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Dynamical systems approach of modelling
Published in A. W. Jayawardena, Environmental and Hydrological Systems Modelling, 2013
In mathematics and physics, the chaos theory describes the behaviour of certain nonlinear dynamical systems that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the ‘butterfly effect’1). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behaviour of chaotic systems appears to be random. This happens even though the systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. Such systems are governed by physical laws and are very difficult to predict accurately. Weather forecasting, which is predictable in the short term if enough information is available but unpredictable with certainty in the long term, is a commonly cited example. The principal characteristics of deterministic chaotic systems include the difficulty or impossibility to predict the long-term behaviour; sensitivity to initial conditions, meaning that two trajectories starting from very close initial conditions very rapidly moving to divergent states; and the exponential amplification of errors.
Overview of dynamical systems and chaos
Published in Marcio Eisencraft, Romis Attux, Ricardo Suyama, Chaotic Signals in Digital Communications, 2018
Numerically, chaos was found by John von Neumann (1903–1957) and Stanislaw M. Ulam (1909–1984) by simulating in a computer the logistic map with p = 4, which was used, in 1947, as a random number generator. The title of their short paper is “On Combination of Stochastic and Deterministic Processes,” because apparently random sequences were obtained from a deterministic equation [44]. In time-continuous systems, numerical chaotic solutions were reported in 1963 by Edward N. Lorenz (1917–2008) in the manuscript “Deterministic Non-periodic Flow,” about a hydrodynamic model for meteorology, which was solved in a computer [22]. In this context, the butterfly effect is a poetic metaphor of sensitivity to initial conditions: the flight of a butterfly certainly causes tiny alterations in the atmosphere; however, if this system produces chaotic solutions, then such minor changes can result in completely different weather scenarios. Observe that, in simulations, rounding and truncation errors always occur; therefore, a numerically calculated chaotic solution will diverge from the true solution with the same initial conditions. Under certain circumstances, the shadowing lemma assures that there is a true solution with slightly different initial coordinates that stays near (“shadows”) the numerically computed solution (e.g., [16]). Thus, the characterization of a chaotic system from data obtained via simulations can be valid. This characterization is usually based on its spatio-temporal “statistical” properties (as in truly stochastic processes). Some tools employed are analysis of power spectrum density, entropy derived from Information Theory, and invariant measure provided by Ergodic Theory (e.g., [11]). Other tools are Lyapunov exponent and Hausdorff dimension, which will be defined in the next sections.
The Human in Complex Dynamic Systems
Published in Guy H. Walker, Neville A. Stanton, Paul M. Salmon, Daniel P. Jenkins, Command and Control: The Sociotechnical Perspective, 2009
Guy H. Walker, Neville A. Stanton, Paul M. Salmon, Daniel P. Jenkins
Another prominent artefact of complex systems is ‘sensitive dependence on initial conditions’. The term often given to this phenomenon is ‘the butterfly effect’, in recognition of a paper by mathematician and meteorologist Edward Lorenz. The title of Lorenz’s paper was ‘Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?’ (see Hilborn, 2004). Lorenz argued that, at least in theory, it could. The supposition was based on the now legendary story of how a primitive computerised simulation of the weather gave birth to an entirely new scientific discipline, that of Chaos Theory. Midway through a weather simulation, some interesting phenomenon had emerged and the simulation was paused in order to look more closely. To avoid restarting the simulation from scratch, the state of the simulation at that moment, which was represented by a single number expressed to six decimal places (.506127), was re-entered but to only three decimal places (.506) in order to save time. The assumption was that any effects of such a small difference would be damped out by the larger and presumably more powerful global phenomenon. This assumption proved to be incorrect. .506127 represented the initial conditions of the simulation and it was highly sensitive to change. This one part in a thousand (and smaller) difference led the simulation, over a fairly short time, to evolve into a completely different state. So why the butterfly? It is because a flap of a butterfly’s wings may represent a one part in a thousandth, millionth, billionth or even smaller of a total weather system, but given enough time it is conceivable (if not especially probable) that its presence and absence could be the difference concerning a tornado in Texas. The location of the butterfly and the ensuing tornado have varied over the years, but the fundamental attribute of ‘sensitive dependence on initial conditions’ has remained.
Fractional optical solitons with stochastic properties of a wick-type stochastic fractional NLSE driven by the Brownian motion
Published in Waves in Random and Complex Media, 2021
Ben-Hai Wang, Yue-Yue Wang, Chao-Qing Dai
Functions and are white noise functions, which are related to the Brownian motion. As we all know, chaos theory is about bifurcation, periodic motion and aperiodic motion entanglement in nonlinear systems under certain parameters, and chaotic motion in natural sciences usually refers to a random behavior in deterministic systems. Considering the random nature of the Brownian motion, we use the Lorentz chaotic system to influence fractional stochastic soliton solutions of wick-type SFNLSE. Lorenz found that the so-called ‘butterfly effect’ is a chaotic phenomenon of climate change caused by simple convection, and chaotic systems are highly sensitive to initial values. Lorentz differential equation system is a typical chaotic system model, and it describes a singular attractor (Lorentz attractor) in motion [38]. In Lorentz equations of the following form, chaos occurs by selecting different parameters [38]. The evolution diagram of a typical singular chaotic attractor is shown in Figure 1(a) when we set the parameter and initial value of Lorentz chaos model as [39,40].
Understanding climate change through Earth’s energy flows
Published in Journal of the Royal Society of New Zealand, 2020
The best example of the tremendous success of this approach is numerical weather prediction, which uses a computer-based model of the atmosphere. Beginning with crude models in the 1950s, these have developed to include 50–100 levels in the vertical and horizontal resolutions of order 20 km. They have improved enormously, as has the observing system, including data and imagery from satellites, that are combined through a data assimilation process. Together they allow the state of the atmosphere at any time to be determined to a known level of accuracy through a combination of past predictions of the model, which essentially carries forward in time all of the past observed data, with the new observations. This observed state is then used as the starting point for new predictions for up to about 2 weeks. The predictability of each analysed situation can be determined by making imperceptible perturbations in the fields, and watching how well the predictions group or spread. Hence it is common now-a-days to use an ensemble of predictions to explore these aspects. Indeed, it is well established that even for a perfect model, the deterministic predictability of the atmosphere is of order 10 days before the chaotic aspects take over (Zhang et al. 2019). Here ‘chaos’ refers to the mathematics of sensitivity to small perturbations, often called ‘the butterfly effect’ after Ed Lorenz, who first discovered and documented mathematical chaos: how ‘a butterfly flapping its wings in Brazil can produce a tornado in Texas’ (see Gleick 1987 and Lorenz 1993).