FIG 3 - uploaded by Madhura Joglekar
Content may be subject to copyright.
Source publication
The character of the time-asymptotic evolution of physical systems can have
complex, singular behavior with variation of a system parameter, particularly
when chaos is involved. A perturbation of the parameter by a small amount
$\epsilon$ can convert an attractor from chaotic to non-chaotic or vice-versa.
We call a parameter value where this can ha...
Citations
... Understanding the self-organizing behavior of systems under changing environmental conditions and improving capabilities to predict large avalanches are priorities for research [56,57]. An influential approach to predicting the onset of large avalanches is to estimate changes in the critical exponent, and good results have been obtained [58][59][60][61][62][63][64][65]. Despite its appeal, the critical exponent in fact has a limitation: The results of the critical exponent usually require high-resolution spatial data, and such data are sometimes not available [60,[66][67][68]. ...
... An influential approach to predicting the onset of large avalanches is to estimate changes in the critical exponent, and good results have been obtained [58][59][60][61][62][63][64][65]. Despite its appeal, the critical exponent in fact has a limitation: The results of the critical exponent usually require high-resolution spatial data, and such data are sometimes not available [60,[66][67][68]. ...
Avalanches are sudden, destructive, and extremely difficult to forecast natural disasters that can result in numerous fatalities and extensive property damage. Given the immense danger posed by avalanches, there is a significant amount of attention paid to accurately predicting these events. We investigate the predictability of large avalanches in a class of self-organizing systems, which change their internal structure or function in response to external circumstances by manipulating or organizing other elements of the same system. Here, we propose a practical relaxation time to replace a traditional recovery time, and importantly, the relaxation time does not require the removal of part of the resources (perturb state variables) in the environment. This work provides examples of the forest fire model and sandpile model as self-organizing systems in which the relaxation time successfully predicts the onset of large avalanches. Furthermore, the relaxation time can show a consistent with the increasing trend in both oscillatory and nonoscillatory bifurcations, suggesting that the relaxation time is more universal than traditional indirect metrics such as the variance and the lag-1 autocorrelation function. We aim to identify early warning signals before the onset of large avalanches and provide scientific evidence and significant information for managers to formulate mitigation countermeasures and strategic decisions.
... Other works have argued that the use of topological criteria are efficient tools for determining parameter sensitivity [2,5,6], especially in the context of quadratic maps, and evidenced that this phenomenon is sufficient to produce divergence patterns close or even equivalent to those caused by variations in the initial conditions of chaotic systems [7]. On the other hand, investigations involving the dependence of Lyapunov exponents on parameters show that variations in system parameters can be reflected in simple power laws 1 ...
... and it is possible to observe this type of solution in numerical works such as Safritri et al. (2020) [11], who evidenced a dependence of Lyapunov exponents with the mass ratio of a double pendulum λ ∼ m2 m1 , or Delis et al. (2015) [9], who explored the dependence of Lyapunov exponents with a mass parameter (λ ∼ m p ) in a galactic potential. The latter also searched for physical origins for the power law associated with the mass parameter, however, according to the equation (6), it is possible to verify that this type of dependence appears simply due to the linearity of the parameters in the analyzed lagrangian function. ...
... where C and k are constants, according to equation (6). This behavior can be seen numerically by considering a interval b ± 0.001 with initial conditions x(0) = 0. We can also consider varying other parameters to see how these properties change. ...
The sensitive dependence of chaos on parameters is a topic of great interest in the study of integrability and stability of dynamical systems. Previous work has proposed ways to identify the sensitive dependence on parameters by topological criteria or large numerical simulations. In this paper, we show that when the Lyapunov exponents of the system vary with a change in the parameters, the system diverges exponentially in the orbits associated with the considered parameters. We use this result to explore the sensitive dependence on parameters in an uncertainty interval and conclude that the characterization of this phenomenon is directly related to our ability to determine the Lyapunov exponents of the system for different parameters.
... In this section, the chaotic maps employed for improving the MRFO are first presented, and then the steps followed for embedding them are described. Ten chaotic variants are considered in this article as: Chebyshev [88], Logistic [89], Piecewise [82], Sine [82], Singer [82], Sinusoidal [82], Tent [90], Lozi [91], Henon [66], Quadratic [92]. Furthermore, there are more maps have been used to improve the suggested approach but have not been presented due to the worst results found. ...
Manta ray foraging optimization (MRFO) algorithm is relatively a novel bio-inspired optimization technique directed to given real-world engineering problems. In this present work, wind turbines layout (WTs) inside a wind farm is considered a real nonlinear optimization problem. In spite of the better convergence of MRFO, it gets stuck into local optima for large problems. The chaotic sequences are among the performed techniques used to tackle this shortcoming and improve the global search ability. Therefore, ten chaotic maps have been embedded into MRFO. To affirm the performance of the suggested chaotic approach CMRFO, it was first assessed using the IEEE CEC-2017 benchmark functions. This examination has been systematically compared to eight well-known optimization algorithms and the original MRFO. The non-parametric Wilcoxon statistical analysis significantly demonstrates the superiority of CMRFO as it ranks first in most test suites. Secondly, the MRFO and its best enhanced chaotic version were tested on the complex problem of finding the optimal locations of wind turbines within a wind farm. Besides, the application of the CMRFO to the wind farm layout optimization (WFLO) problem aims to minimize the cost per unit power output and increase the wind-farm efficiency and the electrical power engendered by all WTs. Two representative scenarios of the problem have been dealt with a square-shaped farm installed on an area of 2 km
$\times2$
km, including variable wind direction with steady wind speed, and both wind direction and speed are variable. The WFLO outcomes reveal the CMRFO capability to find the optimal locations of WTs, which generates a maximum power for the minimum cost compared to three stochastic approaches and other previous studies. At last, the suggested CMRFO with Singer chaotic sequence has been successfully enhanced by accelerating the convergence and providing better accuracy to find the global optimum.
... and it is possible to observe this type of solution in numerical works such as Safitri et al. (2020) [11], who evidenced a dependence of Lyapunov exponents with the mass ratio of a double pendulum λ ∼ m2 m1 , or Delis et al. (2015) [9], who explored the dependence of Lyapunov exponents with a mass parameter (λ ∼ m p ) in a galactic potential. The latter also searched for physical origins for the power law associated with the mass parameter, however, according to the equation (6), it is possible to verify that this type of dependence appears simply due to the linearity of the parameters in the analyzed lagrangian function. ...
... where C and k are constants, according to equation (6). This behavior can be seen numerically by considering a interval b ± 0.001 with initial conditions x(0) = 0. We can also consider varying other parameters to see how these properties change. ...
... It was originally described by Robert May with a direct application in Biology [1][2][3]. Applications of the predictions derived from the logistic map were verified in a set of other rather far complicated models including experiments on fluids with turbulence, chemical reaction oscillations, non-linear electric circuits and a variety of other systems [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. ...
... A important description to represent the growth of biological populations using nonlinear mapping was introduced by May [1]. After his work, many other applications of mappings can be found in problems of the physics, chemistry, biology, engineering, mathematics, and many others [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] The one-dimensional mappings can be described by a dynamical variable and control parameters allowing to built the orbit diagrams. The local bifurcations in the orbit diagrams can be investigated by the stability of either fixed points or from higher period orbits [18,19]. ...
The convergence to the stationary state is described using scaling arguments at a fold and a period doubling bifurcation in a one-dimensional Gauss map. Two procedures are used: (i) a phenomenological investigation leading to a set of critical exponents defining the universality class of the bifurcation and; (ii) analytical investigation that transforms, near the stationary state, the difference equation into an ordinary differential equation that is easily solved. The novelty of the procedure comes from the fact that it is firstly applied to the Gauss map and critical exponents for the fold bifurcations are defined.
... A generalization of this formula to chaotic saddles is the Kantz-Grassberger relation [6], which connects the information dimensions along unstable directions with the associated Lyapunov exponents and the overall rate of escape from the saddle. While some fundamental open problems remain the subject of active research (e.g., the properties and applications of transient chaos [7][8][9][10][11], as well as the robustness [12], the classification [13], and the very definition [14] of chaos), studies of chaos in such systems are relatively mature [15]. ...
... which, as an integral quantity, properly accounts for the cumulative impact of the effective dimension on the relation between initial state accuracy and final state uncertainty in the systems we consider. The equivalent dimension in Eq. (12) can be derived as follows. First, writing the final state uncertainty of the equivalent self-similar system asf (ε ) = C · (ε ) N −Deq , where C is a constant, we have f (L) = CL N −Deq and f (ε) = Cε N −Deq . ...
Traditional studies of chaos in conservative and driven dissipative systems have established a correspondence between sensitive dependence on initial conditions and fractal basin boundaries, but much less is known about the relation between geometry and dynamics in undriven dissipative systems. These systems can exhibit a prevalent form of complex dynamics, dubbed doubly transient chaos because not only typical trajectories but also the (otherwise invariant) chaotic saddles are transient. This property, along with a manifest lack of scale invariance, has hindered the study of the geometric properties of basin boundaries--most remarkably, the very question of whether they are fractal across all scales has yet to be answered. Here we derive a general dynamical condition that answers this question, which we use to demonstrate that the basin boundaries can indeed form a true fractal; in fact, they do so generically in a broad class of transiently chaotic undriven dissipative systems. Using physical examples, we demonstrate that the boundaries typically form a slim fractal, which we define as a set whose dimension at a given resolution decreases when the resolution is increased. To properly characterize such sets, we introduce the notion of equivalent dimension for quantifying their relation with sensitive dependence on initial conditions at all scales. We show that slim fractal boundaries can exhibit complex geometry even when they do not form a true fractal and fractal scaling is observed only above a certain length scale at each boundary point. Thus, our results reveal slim fractals as a geometrical hallmark of transient chaos in undriven dissipative systems.
... Here J ij , i, j = 1, 2 are as in eq. (21). ...
... There is probably a delicate interplay here between the parameters in the problem, and the size of this invariant set. Recently, Joglekar et al. [21,22] discuss this issue by defining ":-uncertainty" in case of dynamical system and explained that how a small perturbation in parameter value changes the asymptotic behavior of the system. This is still an interesting open question. ...
In the current manuscript, an attempt has been made to understand the dynamics of a time-delayed predator-prey system with modified Leslie-Gower and Beddington-DeAngelis type functional responses for large initial data. In Ref. Upadhyay and Agrawal, 83(2016) 821–837, it was shown that the model possesses globally bounded solutions, for small initial conditions, under certain parametric restrictions. Here, we show that actually solutions to this model system can blow-up in finite time, for large initial condition
... In particular, we showed by calculating the Lyapunov exponents numerically and experimentally that this parameter space presents self-similar periodic structures, the shrimps, embedded in a domain of chaos [1][2][3][4][5][6][7][8]10,11,13,14 . We also show experimentally that those self-similar periodic regions organize themselves in period-adding bifurcation cascades, and whose sizes decrease exponentially as their period grows 1,9,17,18 . ...
We report high-resolution measurements that experimentally confirm a spiral cascade structure and a scaling relationship of shrimps in the Chua's circuit. Circuits constructed using this component allow for a comprehensive characterization of the circuit behaviors through high resolution parameter spaces. To illustrate the power of our technological development for the creation and the study of chaotic circuits, we constructed a Chua circuit and study its high resolution parameter space. The reliability and stability of the designed component allowed us to obtain data for long periods of time (∼21 weeks), a data set from which an accurate estimation of Lyapunov exponents for the circuit characterization was possible. Moreover, this data, rigorously characterized by the Lyapunov exponents, allows us to reassure experimentally that the shrimps, stable islands embedded in a domain of chaos in the parameter spaces, can be observed in the laboratory. Finally, we confirm that their sizes decay exponentially with the period of the attractor, a result expected to be found in maps of the quadratic family.
... In figure 1, we plot, for the case (I), the energy Q(z) = bifurcations followed by period-doubling bifurcations are leading to order (periodic explosions) and chaos (non-periodic explosions), respectively. Recently, Ott and co-workers [25] addressed the following important question: how certain is it that an attractor is chaotic? In figure 3, we plot the peak of the amplitude |ψ| peak versus the energy Q for two frequencies (figure 3a), four frequencies (figure 3b) and chaos (figure 3c), corresponding to the plots shown in figure 1. ...
We show the existence of periodic exploding dissipative solitons. These non-chaotic explosions appear when higher-order nonlinear and dispersive effects are added to the complex cubic quintic Ginzburg Landau equation modelling soliton transmission lines. This counterintuitive phenomenon is the result of period-halving bifurcations leading to order (periodic explosions), followed by perioddoubling bifurcations (or intermittency) leading to chaos (non-periodic explosions). © 2015 The Author(s) Published by the Royal Society. All rights reserved.
