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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)]
Chaotic Soliton Oscillator and Chaotic Communications
Published in David S. Ricketts, Donhee Ham, Electrical Solitons, 2011
O. Ozgur Yildirim, Nan Sun, Xiaofeng Li
As explained in the beginning of this chapter, a positive Lyapunov exponent is an indication of chaos. We simulated the Lyapunov exponent of our circuit as follows. We run two simulations with very close initial conditions and record all the twenty node voltages on the NLTL. We calculate the evolution of the absolute difference [log(Δ(t))] between the NLTL node voltages of the two simulations as time progress. As shown in Fig. 11.7, log(Δ(t)) increases linearly with time, confirming that Δ(t) exponentially grows and the oscillation is indeed chaotic.
The logistic map and elements of complex system dynamics
Published in Arturo Buscarino, Luigi Fortuna, Mattia Frasca, ®and Laboratory Experiments, 2017
Arturo Buscarino, Luigi Fortuna, Mattia Frasca
The Lyapunov exponents are a measure quantifying the high sensitivity of chaotic systems to initial conditions. Recall Example 2.6; in that case the logistic map was started from two initial conditions differing for a small quantity ε = 10–5, but the resulting trajectories were very different. Lyapunov exponents aim at measuring the rate of divergence of nearby trajectories. They are in a number equal to that of the state variables, so in the case of the logistic map one Lyapunov exponent, indicated as λ, is calculated.
Sustainability and robust decision-support strategy for multi-echelon supply chain system against disruptions
Published in International Journal of Logistics Research and Applications, 2023
Le Ngoc Bao Long, Truong Ngoc Cuong, Hwan-Seong Kim, Sam-Sang You
Lyapunov exponent (LE) is a quantitative measure that expresses the convergent rate of two relatively close trajectories with a slight difference in initial conditions. For n-dimensional phase space, let be the distance of two trajectories along direction i at time step k. By using the norm (or Euclidean distance), the exponential growth with LEi can be estimated as follows: where is initial separation, with a chaotic motion of LEi > 0 making behaviour unpredictable, whereas for regular motion LEi ≤ 0. An essential property of chaos is its sensitive dependence on the initial conditions, in which a system's behaviour may evolve differently from slightly different initial conditions. The Lyapunov exponent is a useful analytical metric for characterising various dynamical behaviours of a system or chaos.
Analysis of Fourth-order Chaotic Circuit Based on the Memristor Model for Wireless Communication
Published in IETE Journal of Research, 2023
Sam Thomas, Savarimuthu Prakash, Sankara Malliga Gopalan
Figure 2 shows spectra of the Lyapunov exponents (for C1 = 7.5 nF). The Lyapunov exponent characterizes the divergence or convergence of neighbouring trajectories of a dynamical system. It explains the dynamics for a small space of initial conditions. The amount of Lyapunov exponents equals the dimensions of the system in the phase space. Thus, the proposed system (VC1, VC2, iL, w) has four Lyapunov exponents. A Lyapunov exponent that is positive indicates that the system trajectory expands in the phase space and, the total of the Lyapunov exponents is negative, system trajectory shrinks in the phase space. The expansion and contraction in the phase space correspond to stretching and folding of trajectories, which is the characteristic of chaos. The Lyapunov exponent was scaled by a factor of √(LC2).
Analysis of mixing structures in the Adriatic Sea using finite-size Lyapunov exponents
Published in Geophysical & Astrophysical Fluid Dynamics, 2022
In last decade, powerful tools have been introduced to provide a better understanding of transport barriers (exceptional material lines that deform less than their neighbours) in the ocean. One of these methods, for describing local dispersion properties in time-dependent flows, is the finite-size Lyapunov exponent (FSLE), which is based on the time-dependent velocity field, and has great accuracy. Note that the fundamental definition of the Lyapunov exponent is that it is a global measure of the divergence rates of nearby trajectories. Furthermore, hyperbolic Lagrangian coherent structures (LCSs), which define transport pathways in the ocean, especially over the mesoscale range, can be defined as ridges in the FSLE fields, although a mathematical link between the FSLE and LCSs is yet to be determined.