# Dynamical Systems A dynamical system is actually a concept in math in which a fixed rule explains the time dependence of a point inside a geometrical space. Examples contain the mathematical models that explain the circulation of water in a tube, the swinging of a clock pendulum and the number of fish every springtime in a river.

At any given moment a dynamical system features a state given by a couple of real numbers (a vector) that could be symbolized by a point in a proper state space (a geometrical manifold). Tiny changes in the state of the system generate small adjustments in the numbers. The evolution principle of the dynamical system is a set rule that explains what future states follow through the present state. The rule is deterministic; quite simply, for a given period interval only a single future state follows from the existing state.

## Dynamic Systems Theory

Dynamical systems theory is actually an area of mathematics employed to describe the habits of complicated dynamical systems, generally by utilizing difference equations or differential equations. When difference equations are used, the theory is referred to as discrete dynamical systems. Whenever differential equations are used, the theory is known as continuous dynamical systems. Once the time variable operates over a set which is discrete over a few intervals and constant over other intervals or is any arbitrary time-set including a cantor set one receives dynamic equations on time scales. Some circumstances may be also modeled by combined operators, for example differential-difference equations.

This theory handles the long-term qualitative habits of dynamical systems and the research of the solutions for the equations of motion of systems which are primarily mechanical in dynamics; although this contains both behaviour of electronic circuits and the planetary orbits and the solutions for partial differential equations that occur in biology. A lot of modern research is centered on the analysis of chaotic systems. This area of study is also known as just Dynamical systems, Systems theory or lengthier as Mathematical theory of dynamical systems and the Mathematical Dynamical Systems Theory.

## Modeling and Analysis of Dynamic Systems

Modeling and analysis of dynamic systems which includes electrical, mechanical, hydraulic systems and electro�mechanical. Usage of Laplace changes and complicated algebra. Mathematical modeling of dynamic systems within state�space. Linear systems analysis in frequency domains and time.

Dependability evaluation is a significant, usually indispensable, step in (critical) analysis processes and systems design. The introduction of control and/or computing systems for automate procedures increases the entire system complexity and therefore has an effect in terms of dependability. Furthermore it is of interest to assess redundancy and upkeep policies. In those instances it is difficult to recur to notations like fault trees, reliability block diagrams or reliability graphs to symbolize the system, because the statistical independence presumption is not satisfied. Also more improved formalisms as dynamic FT (DFT) could outcome not adequate to the goal.

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