada.
- Nonlinear Dynamics Workgroup, Institute of Computer Science, Czech Academy of Sciences.
- UPC Dynamical Systems Group Barcelona, Polytechnical University of Catalonia.
- Center for Control, Dynamical Systems, and Computation, University of California, Santa Barbara.

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