Dynamical system

2020 Mathematics Subject Classification: Primary: 37-01 [MSN][ZBL]

In the original meaning of the term a dynamical system is a mechanical system with a finite number of degrees of freedom. The state of such a system is usually characterized by its position (configuration, location) and the rate of change of this position, while a law of motion describes the rate of change of the state of the system.

In the simplest cases the state may be characterized by quantities $ w _ {1} \dots w _ {m} $ which may assume arbitrary (real) values; two different instances of the quantities $ w _ {1} \dots w _ {m} $ and $ w _ {1} ^ \prime \dots w _ {m} ^ \prime $ correspond to different states and vice versa, while if the values of $ w _ {i} ^ \prime $ and $ w _ {i} $ are close for all $ i $, then the respective states of the system are close. In this case the law of motion can be written down as an autonomous system of ordinary differential equations:

$$ \tag{1 } {\dot{w} } _ {i} = f _ {i} ( w _ {1} \dots w _ {m} ) ,\ \ i= 1 \dots m . $$

If the quantities $ w _ {1} \dots w _ {m} $ are considered as coordinates of a point $ w $ in an $ m $- dimensional space, then the corresponding state of the dynamical system may be represented by this point $ w $. This point is known as the phase (sometimes, the representing) point, while the space is called the phase space of the system. (The reason for the adjective "phase" is that in the past the state of a system was often denoted as its "phase" .) The change of the state with time is represented as a motion of the phase point along a certain curve (the so-called phase trajectory, often simply called the trajectory) in the phase space. In this space a vector field can be defined by associating with each point $ w $ the vector $ f ( w) $ with components

$$ \tag{2 } ( f _ {1} ( w _ {1} \dots w _ {m} ) \dots f _ {m} ( w _ {1} \dots w _ {m} ) ) . $$

The differential equations (1) may be written in abbreviated form as

$$ \tag{3 } \dot{w} = f ( w) , $$

using the terms introduced above. They mean that at any moment of time the velocity vector of the motion of the phase point (or, as is often said, the phase velocity vector; not to be confused with the same term as used in optics and in the study of various wave processes in general) is equal to the vector $ f ( w) $ issuing from the point $ w $ of the phase space occupied by the moving phase point at that moment of time. This is the so-called kinematic interpretation of the system of differential equations (1).

For instance, the state of a particle with no internal degrees of freedom (or, as it is called in mechanics, a material point) moving in a potential field with potential $ U ( x _ {1} , x _ {2} , x _ {3} ) $ is characterized by its position $ x = ( x _ {1} , x _ {2} , x _ {3} ) $ and velocity $ \dot{x} $; the last quantity may be replaced by the (linear) momentum $ p = m \dot{x} $, where $ m $ is the mass of the particle. The law of motion may be written in the form

$$ \tag{4 } \dot{x} = \frac{1}{m} p ,\ \dot{p} = - \mathop{\rm grad} U ( x) . $$

The formulas (4) are an abbreviation of a system of six ordinary differential equations of the first order. Here, the phase space is six-dimensional Euclidean space, the six components of the phase velocity vector are the components of the ordinary velocity and of the force, while the projection of the phase trajectory on the space $ x _ {i} $( parallel to the momentum space) is the trajectory of the particle in the ordinary sense of the word.

In a number of cases it is not possible to establish a correspondence between all the states of a dynamical system and the points of a Euclidean space which would have the desired properties, whereas such a correspondence may be established locally, i.e. for states sufficiently close to each other. If one reserves the term "phase space" for the totality of all states of a dynamical system, one may say that, in the general case, the phase space is not a Euclidean space, but rather a differentiable manifold $ W ^ {m} $. Locally, i.e. in any chart (local coordinate system) of $ W ^ {m} $, the motion of the dynamical system is described by a system of differential equations such as (1). On the other hand, a global (i.e. suitable for all states of the dynamical system) and invariant (i.e. independent of the choice of the chart) description of motion is given by (3), where $ f $ is a vector field defined on $ W ^ {m} $, which associates with each point $ w $ a vector $ f ( w) $ in the tangent space of the manifold at that point; equation (3) means that, in the process of motion, a phase point which coincides at a given moment of time with the point $ w \in W ^ {m} $, has velocity $ f ( w) $ at that moment. In local coordinates the vector $ f ( w) $ is represented in terms of its components (2), while (3) is reduced to (1).

Even in many cases in which the phase space is Euclidean, part of the motion of the dynamical system under study may be described by means of a vector field on some invariant manifold $ W $, i.e. a submanifold of the phase space such that the entire trajectory passing through an arbitrary point $ w \in W $ lies in $ W $. Thus, in the preceding example, if the discussion concerns motions with a definite value of the energy $ E $, the system (4) should not be studied throughout the $ 6 $- dimensional Euclidean space of the variables $ ( x , p) $, but in its $ 5 $- dimensional submanifold defined by the equation

$$ \frac{p ^ {2} }{2 m } + U ( x) = E , $$

where $ p ^ {2} = p _ {1} ^ {2} + p _ {2} ^ {2} + p _ {3} ^ {2} $. The invariance of this manifold reflects the fact that the energy of a particle moving in a potential field is conserved, i.e. $ p ^ {2} / 2m + U ( x) $ is a first integral of the system (4) (the so-called energy integral). Many similar examples are related to cyclic coordinates.

An example of a dynamical system with a non-Euclidean phase space is a solid body with a stationary point $ 0 $. If one introduces two orthogonal coordinate systems with origin at $ 0 $, one of which is fixed while the other is rigidly bound to the body, then it is clear that the position of the solid body will be characterized by the position of the second coordinate system with respect to the first — i.e. by an orthogonal matrix of order three with determinant 1 (or in some other, equivalent, way; cf. Euler angles; Cayley–Klein parameters). Accordingly, the totality of all possible positions of the given mechanical system (or its configuration space) is the special orthogonal group of order three SO(3). The phase space $ W ^ {6} $ is the tangent bundle of SO(3), for the rate of position change is characterized by a vector tangent to SO(3). For local coordinates in SO(3) (whose choice automatically determines certain local coordinates in $ W ^ {6} $) one usually takes the Euler angles; the equation of motion is in this case known as the Euler equation (of motion of a solid body).

In the above kinematic interpretation of an autonomous system of ordinary differential equations (1) (or the picture of motions of the phase points in the phase manifold according to equation (3)) it is irrelevant whether these equations do or do not describe some mechanical system. Accordingly, the term "dynamical system" came to be used in the wider sense of an arbitrary physical system (such as an electric circuit) described by differential equations such as (1) or (3) and subsequently simply of a system of differential equations of that form, irrespective of its origin. Mechanical dynamical systems are distinguished from dynamical systems in this wider sense by certain specific properties: most of them belong to the special class of Hamiltonian systems (cf. Hamiltonian system). (However, also systems not in this class are considered in mechanics, e.g. most non-holonomic systems. Conversely, Hamiltonian systems are also encountered in several problems in physics.)

In this sense the concept of a dynamical system is equivalent to that of an autonomous system of differential equations of the form (1) or (3). In practice, however, one speaks of dynamical systems when studying the qualitative picture of the behaviour of all trajectories in the phase space (global theory) or at least in some part of it (local theory). In the theory of dynamical systems much attention is paid to the behaviour of phase trajectories as time increases indefinitely. The trajectories which are considered most interesting in the theory of dynamical systems are those with properties which to a large degree determine the qualitative (even only local) picture. This includes an equilibrium position (or singular point), a periodic trajectory (see also Limit cycle), and separatrices (cf. Separatrix).

For systems of two equations of the form (1) ( $ m= 2 $) the kinematic interpretation provides an illustrative and effective method of investigation, since the vector field $ f ( w) $ and the phase trajectories can in fact be mapped on a phase plane. If the number of equations is three $ ( m = 3 ) $, the respective constructions would have to be performed in a three-dimensional space, which is a difficult task, while if $ m > 3 $ such an approach is altogether impractical. Accordingly, if $ m \geq 3 $, and in many cases even if $ m= 2 $, the kinematic interpretation enables one to use geometrical concepts, methods and language which generalize to some extent the everyday geometrical ideas in the study of differential equations.

Even if relatively weak assumptions are made about the vector field $ f ( w) $( e.g. that it is differentiable), there exists for each point $ w _ {0} \in W ^ {m} $ exactly one solution $ w ( t) $ of (3) with initial value $ w _ {0} $: $ w ( 0) = w _ {0} $. The physical meaning of this result is that if the law of motion (3) is given, the state of the system at any moment of time is fully determined by its initial state. Generally speaking, the solution need not be defined for all $ t $, but only on a certain time interval. In the global theory of dynamical systems one makes the additional assumption that for any initial value the corresponding solution is defined for all $ t $, while in local problems it is usually unnecessary to make any assumption on the subsequent behaviour of the trajectories which leave the domain of the phase space under study.

If the above assumption is met, to each $ w _ {0} \in W ^ {m} $ one assigns the state $ w ( t) $ after time $ t $ of the phase point moving according to (3) starting at $ w _ {0} $ at $ t = 0 $, and one obtains a mapping $ S _ {t} $ of the phase space $ W ^ {m} $ into itself:

$$ S _ {t} w _ {0} = w ( t) , $$

where $ w ( t) $ is the solution of (3) and $ w ( 0) = w _ {0} $. The mappings $ S _ {t} $ form a continuous one-parameter group of diffeomorphisms (cf. Diffeomorphism) of the phase manifold $ W ^ {m} $( the group property $ S _ {t} S _ {s} = S _ {t + s } $ follows from the fact that the system (3) is autonomous). As an illustration, the analogy is often drawn in the literature with an example which is familiar from everyday life and which was the first studied in science. In this example there arises a similar family of transformations of space. The example is: a stationary flow of a liquid or a gas in which a liquid particle flows from the point $ w _ {0} $ to $ S _ {t} w _ {0} $ during time $ t $. (It may be remarked in this connection that such an analogy is rather superficial, since the "phase liquid" which is "flowing" in the phase space differs from real continuous media in that there is no interaction between the neighbouring particles.) Accordingly, the term flow (continuous-time dynamical system) is employed as a synonym for the term "dynamical system" .

In physical literature it is customary to speak of ensembles of dynamical systems. This means that each possible given physical system (i.e. each point in the phase space) is thought to represent some physical system described by equation (3) which is in that state; the resulting set of systems of the same type, which are non-interacting and which only differ in their state at the given moment, is referred to as an "ensemble" . In this language, the transformations $ S _ {t} $ of the phase space correspond to the evolution of the "ensemble" , consisting in changes of state of its constituent systems.

In developing the global theory of dynamical systems the concept of such a system is further generalized. In the widest sense of the word, a dynamical system is understood to mean an arbitrary action of a group (or even of a semi-group) $ G $ on a certain set $ W $ which is named the "phase space" . This means that for every $ g \in G $ a mapping $ S _ {g} : W \rightarrow W $ is defined such that $ S _ {g _ {1} } S _ {g _ {2} } = S _ {g _ {1} g _ {2} } $ and that, if $ e $ is the unit of $ G $, then $ S _ {e} $ is the identity transformation (i.e. $ S _ {e} ( w) = w $ for all $ w $). The set of points $ S _ {g} ( w _ {0} ) $, where $ w _ {0} $ is fixed while $ g $ runs through $ G $, is called the trajectory (or orbit) passing through the point $ w _ {0} $ or, briefly, the trajectory of this point. The group $ G $ is usually considered to be a topological group, the phase space is considered to be a topological space or a measure space, while the mapping

$$ \tag{5 } G \times W \rightarrow W ,\ ( g , w ) \rightarrow S _ {g} ( w) $$

is assumed to be, respectively, continuous or measurable, and in the latter case it is usual to assume that the mappings $ S _ {g} $ preserve the measure (i.e. the pre-image of a measurable subset of the phase space is measurable and has the same measure). The respective branches in the theory of dynamical systems are known as topological dynamics [GH], [Sib] and ergodic theory [Hal], , [A], [Sin]. If $ G $ is a Lie group and $ W $ is a smooth manifold, while the mapping (5) is smooth, one speaks of a smooth dynamical system.

The principal cases in all three approaches are those in which $ G $ is either the group $ \mathbf R $ of real numbers, in which case the dynamical system is referred to as a "flow" (even though this term is sometimes used as a synonym of the term "dynamical system" in its widest meaning) or else $ G $ is the group $ \mathbf Z $ of integers (or a variant: the additive semi-group of non-negative integers), in which case the name cascade has been proposed (one also speaks of a dynamical system with discrete time, but this term may also merely mean that a discrete topology on $ G $ is taken). A smooth flow for which the mapping (5) is of class $ C ^ {2} $ is defined by a smooth vector field, namely by

$$ f ( w) = \left . \frac{d}{dt} S _ {t} ( w) \right | _ {t= 0 } ; $$

when $ t $ varies, the point $ w = S _ {t} ( w _ {0} ) $ moves according to (3). In the case of a cascade, the mappings $ S _ {n} $ are obtained by iteration of the transformation $ S _ {1} $ and of its inverse (variant: iteration of only the mapping $ S _ {1} $); for a smooth cascade all $ S _ {n} $ are diffeomorphisms (variant: continuously differentiable mappings).

Of the three approaches in the theory of dynamical systems mentioned above, topological dynamics has a distinct set-theoretic character, and at first its role was more auxiliary. This was because a number of concepts (non-wandering point; limit set of a trajectory; minimal set, almost-periodicity, distality, Lagrange stability; Poisson stability, etc.) and their interconnections, which are important in treating more concrete objects such as smooth dynamical systems, are more conveniently studied under more abstract conditions, without specifying the dynamical system with the aid of a diffeomorphism or by equation (3). This is in fact done in topological dynamics. Subsequently, major advances were made in topological dynamics, mainly concerning the study of certain minimal sets and their extensions (cf. [B], [E], [V] and also Distal dynamical system).

The genesis of ergodic theory is connected with classical (pre-quantum) statistical physics. Its foundation involved the following problem. Is it possible to find statistical properties of the behaviour of all or almost-all phase trajectories as $ t \rightarrow \infty $ without solving the Hamiltonian system of differential equations which describes the motion of the particles constituting the macroscopic body under study, and even without the knowledge of the initial values for its solution (which would mean specifying the instantaneous positions and velocities of all these particles)? For example, does the limit of the time-average

$$ \tag{6 } \lim\limits _ {T \rightarrow \infty } \frac{1}{T} \int\limits _ { 0 } ^ { T } g ( S _ {t} w ) dt $$

exist for almost-all $ w $, where $ g ( w) $ is a function defined on the phase space, and does this limit depend on $ w $? The definition of a dynamical system given above and adopted in ergodic theory is the result of abstracting from the concrete origin of the systems of statistical physics, in particular from their Hamiltonian structure; only one consequence of this structure — to wit, conservation of the measure by the transformations $ S _ {t} $— is retained. In this case such an abstraction is not merely a tool in the logical analysis of concepts, but it results in a much more general theory, which includes material connected with probability theory, functional analysis, number theory, and topological algebra. Owing to these connections with various fields of mathematics the contents of ergodic theory are sufficiently extensive to ensure its development as an independent scientific discipline. The theory comprises not only the study of the statistics of solutions for $ t\rightarrow \infty $, which includes both the proof of the existence of the limit (6) for almost-all $ w $ and the derivation of the conditions under which it is independent of $ w $, but also several other conditions (e.g. mixing), the important role played in ergodic theory by the problem of isomorphism of dynamical systems, the study of which resulted in the construction of a number of invariants of dynamical systems and an identification of certain classes of dynamical systems with interesting properties.

The theory of smooth dynamical systems [A], [Sm], [N] merges, to a considerable extent, with the qualitative theory of differential equations, especially so if a concretely specified system (1) is being studied or if (irrespective of the way the dynamical system being studied has been specified) the study involves ideas connected with differential equations written down in a more or less explicit manner. Smooth dynamical systems are studied both locally and globally. The local properties include the study of equilibrium positions and the above-mentioned special types of trajectories for flows and their analogues for cascades, quasi-periodic motions (cf. Quasi-periodic motion) and invariant manifolds for these and other types of motion, as well as certain classes of invariant sets (cf. Invariant set). The study of these objects comprises their detection and localization, and the study of the behaviour of other trajectories of the dynamical system in their neighbourhood. Both analytical and topological methods [CL], [L], [Har] are used in the study of fixed points of cascades, equilibrium positions and periodic solutions of flows; primarily analytical methods are employed for other objects (see, however, [C]). Many of these methods are connected with the following problem: What is, on a flow or a cascade with specified local or global properties, the effect of a small change in the vector field or diffeomorphism defining the flow or the cascade? Such an approach is also connected with certain results and concepts of the global theory, particularly those aiming at the search for properties of classes of dynamical systems which would be in some sense "typical" (see, for example, Rough system). Other results of a global nature concern certain classes of dynamical systems which frequently occur in related disciplines.

In the special case of flows on two-dimensional surfaces it is possible to obtain fairly satisfactory information on the different possibilities of the behaviour of phase trajectories which may occur; this applies, in particular, to systems (1) with $ m = 2 $( two equations) (the Poincaré–Bendixson theory [NS], [CL], [L], [Har]) and flows on a torus without stationary points [CL], [Har], [N]. However, this theory does not provide an answer to the question of exactly how the trajectories behave in a concrete system. A large number of studies deals with this problem for various classes of equations. The special situation of flows on two-dimensional surfaces is due to the fact that a trajectory then locally subdivides the phase space. Accordingly, the natural higher-dimensional generalization of the relevant theory does not concern dynamical systems, but rather foliations of codimension one.

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A rough system is sometimes called a structurally stable system or a robust system.

See also Topological dynamical system; $ Y $- system; Bendixson criterion (absence of closed trajectories); Poincaré–Bendixson theory.

A well-documented survey on (mainly differentiable) dynamical systems is [Hi]. Many recent developments are discussed in the various volumes of [DS].

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