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The state space or phase space is the geometric space in which the axes are the state variables. The system state can be represented as a vector , the state vector . If the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form.
If the size of the state space is finite, calculating the size of the state space is a combinatorial problem. [4] For example, in the Eight queens puzzle, the state space can be calculated by counting all possible ways to place 8 pieces on an 8x8 chessboard. This is the same as choosing 8 positions without replacement from a set of 64, or
At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manifold. The evolution rule of the dynamical system is a function that describes what future states follow from the current state.
A solution to a discretized partial differential equation, obtained with the finite element method. In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical ...
The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the phase space usually consists of all possible values of the position and momentum parameters.
The equation of motion is ¨ + ˙ + = In mathematics , a phase portrait is a geometric representation of the orbits of a dynamical system in the phase plane . Each set of initial conditions is represented by a different point or curve .
The parameters of a dynamic equation evolve as the equation is iterated, and the specific values may depend on the starting parameters. An example is the well-studied logistic map , x n + 1 = r x n ( 1 − x n ) {\displaystyle x_{n+1}=rx_{n}(1-x_{n})} , whose basins of attraction for various values of the parameter r {\displaystyle r} are shown ...
Thus the set of all states of M with the weak-* topology forms a compact Hausdorff space, known as the state space of M. In the C*-algebraic formulation of quantum mechanics, states in this previous sense correspond to physical states, i.e. mappings from physical observables (self-adjoint elements of the C*-algebra) to their expected ...
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