A double rod pendulum animation showing chaotic behavior. Starting the pendulum from a slightly different initial condition would result in a completely different trajectory. The double rod pendulum is one of the simplest dynamical systems with chaotic solutions. Chaos chaos and time series analysis sprott pdf is a branch of mathematics and it is focused on the behavior of dynamical systems that are highly sensitive to initial conditions.
Chaos’ is an interdisciplinary theory stating that within the apparent randomness of chaotic complex systems, there are underlying patterns, constant feedback loops, repetition, self-similarity, fractals, self-organization, and reliance on programming at the initial point known as sensitive dependence on initial conditions. The butterfly effect describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state, e. Brazil can cause a hurricane in Texas. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved.
In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos.
Chaos: When the present determines the future, but the approximate present does not approximately determine the future. Chaotic behavior exists in many natural systems, such as weather and climate. It also occurs spontaneously in some systems with artificial components, such as road traffic.
This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Chaos theory has applications in several disciplines, including meteorology, anthropology, sociology, physics, environmental science, computer science, engineering, economics, biology, ecology, and philosophy. The theory formed the basis for such fields of study as complex dynamical systems, edge of chaos theory, and self-assembly processes. Chaos theory concerns deterministic systems whose behavior can in principle be predicted.
Chaotic systems are predictable for a while and then ‘appear’ to become random. The amount of time that the behavior of a chaotic system can be effectively predicted depends on three things: How much uncertainty can be tolerated in the forecast, how accurately its current state can be measured and a time scale depending on the dynamics of the system, called the Lyapunov time. In chaotic systems, the uncertainty in a forecast increases exponentially with elapsed time. Hence, mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast.
This means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made, the system appears random. Here, two series of x and y values diverge markedly over time from a tiny initial difference. Note, however, that the y coordinate is defined modulo one at each step, so the square region is actually depicting a cylinder, and the two points are closer than they look.