Loading...

Articles

A **dynamical** system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space. Small changes in the state of the system correspond to small changes in the numbers. The numbers are also the coordinates of a geometrical space—a manifold. The evolution rule of the **dynamical** system is a fixed rule that describes what future states follow from the current state. The rule may be deterministic (for a given time interval only one future state follows from the current state) or stochastic (the evolution of the state is subject to random shocks).

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. 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 spring in a lake.

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).

A major theme in the mathematical and computational analysis of graph **dynamical** systems is to relate their structural properties (e.g. the network connectivity) and the global dynamics that result.

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 real numbers R, M a manifold locally diffeomorphic to a Banach space, and Φ a continuous function. If T=R we call the system global, if T is restricted to the non-negative reals we call the system a semi-flow. If Φ is continuously differentiable we say the system is a differentiable **dynamical** system. If the manifold M is locally diffeomorphic to R n, the **dynamical** system is finite-dimensional; if not, the **dynamical** system is infinite-dimensional. Note that this does not assume a symplectic structure.

The full Chandrasekhar **dynamical** friction formula for the change in velocity of the object involves integrating over the phase space density of the field of matter and is far from transparent. The Chandrasekhar **dynamical** friction formula reads as

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

The discipline shares connections to and applications with both the static world of optimization and equilibrium problems and the **dynamical** world of ordinary differential equations. A projected **dynamical** system is given by the flow to the projected differential equation.

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).

Graph **dynamical** system - Stochastic graph **dynamical** systems

Every element of a graph **dynamical** system can be made stochastic in several ways. For example, in a sequential **dynamical** system the update sequence can be made stochastic. At each iteration step one may choose the update sequence w at random from a given distribution of update sequences with corresponding probabilities. The matching probability space of update sequences induces a probability space of SDS maps. A natural object to study in this regard is the Markov chain on state space induced by this collection of SDS maps. This case is referred to as update sequence stochastic GDS and is motivated by, e.g., processes where "events" occur at random according to certain rates (e.g. chemical reactions), synchronization in parallel computation/discrete event simulations, and in computational paradigms described later.

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?"

Graph **dynamical** system - Stochastic graph **dynamical** systems

This specific example with stochastic update sequence illustrates two general facts for such systems: when passing to a stochastic graph **dynamical** system one is generally led to (1) a study of Markov chains (with specific structure governed by the constituents of the GDS), and (2) the resulting Markov chains tend to be large having an exponential number of states. A central goal in the study of stochastic GDS is to be able to derive reduced models.

Graph **dynamical** system - Sequential **dynamical** systems (SDS)

If the vertex functions are applied asynchronously in the sequence specified by a word w = (w 1, w 2 , ... , w m ) or permutation \pi = ( \pi_1, ) of v[Y] one obtains the class of Sequential **dynamical** systems (SDS). In this case it is convenient to introduce the Y-local maps F i constructed from the vertex functions by

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

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, that is the study of the initial value problems for their describing systems of ordinary differential equations.

Graph **dynamical** system - Sequential **dynamical** systems (SDS)

Example: Let Y be the circle graph on vertices {1,2,3,4} with edges {1,2}, {2,3}, {3,4} and {1,4}, denoted Circ 4. Let K={0,1} be the state space for each vertex and use the function nor 3 : K 3 → K defined by nor 3 (x, y, z) = (1 + x)(1 + y)(1 + z) with arithmetic modulo 2 for all vertex functions. Using the update sequence (1,2,3,4) then the system state (0, 1, 0, 0) is mapped to (0, 0, 1, 0). All the system state transitions for this sequential **dynamical** system are shown in the phase space below.

Given a global **dynamical** system (R, X, Φ) on a locally compact and Hausdorff topological space X, it is often useful to study the continuous extension Φ* of Φ to the one-point compactification X* of X. Although we lose the differential structure of the original system we can now use compactness arguments to analyze the new system (R, X*, Φ*).

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

Projected **dynamical** system - History of projected **dynamical** systems

The formalization of projected **dynamical** systems began in the 1990s. However, similar concepts can be found in the mathematical literature which predate this, especially in connection with variational inequalities and differential inclusions.

Projected **dynamical** system - History of projected **dynamical** systems

Projected **dynamical** systems have evolved out of the desire to dynamically model the behaviour of nonstatic solutions in equilibrium problems over some parameter, typically take to be time. This dynamics differs from that of ordinary differential equations in that solutions are still restricted to whatever constraint set the underlying equilibrium problem was working on, e.g. nonnegativity of investments in financial modeling, convex polyhedral sets in operations research, etc. One particularly important class of equilibrium problems which has aided in the rise of projected **dynamical** systems has been that of variational inequalities.