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Introduction to Dynamical System

A dynamical system is a notion in mathematics where a fixed rule explains the time dependence of a point in a geometrical space. The dynamical system's illustrations involve the mathematical models which explain the swinging of a clock pendulum, flow of water in a pipe, and the number of fish each spring time in a lake.

At any given time a dynamical system has a state given by a set of real numbers which can be represented by a point in a suitable state space (i.e., a geometrical manifold). A little modification in the state of the system generates small changes in the numbers. The evolution rule of dynamical system is a fixed rule which explains what future states follow from the present state. The rule is deterministic, for a given time interval only one future state follows from the present state.

**Chaos theory**

The Chaos theory is a field of study in mathematics, with applications in different disciplines that involves engineering, philosophy, physics, economics, and biology. Chaos theory studies the behavior of dynamical systems which are highly sensitive to initial conditions, an effect that is generally termed to as the butterfly effect. The small differences in main conditions (like those due to rounding errors in numerical computation) results widely diverging outcomes for chaotic systems, depiction long-term prediction not possible in common. This occurs even though these systems are deterministic, meaning that their future behavior is entirely determined by their initial conditions with no arbitrary elements involved. In another meaning the deterministic nature of these systems does not make them conventional. This behavior is termed as deterministic chaos or simply chaos.

The Chaotic behavior can be observed in numerous natural systems, such as weather. Elucidation of that type behavior might be sought through analysis of a chaotic mathematical model or by analytical methods such as recurrence plots and Poincare maps.

In the hypothesis of chaos the butterfly effect is the sensitive dependence on initial conditions where a little change at one place in a deterministic nonlinear system can outcome in large differences to a later state. The effect's name is introduced by Edward Lorenz that is derived from the theoretical illustration of a hurricane's formation being contingent on whether or not a distant butterfly had flapped its wings a number of weeks before.

However the butterfly effect might appear to be an esoteric and unlikely behavior, it is displayed by very simple systems: for illustration, a ball placed at the crest of a hill may roll into any of numerous valleys depending on, among other things, slight differences in initial position.

The butterfly effect is a general trope in fiction when presenting scenarios involving time travel and with hypotheses where one storyline diverges at the moment of a seemingly minor event resulting in two significantly different outcomes.

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