Dynamical systems – Scholarpedia

A includes an abstract or condition space, whose coordinates describe the condition at any instant, along with a dynamical rule that specifies the immediate way forward for all condition variables, given just the present values of individuals same condition variables. As an example the condition of the pendulum is its position and angular velocity, and also the evolution rule is Newton’s equation (F = ma .)Introduction

In past statistics, a dynamical product is explained a preliminary value problem. The implication is the fact that there’s an idea of time which a condition previously evolves to some condition or maybe a assortment of states at another time. Thus states could be purchased by time, and time could be regarded as just one quantity.

Dynamical systems are deterministic if there’s a distinctive consequent to each condition, or stochastic or random if there’s a probability distribution of possible consequents (the idealized gold coin toss has two consequents with equal probability for every initial condition).

A dynamical system might have discrete or continuous time. A deterministic system with discrete time is determined with a map, [ x_1 = f(x_) , ] that provides the condition (x_1 ) caused by the first condition (x_ ) at next time value. After time (n) you have [ x_n = f^n(x_) , ] where (f^n) may be the (n)-th iterate of (f .) A deterministic system with continuous time is determined with a flow, [ x(t) = varphi_t(x()), ] that provides the condition sometimes (t ,) since condition was (x()) sometimes . An even flow could be differentiated regarding time for you to provide a differential equation, (dx/dt = X(x) .) The part (X(x)) is known as a vector field, it provides a vector pointing in direction of the rate at each reason for phase space.


A dynamical product is a condition space (S ,) some occasions (T) along with a rule (R) for evolution, (R: S occasions T rightarrow S) that provides the consequent(s) to some condition (s in S .) A dynamical system can be viewed as to become a model describing the temporal evolution of the system.

Condition space

The condition space is an accumulation of coordinates that describe all of the modeler feels is required to provide a complete description from the system. Because of the current condition from the system, the evolution rule predicts the following condition or states. Additionally towards the condition, which evolves over time, one might also rely on parameters which are constant or possibly known functions of your time, as an example the mass of physiques inside a mechanical model or even the birth rate and transporting capacity inside a population model.

A condition space could be discrete or continuous. As an example the gold coin toss may be modeled with a condition space composed of two states, heads and tails. Thus the condition (s) each and every time is included in the set ( .) A discrete space may also have infinitely many states for instance, an arbitrary walk might be limited to a lattice of points, and also the system condition is just which lattice point is presently occupied.

Once the condition space is continuous it’s frequently an even manifold. Within this situation it’s known as the phase space. For instance, an easy is modeled like a rigid fishing rod that’s suspended inside a vertical gravitational field from the pivot that enables the pendulum to oscillate inside a plane. Based on Newton, understanding from the position from the fishing rod in accordance with the vertical, (theta ,) and also the angular velocity, (nu = dtheta/dt) will describe the pendulum’s condition. Thus the phase space from the pendulum may be the assortment of possible values of (theta) and (nu ,) a 2-dimensional manifold. This manifold may be the cylinder since (theta) is periodic. Additionally towards the pendulum’s condition, the model also is determined by two parameters, the pendulum’s length and the effectiveness of gravity.

A phase space may also be infinite dimensional, e.g. the purpose space. This is actually the situation for dynamics that’s modeled by partial differential equations.


Time can also be discrete or continuous or even more generally be symbolized with a topological group. Dynamical systems with discrete time, such as the ideal gold coin toss, get their states evaluated once certain discrete times. Within the situation from the gold coin toss, the graceful tumbling and bouncing from the gold coin is overlooked, and it is condition is just viewed if this originates to equilibrium. Others which are frequently modeled with discrete time include population dynamics (the discreteness talking about subsequent generations) and impacting systems just like a billiard where just the condition at impact can be used. It’s quite common to scale the discrete time interval to 1, therefore the group of permitted occasions becomes (T = mathbb) or even only nonnegative integers, (T = mathbb .) This really is convenient even just in cases such as the billiard in which the bodily time interval between impacts might not be constant.

Figure 1: Henri Poincaré, the daddy of dynamical systems

Dynamical systems first made an appearance when Newton introduced the idea of (ODEs) into Mechanics. Within this situation, (T = mathbb .) However, may be the father from the modern, qualitative theory of dynamical systems. He recognized that even differential equations may very well be a discrete-time systems by strobing, i.e. only recording the answer at some discrete occasions, or by . This, obviously, is needed in almost any computational formula and in any experimental measurement as it is only easy to measure finitely many values.

Evolution rule

The evolution rule supplies a conjecture from the next condition or claims that follow in the current condition space value. An evolution rule is deterministic if each condition includes a unique consequent, and it is stochastic (or “random”) if there’s several possible consequent for any given condition.

The forward orbit or trajectory of the condition (s) it’s time-purchased assortment of claims that follow from (s) while using evolution rule. For any deterministic rule with discrete time the forward orbit of (s_) may be the sequence (s_ , s_1 , s_2 , ldots .) When both condition space and time are continuous, the forward orbit is really a curve (s(t), t ge .)

Deterministic evolution rules are invertible if each condition includes a unique precedent or preimage. Within this situation the entire orbit from the product is the bi-infinite sequence or curve that starts at (s_ ) or (s()) and extends both in directions of your time.



A deterministic evolution rule with discrete time, along with a continuous condition space is known as a , [f: S rightarrow S .] The evolution is determined by iteration (s_ = f(s_t ) .) A roadmap could be one-to-one (invertible) or otherwise. Invertible maps could be continuous with continuous inverses () or perhaps be smooth and easily invertible ().

An easy example may be the of population dynamics. Here the condition space is (mathbb^+ ,) the nonnegative reals, representing a continuing approximation to some population size. The map is [tag f(x) = rxleft( 1 – fracright) ]

where (r) may be the rate of growth per individual and (K) may be the transporting capacity. The map () isn’t invertible because most states within the interval ([,K]) have two preimages.


A flow is really a deterministic dynamical system on the manifold, (M) that’s continuously differentiable regarding time. It’s based on the purpose [varphi : Rtimes M to M ,] so the orbit is offered by [tag x(t) = varphi_t(x()) ]

Flows obey the qualities

  • Identity[varphi_(x) = x]
  • Group[varphi_(x) = varphi_t(varphi_s(x))]
  • Differentiability[ frac varphi_t(x)_ = X(x)]

The 2nd property is called the group property it expresses the notion that the dynamics could be restarted at any time (x(s)) along its trajectory to obtain the same result (x(t+s)) as flowing forward for time (t+s) from (x() .) The final property, differentiability, defines a vector field (X) that’s connected with any flow. Due to the audience property would be that the orbits of the flow are solutions from the ordinary differential equation [ frac x = X(x) ]

It’s easy to define the dynamics connected with differential equations with the flow concept since the problems with existence and uniqueness from the solutions from the ODE may then be prevented: the orbits of the flow are unique (just one orbit goes through each reason for (M)) and exist forever. This isn’t true generally for ODEs.

A semi-flow is really a flow defined just for nonnegative values of your time. Semi-flows generally arise for partial differential equations.

Iterated function system

A stochastic evolution with discrete time but continuous phase space is definitely an . Within this situation there’s an accumulation of functions (f_alpha) listed in parameters (alpha .) The evolution is random using the next condition (s_ = f_alpha (s_t )) where (alpha) is chosen from the probability distribution.

Iterated function systems can generate interesting dynamics even if your functions are . Within this situation the orbits are frequently drawn to some fractal set.

Cellular automata

A dynamical system having a deterministic rule, discrete some time and discrete condition space is really a cellular automata. The evolution rule assigns a brand new condition to some cell like a purpose of that old condition of the cell and finitely a lot of its neighbors. The (relative) rule is identical for every cell.

A good example may be the bet on existence, where there’s a square grid on the flight and every cell can assume 2 states: alive or dead (but there are just finitely many live cells).


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Dynamical Systems – Stefano Luzzatto – Lecture 01