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Continuous Models Using Ordinary Differential Equations
Published in Sandip Banerjee, Mathematical Modeling, 2021
Chaos, in an autonomous deterministic system, is defined as an aperiodic long-term behaviour (phase space trajectories do not converge to a point or a periodic orbit) that is sensitive to initial conditions (trajectories that start nearby initially, separate exponentially fast) [154]. Thus, chaos requires these three ingredients, namely, a periodic long-term behaviour, a deterministic system and sensitive dependence on initial conditions. The last condition implies that nearby trajectories diverge exponentially (positive Lyapunov exponent). Continuous systems in a two-dimensional phase space cannot experience such a divergence, hence chaotic behaviours can only be observed in a deterministic continuous systems with a phase space of dimension 3, at least. However, non-autonomous two-dimensional systems are equivalent to autonomous three-dimensional systems, which can exhibit chaotic behavior. Similarly, delay systems are infinite-dimensional, therefore, chaos can be observed in such systems. Discrete systems can exhibit chaotic behaviors even if they are one-dimensional.
Fractal and chaotic characteristics of alluvial rivers
Published in Zhao-Yin Wang, Shi-Xiong Hu, Stochastic Hydraulics 2000, 2020
Deng Zhi-Qiang, Vijay P. Singh
Lyapunov exponents are of interests in the study of dynamic systems in order to characterize quantitatively the average rates of convergence or divergence of nearby trajectories in phase space, and, therefore, they measure how predictable or unpredictable the system is. Since they can be computed either from a mathematical model or from experimental data, they are widely used for the classification of attractors (Szemplinska-Stupnicka et al, 1991). Negative Lyapunov exponents signal periodic orbits; constant Lyapunov exponents imply that nearby trajectories do not diverge or converge; whilst at least one positive exponent indicates a chaotic orbit and the divergence of initially neighboring trajectories. Tsonis (1992) proposed the following definition of Lyapunov exponents : λi=limT→∞1Tln[Pi(T)Pi(0)]
State-Space Reconstruction
Published in Nicholas Stergiou, Nonlinear Analysis for Human Movement Variability, 2018
These equations provide us with the values for the X, Y, and Z coordinates. Thus, we have three sequences of numbers called X(t), Y(t), and Z(t). The plots of each of these sequences are shown in Figure 3.3a through c. All three sequences can be plotted together in a three-dimensional plot (Figure 3.3d). Each point in this three-dimensional graph also defines a vector and contains the three values from the coordinates (X, Y, and Z; Figure 3.4). This vector is the state that describes what is going on with the system at a specific time. For example, at time t = t1, we have a vector (X(t1), Y(t1), Z(t1)) representing the state of the system at time t1 (Figure 3.4). A visual representation of the behavior of the system in the state-space is called a state-space plot or phase-space plot. A phase space can be considered as a special case of state-space, and, in general, the term is used to describe a mechanical system where state-space consists of all possible values of position and momentum variables (the first derivative and successive derivatives). However, these two terms are often used interchangeably.
Frictional Vibration Performances of Dry Gas Seal Rings with DLC Film Textured Surface via Chaos Theory
Published in Tribology Transactions, 2021
Jinlin Chen, Liping Tang, Xuexing Ding, Jiaxin Si, Delin Chen, Baocai Sun
The phase space refers to a geometric space that a system state is in. Generally speaking, a quite high phase space dimension results from a nonlinear dynamic system. Nevertheless, data acquired from the tests were actually single-variable time series that were received from the cooperation of various parameters in the system. In this case, for more dynamic information to be collected, a reconstruction of the test data into a high-dimensional space was necessary. By means of phase space reconstruction, Takens (28) extended a 1D chaotic time series into a 3D or higher-dimensional phase space to gather more information that the time series hid. Thus, a matrix is constructed in phase space: where m is the embedding dimension, τ reflects the delay time, and N is used to represent the number of vectors in the reconstructed phase space. It can be concluded that selecting an appropriate τ and m is essential for reconstructing the phase space.
The variable and chaotic nature of professional golf performance
Published in Journal of Sports Sciences, 2018
In general, the behaviour of a dynamical system is represented by a trajectory in phase space. In our case the behaviour of the system is represented by the one-dimensional stroke series of a player’s Shots Saved values from the whole tournament or from all shots within a shot category. According to Takens (1981) the phase space needs to be reconstructed by embedding the measured trajectory into a higher dimensional phase space which “guarantee[s] the existence of a diffeomorphism between the original and the reconstructed” (Marwan et al., 2007, p. 246) phase space. The calculation of an RP is based on the respective dynamical system’s trajectory in the reconstructed phase space. To achieve the phase space reconstruction we used the time delay embedding method suggested by Marwan et al. (2007).
Wine and maths: mathematical solutions to wine–inspired problems
Published in International Journal of Mathematical Education in Science and Technology, 2018
There are plenty of papers on fermentation research like [11–14], where one can find other geometric details about stacking wine bottles, some math about the link between food and wine and many other applications in the wine industry. For a more recent application of mathematical models to wine tasting and analysis, we refer to [15] where wine is interpreted as a chaotic dynamical system. We know that a dynamical system is described by its state at the time t, where x1,… , xn are measurable quantities. The evolution of the system x(t) = F(t0, x0; t) is a function called trajectory that allows to uniquely determine the state of the system in any successive time t when the initial state x0 at t0 is known. The operator F is replaced by some differential equations representing the local evolution of the system when F is not easy to calculate. But, only in simple cases as linear differential equations, one can find the analytic solution of the system satisfying the initial conditions. A dynamical system is chaotic if an arbitrary small perturbation of the current trajectory may lead to significantly different future behaviour; if the system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region; if its periodic orbits are dense.