Скачать или смотреть Jerk Chaos - Free Sound Effects Download FSED

Jerk Chaos - Free Sound Effects Download FSED
In recent years, it is observed that the third-order explicit autonomous differential equation, named as jerk equation, represents an interesting sub-class of dynamical systems that can exhibit many major features of the regular and chaotic motion. In this paper, we investigate the global dynamics of a special family of jerk systems {ie075-01}, whereG(x) is a non-linear function, which are known to exhibit chaotic behaviour at some parameter values. We particularly identify the regions of parameter space with different asymptotic dynamics using some analytical methods as well as extensive Lyapunov spectra calculation in complete parameter space. We also investigate the effect of weakening as well as strengthening of the non-linearity in theG(x) function on the global dynamics of these jerk dynamical systems. As a result, we reach to an important conclusion for these jerk dynamical systems that a certain amount of non-linearity is sufficient for exhibiting chaotic behaviour but increasing the non-linearity does not lead to larger regions of parameter space exhibiting chaos.
Chaos theory is a branch of mathematics focusing on the study of chaos—states of dynamical systems whose apparently random states of disorder and irregularities are often governed by deterministic laws that are highly sensitive to initial conditions.[1][2] Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization.[3] The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions).[4] A metaphor for this behavior is that a butterfly flapping its wings in China can cause a hurricane in Texas.[5]

Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general.[6] This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution[7] and is fully determined by their initial conditions, with no random elements involved.[8] In other words, the deterministic nature of these systems does not make them predictable.[9][10] This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:[11]

Chaos: When the present determines the future, but the approximate present does not approximately determine the future.

Chaotic behavior exists in many natural systems, including fluid flow, heartbeat irregularities, weather and climate.[12][13][7] It also occurs spontaneously in some systems with artificial components, such as the stock market and road traffic.[14][3] This behavior can be studied through the analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Chaos theory has applications in a variety of disciplines, including meteorology,[7] anthropology,[15] sociology, physics,[16] environmental science, computer science, engineering, economics, biology, ecology, pandemic crisis management,[17][18] and philosophy. The theory formed the basis for such fields of study as complex dynamical systems, edge of chaos theory, and self-assembly processes.
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