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An Introduction to Chaos: Physics and Mathematics of Chaotic Phenomena
... Chaos measured by Lyapunov exponents, (also called Lyapunov characteristic components or LCEs); LCE > 0 indicates existence of chaos and LCE < 0 indicates regularity, [52][53][54][55][56][57][58][59][60][61][62]. A complex system can better understood by measuring (i) chaos, (ii) Topological entropies and (iii) correlation dimension. ...
... More topological entropy in any system signifies more complexity in it. Actually, it measures the evolution of distinguishable orbits over time, thereby providing an idea of how complex the orbit structure of a system is, [48][49][50][61][62][63][64][65][66][67][68][69]. A system may be chaotic with zero topological entropy. ...
... In addition, a significant increase in topological entropy does not justify that it is chaotic. The book by Nagashima and Baba,[62], gives a very clear definition of topological entropy. The correlation dimension provides the dimensionality of the chaotic attractor. ...
Chaotic phenomena and presence of complexity in various nonlinear dynamical systems extensively discussed in the context of recent researches. Discrete as well as continuous dynamical systems both considered here. Visualization of regularity and chaotic motion presented through bifurcation diagrams by varying a parameter of the system while keeping other parameters constant. In the processes, some perfect indicator of regularity and chaos discussed with appropriate examples. Measure of chaos in terms of Lyapunov exponents and that of complexity as increase in topological entropies discussed. The methodology to calculate these explained in details with exciting examples. Regular and chaotic attractors emerging during the study are drawn and analyzed. Correlation dimension, which provides the dimensionality of a chaotic attractor discussed in detail and calculated for different systems. Results obtained presented through graphics and in tabular form. Two techniques of chaos control, pulsive feedback control and asymptotic stability analysis, discussed and applied to control chaotic motion for certain cases. Finally, a brief discussion held for the concluded investigation.
... Positivity of LCEs and topological entropy are characteristic of chaos in the system. A mathematical definition of topological entropy can be found in the book by Nagashima and Baba, (2005). A compact and complex pattern of diagram appears in bifurcation diagram for a system evolving chaotically. ...
... Correlation dimension provides such dimension of the attractor and here, we have calculated after drawing the correlation curves for each system. For this, first, we have collected data for the correlation curves plotted for each model and then used the method of least square linear fit described by Martelli (1999) and Nagashima and Baba (2005) to obtain correlation dimensions. Actually, the correlation dimension gives the measure of complexity whenever a system evolves chaotically. ...
The state of chaos exhibited in nonlinear system may appear through various roots. However, for mostly one dimensional system, it appears through period doubling roots. Phenomena of bifurcations, drawn by varying certain parameter of the system, explain clearly regular as well as chaotic evolution of the system. As a measure of chaos, the most suitable tools to be considered are: Lyapunov characteristic exponents (LCE) and topological entropies. Though the plots of both of the LCE's and topological entropies are similar, but both have certain limitations. LCE would not work for systems having relativistic considerations but the topological entropy can work. In fact, topological entropy can nicely provide the measure of complexity of the system in the sense that the more complexity in the system means more topological entropy it will have. A chaotic system exhibits a chaotic set, strange attractor, having fractal property. Calculation of correlation dimension is required to obtain the dimension of such attractor. The work presented here explains the appearance of chaos through bifurcation in some nonlinear one dimensional discrete system. The stability of the steady state, (i.e. fixed points), have been examined for each system and then plots of Lyapunov exponents and topological entropies for their evolution have been 4198 L. M. Saha, Sadanand Prasad and R. K. Mohanty obtained. Then, the calculations have been extended to find the correlation dimensions chaotic sets, chaotic attractors, seen for different systems. Graphical results reveal some interesting information.
... It is featured by highly unstable motion of deterministic systems in a bounded region of the phase space. High instability means that the distance of two nearby orbits increases exponentially with time [11], which is a result of the extreme sensitivity of chaotic systems to the initial conditions. The Lyapunov exponents quantify this property. ...
... Chaos is then characterized by the boundedness of the system trajectories with a positive Lyapunov exponent, which implies that the average gradient of the map is greater than unity, and accordingly two nearby orbits in phase space diverge at an exponential rate. It was emphasized that the sensitivity to the initial value suggests the irregularity of the series fx i g generated by chaos [11]. Consider the ith number x i of the series. ...
As an emerging effective approach to nonlinear robust control, simplex sliding mode control demonstrates some attractive features not possessed by the conventional sliding mode control method, from both theoretical and practical points of view. However, no systematic approach is currently available for computing the simplex control vectors in nonlinear sliding mode control. In this paper, chaos-based optimization is exploited so as to develop a systematic approach to seeking the simplex control vectors; particularly, the flexibility of simplex control is enhanced by making the simplex control vectors dependent on the Euclidean norm of the sliding vector rather than being constant, which result in both reduction of the chattering and speedup of the convergence. Computer simulation on a nonlinear uncertain system is given to illustrate the effectiveness of the proposed control method.
... Chaos theory, as a mathematical discipline aims to study the dynamic behavior of systems that are highly sensitive to the initial conditions and parameter values (Grassi 2021). Chaos can be found basically in almost all fields from natural and social sciences, to engineering, and medicine, even economics (Nagashima et al. 2019). As a result chaos theory has evolved to a large attraction for researchers, and the past decades is continuously being studied, due to a number of appealing characteristics such as randomness, and unpredictability, nonlinearity, and initial condition sensitivity, which over the years led to many interesting and varying applications. ...
Many drawbacks in chaos-based applications emerge from the chaotic maps' poor dynamic properties. To address this problem, in this paper a chaotification model based on modulo operator and secant functions to augment the dynamic properties of existing chaotic maps is proposed. It is demonstrated that by selecting appropriate parameters, the resulting map can achieve a higher Lyapunov exponent than its seed map. This chaotification method is applied to several well-known maps from the literature, and it produces increased chaotic behavior in all cases, as evidenced by their bifurcation and Lyapunov exponent diagrams. Furthermore, to illustrate that the proposed chaotification model can be considered in chaos-based encryption and related applications, a voice signal encryption process is considered, and different tests are being used with respect to attacks, like brute force, entropy, correlation, and histogram analysis.
... are well-known and typical examples of one-dimensional chaotic maps; such simple maps can show very complicated and interesting behaviors (see e.g. [1,4,6]). One of important problems is to classify periodic and nonperiodic orbits of those maps by the initial values. ...
We study periodic orbits starting from finite decimals under p-adic transformations and stylized Chebyshev polynomials. Among others, the structure of periodic orbits for 5-adic transformation is established.
... However, as shown in Fig. 2(b), the tangent bifurcation (μ l = 1) arises at d ≈ −11.9 and the chaotic state d ≲ −11.9 (λ 1 > 0, λ 2 = 0) appears. This chaotic state produced by tangent bifurcation indicates the alternating laminar and turbulent modes of intermittency chaos in a general way 46 ; this dynamic was demonstrated in our previous work 34 . That is, the intermittency route to chaos exists in this region. ...
Chaotic resonance (CR), in which a system responds to a weak signal through the effects of chaotic activities, is a known function of chaos in neural systems. The current belief suggests that chaotic states are induced by different routes to chaos in spiking neural systems. However, few studies have compared the efficiency of signal responses in CR across the different chaotic states in spiking neural systems. We focused herein on the Izhikevich neuron model, comparing the characteristics of CR in the chaotic states arising through the period-doubling or tangent bifurcation routes. We found that the signal response in CR had a unimodal maximum with respect to the stability of chaotic orbits in the tested chaotic states. Furthermore, the efficiency of signal responses at the edge of chaos became especially high as a result of synchronization between the input signal and the periodic component in chaotic spiking activity.
... where λ m is defined byA snap-back repeller is a kind of unstable fixed point, and its existence guarantees that the system (C1) is chaotic in the sense of Li–Yorke. Then, it is well-known that there are an infinite number of orbits which are repelled by the snap-back repeller but which are attracted to its neighborhood[12]. Here, note that the selection of ω i j ...
... Through this tangent bifurcation, at d = −12 ((b)), the orbit of u i exhibited sluggish movement (called laminar state) in the region where the slope of ψ 2 was approximately 1.0 (−102 ≲ u i ≲ −94), and irregularly active movement (called turbulent or burst state) in regions with larger slopes () 1). Such chaotic, dynamic alternation between laminar and turbulent states is called intermittency chaos [34,35]. Note that the term of burst is not used in this paper to avoid confusion between the chaotic movement and the neural spike patterns in neurodynamics such as intrinsically bursting and chattering bursting. ...
In stochastic resonance (SR), the presence of noise helps a nonlinear system amplify a weak (sub-threshold) signal. Chaotic resonance (CR) is a phenomenon similar to SR but without stochastic noise, which has been observed in neural systems. However, no study to date has investigated and compared the characteristics and performance of the signal responses of a spiking neural system in some chaotic states in CR. In this paper, we focus on the Izhikevich neuron model, which can reproduce major spike patterns that have been experimentally observed. We examine and classify the chaotic characteristics of this model by using Lyapunov exponents with a saltation matrix and Poincaré section methods in order to address the measurement challenge posed by the state-dependent jump in the resetting process. We found the existence of two distinctive states, a chaotic state involving primarily turbulent movement and an intermittent chaotic state. In order to assess the signal responses of CR in these classified states, we introduced an extended Izhikevich neuron model by considering weak periodic signals, and defined the cycle histogram of neuron spikes as well as the corresponding mutual correlation and information. Through computer simulations, we confirmed that both chaotic states in CR can sensitively respond to weak signals. Moreover, we found that the intermittent chaotic state exhibited a prompter response than the chaotic state with primarily turbulent movement.
... where λ m is defined byA snap-back repeller is a kind of unstable fixed point, and its existence guarantees that the system (C1) is chaotic in the sense of Li–Yorke. Then, it is well-known that there are an infinite number of orbits which are repelled by the snap-back repeller but which are attracted to its neighborhood[12]. Here, note that the selection of ω i j ...
In this paper, we consider the particle swarm optimization (PSO). In particular, we focus on an improved PSO called the CPSO-VQO, which uses a perturbation-based chaotic system and a threshold-based method of selecting from the standard and chaotic updating systems for each particle on the basis of the difference vector between its pbest and the gbest. Although it was reported that the CPSO-VQO performs well, it is not easy to select an amplitude of the perturbation and a threshold appropriately for an effective search. This is because the bifurcation structure of the chaotic system depends on the difference vector, and the difference vector varies widely between different stages of the search and between different problems.
Therefore, we improve the CPSO-VQO by proposing a modified chaotic system whose bifurcation structure is irrelevant to the difference vector, and show theoretically desirable properties of the modified system. We also propose a new stochastic method that selects the updating system according to the ratio between the components of the difference vector for each particle, and restarting and acceleration techniques to develop the standard updating system used in the proposed PSO model. The proposed methods can maintain an appropriate balance between the identification and diversification aspects of the search. Moreover, we perform numerical experiments to evaluate the performance of the proposed PSOs: PSO-TPC, PSO-SPC, PSO-SDPC, IPSO-SPC and IPSO-SDPC. In particular, we demonstrate that the IPSO-SDPC finds high-quality solutions and is robust against variations in its parameter values.
... However, at d = −12 ((b)), the orbit of u i exhibits sluggish movement (laminar mode) in the region where the slope of ϕ 2 around 1.0 (−102 u i −94) and irregularly active movement (turbulent mode) in the other region having larger slope (≫ 1). This chaotic dynamics alternating between the laminar and turbulent modes is called intermittency chaos [23,24]. As the value of d decreases, the region producing the laminar mode, where the ϕ 2 slope indicates around 1.0, reduces and then the turbulent mode is dominant in the dynamics as shown in Figs.9 ...
Several hybrid neuron models, which combine continuous spike-generation mechanisms and discontinuous resetting process after spiking, have been proposed as a simple transition scheme for membrane potential between spike and hyperpolarization. As one of the hybrid spiking neuron models, Izhikevich neuron model can reproduce major spike patterns observed in the cerebral cortex only by tuning a few parameters and also exhibit chaotic states in specific conditions. However, there are a few studies concerning the chaotic states over a large range of parameters due to the difficulty of dealing with the state dependent jump on the resetting process in this model. In this study, we examine the dependence of the system behavior on the resetting parameters by using Lyapunov exponent with saltation matrix and Poincaré section methods, and classify the routes to chaos.
... An expedient is to draw M samples (x(t n( j) ), j = 1, 2, . . . , M) out of N and evaluate the following [36]: ...
This paper describes similarities between the narrow-band response of a resonant system excited by random inputs and low-dimensional chaos, with particular emphasis on the geometric characteristics of trajectories reconstructed in m-dimensional phase space from measured scalar time series data with a time-delay coordinate system. In this study, the time series data of ship roll angle in irregular waves were analyzed as an example of the narrow-band response of a resonant system. These time series data were measured by one of the authors in Tokyo Bay. The similarities between the narrow-band response of a resonant system excited by random inputs and low-dimensional chaos are verified by numerical simulation data.
... In this paper, we apply the forward Euler scheme to discrete the model and investigate the dynamical behaviors in detail as a discrete dynamical system in R 2 by using bifurcation theory of continuous and discrete systems in [3,[7][8][9][10][11][12]15,[17][18][19]21,22] (the similar method applied to the discrete BVP oscillator can refer to [14]). The stability of the fixed points and bifurcations including fold bifurcation and flip bifurcation are found. ...
A discrete genetic toggle switch system obtained by Euler method is first investigated. The conditions of existence for fold bifurcation and flip bifurcation are derived by using center manifold theorem and bifurcation theory. The numerical simulations, including bifurcation diagrams, phase portraits, and computation of Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behavior. We show the period 3 to 13 windows in different chaotic regions, period-doubling bifurcation or inverse period-doubling bifurcation from period-2 to 12 orbits leading to chaos, different kind of interior crisis and boundary crisis, intermittency behavior, chaotic set, chaotic non-attracting set, coexistence of period points with invariant cycles, and so on. The influence of the amplitude and frequency of excitable forcing on the system are also first considered by using numerical simulation. A different type of quasiperiodic orbits, jumping behaviors of quasiperiodic set from one set to another set, and the processes from quasiperiodic orbits to strange non-chaotic attractor are found.
... In physics, positivity of Lyapunov exponents is often taken as a criterion of chaos. For more on these definitions, see for instance [NB99]. ...
... In a chaotic dynamical system, miniscule differences in initial conditions yield widely diverging outcomes, thereby generally rendering long-term predictions impossible [8,9]. Chaos is abundant in nature [9,10,11], and such complex, non-linear systems are widely studied in mathematics [12,13,14], physics [15,16,17], gravitation and cosmology [18,19], astrophysics [20,21], chemistry [22,23,24], biology [25,26,27,28,29,30,31,32], neuro-science [33,34,35], medicine [36,37], computation [38,39,40], cryptology [41,42,43], economics [44,45], and warfare [46,47,48]. Chaos theory has significantly enhanced our general scientific comprehension of a wide range of phenomena, from structural dynamics to turbulence [8,9,46,49] which applies to, for example, aquatic ecosystems [50], weather [51], black holes [15], the cosmic microwave background [19], galaxy distributions [20,21], population biology [25], viruses and pathogens [26], cancers and genetics [27,52], military strategy [46,47,48], volcanoes [53], earthquakes [54,55], and the global stock market [44]. ...
We propose a preliminary algorithm which is designed to reduce aspects of the n-body problem to a 2-body problem for holographic principle compliance. The objective is to share an alternative view-point on the n-body problem to try and generate a simpler solution in the future. The algorithm operates 2D and 3D data structures to initiate the encoding of the chaotic dynamical system equipped with modified superfluid order parameter fields in both 3D and 4D versions of the Inopin holographic ring (IHR) topology. For the algorithm, we arbitrarily select one point-mass to be the origin and, from that reference frame, we subsequently engage a series of instructions to consolidate the residual (n-1)-bodies to the IHR. Through a step-by-step example, we demonstrate that the algorithm yields "IHR effective" (IHRE) net quantities that enable us to hypothetically define an IHRE potential, kinetic, and Lagrangian.
... In general, while many nonlinear systems are not chaotic, the reverse is not true. That is, all chaotic systems are nonlinear (Nagashima and Baba, 1999). Thus, one can hypothesize that non-Gaussian behavior can represent the underlying distribution of chaotic time series. ...
... which is mentioned in [19]. (12) implies that the width of the window is in inverse proportion to jaðaÞj. ...
Recently, global optimization methods using chaotic dynamics have been investigated. In those methods, it is significant what kind of chaotic dynamical system is selected. However, the system used in most existing methods for generating a chaotic sequence is sometimes not suitable for solving the problem because the system often has some windows and a generated sequence tends to overconcentrate around the boundary of the feasible region. In this paper, in order to improve them, we propose a new dynamical system which generates a chaotic sequence by the steepest descent method for minimizing an objective function with additional sinusoidal perturbation terms. In addition, we theoretically show the sufficient condition under which an approximated dynamical system of the proposed model at any local minimum or the global minimum is chaotic. Through numerical experiments we analyze properties of the proposed model for optimization to overcome these drawbacks. Furthermore, we compare the proposed method with the existing method through computational experiments by applying them to some global optimization problems.
A predator–prey system with consideration of certain specific environmental modulations applied to the natural death of predator parameter was investigated here. Visualization of regular and chaotic motion was presented through bifurcation diagrams by varying a parameter of the system while keeping other parameters constant for different cases of environmental modulation. Complex bifurcation pattern was observed with the application of stated modulation. Within chaotic region, in addition to periodic windows, phenomena of period doubling followed by second phase of chaos were observed. Taking views of some recent articles, a proper analysis on these bifurcations was also presented. Regular and chaotic attractors were obtained by proper numerical simulations. Lyapunov exponents and topological entropies, respectively, were calculated as measures of chaos and complexity. Correlation dimensions were computed to offer a dimensional perspective on chaotic attractors, and the obtained results were visually depicted through graphics. A concise discussion was incorporated into the conclusion section.
Izhikevich neuron model, which combines continuous spike-generation mechanisms and discontinuous resetting process after spiking, can reproduce almost all spiking activities including chaotic spiking in actual neural systems. When the chaotic state is evaluated in this model, it is known that conventional Lyapunov exponent where the continuous trajectory is presupposed cannot be applied due to the state dependent jump in the resetting process. To evaluate Lyapunov exponent in the system with the resetting process, the accurate numerical calculation for the trajectory by Newton method and the consideration for saltation matrix are needed. By virtue of this method, several routes to chaos have been found in Izhikevich neuron model. While on the other hand, in this study,we have proposed the method combining Euler method and Lyapunov exponent on Poincaré section and evaluated the chaotic state in Izhikevich neuron model. As the result, it has been confirmed that this method can also judge the chaotic state by tuning the initial perturbation against the trajectory.
Evolutionary motions in a bouncing ball system consisting of a ball having a
free fall in the Earth's gravitational field have been studied systematically.
Because of nonlinear form of the equations of motion, evolutions show chaos for
certain set of parameters for certain initial conditions. Bifurcation diagram
has been drawn to study regular and chaotic behavior. Numerical calculations
have been performed to calculate Lyapunov exponents, topological entropies and
correlation dimension as measures of complexity. Numerical results are shown
through interesting graphics
Regular and chaotic oscillations in a modified discrete two dimensional coupled predator-prey model, proposed recently, is re-investigated by observing the bifurcation diagram, calculating Lyapunov exponents and correlation dimensions for various orbits. The map evolves from one cycle to three cycles followed by period doubling and then to a chaotic regime and show bistability as its parameter λ varies, 0 < ≤ 1.211. Initial population size of the species and the value their coupling coefficients play crucial role for subsequent evolutionary phenomena of the system. Regular and chaotic evolutions depend completely on the coefficient ,λ, and on the initial prey and predator population. Lyapunov exponents and correlation dimensions provide true measure of chaos and also, to identify the chaotic orbits. Some recently discovered indicators, FLI, SALI and DLI have also been used to understand the nature of orbits.Mathematical analysis have been carried out to find stable and unstable fixed points and to obtain numerical value of the parameter λ for sensitive state of the system to initial conditions i.e. emergence of chaos. Bistability condition have been discussed for variation of λ values. Graphical representation of Lyapunov exponents and correlation dimensions provide better understanding of the nature of orbits which are further justified by the use of FLI, SALI and DLI.
This chapter gives an overview of chaos, including the definition of chaos, the development of chaology, and the research of chaos. In particular, the research and development of chaos in the field of electrical engineering - with an emphasis on electric drive systems are discussed in detail. The development of chaology started in mathematics and physics, and expanded into chemistry, biology, engineering, and social sciences. In electrical engineering, research on chaos has covered a very wide range of areas, including, but not limited to, electronic circuits, telecommunications, power electronics, power systems, and electric drive systems. The investigation of chaos in electronic circuits can be grouped as one dimensional map circuits, higher-dimensional map circuits, continuous-time autonomous circuits, and continuous-time non-autonomous circuits. The chaos-based telecommunications can be grouped into three main types: chaotic masking, chaotic modulation, and chaotic switching. chaos generators; electronic circuits; power electronics; power systems; telecommunications
This thesis is devoted to time series analysis, in particular, to segmentation of time series and to discrimination of time series segments. The problem of measuring time series complexity is also addressed. As theoretical models for time series we consider discrete-time dynamical systems and stochastic processes. To solve the above-mentioned problems we use ordinal pattern analysis, a novel approach based on considering order relations between values of time series instead of the values themselves. The central objects of ordinal pattern analysis are ordinal patterns of order d that describe order
relations between (d + 1) successive points of a time series.
The main results of this thesis are the following.
• We establish a new ordinal-patterns-based complexity measure, the conditional entropy of ordinal patterns, and investigate the properties of this quantity.
• We suggest a new method for segmentation of time series on the basis of the conditional entropy of ordinal patterns.
• We develop a method for discrimination of time series segments based on clustering and successfully apply this method to EEG time series.
Conceptual analysis is to pursue a mathematical formulation of things that is maximally consistent with our intuition, but it is not an easy task. This chapter utilizes ‘chaos’ to illustrate conceptual analysis. It turns out that this requires conceptual analyses of other fundamental concepts such as ‘measure,’ ‘probability,’ ‘information,’ ‘computation,’ ‘randomness,’ etc. After elementary expositions of these concepts from the conceptual analysis point of view, a close relation between chaos and algorithmic randomness (Brudno’s theorem) is explained to exhibit the goodness of our conceptual analytical result for ‘chaos.’
In this brief note, we introduce the new, emerging sub-discipline of iso-fractals by highlighting and discussing the preliminary results of recent works. First, we note the abundance of fractal, chaotic, non-linear, and self-similar structures in nature while emphasizing the importance of studying such systems because fractal geometry is the language of chaos. Second, we outline the iso-fractal generalization of the Mandelbrot set to exemplify the newly generated
Mandelbrot iso-sets. Third, we present the cutting-edge notion of dynamic iso-spaces and explain how a mathematical space can be iso-topically lifted with iso-unit functions that (continuously or discretely) change; in the discrete case examples, we mention that iteratively generated
sequences like Fibonacci’s numbers and (the complex moduli of) Mandelbrot’s numbers can supply a deterministic chain of iso-units to construct an ordered series of (magnified and/or de-magnified) iso-spaces that are locally iso-morphic. Fourth, we consider the initiation of iso-fractals with Inopin’s holographic ring (IHR) topology and fractional statistics for 2D and 3D iso-spaces. In total, the reviewed iso-fractal results are a significant improvement over traditional fractals because the application of Santilli’s iso-mathematics arms us an extra degree of freedom for attacking problems in chaos. Finally, we conclude by proposing some questions and ideas for future research work.
In this paper, we discuss the particle swarm optimization method (PSO). In particular, we focus on the CPSO-VQO, a PSO with a perturbation-based chaotic system derived from the steepest descent method for a virtual quartic function having global minima at the lbest and gbest, which selects the updating system of the particle's position from the standard system of the original PSO and the chaotic one on the basis of a threshold distance between two bests. Although the good performance of CPSO-VQO is reported, it is not so easy to select appropriate parameter values of its chaotic system for each problem because the bifurcation structure of the chaotic system depends on the distance of two bests, and, moreover, it is required an appropriate threshold for selecting the updating system. Therefore, we improve the CPSO-VQO by proposing a modified chaotic system having the bifurcation structure irrelevant to the distance of two bests, and a new stochastic selection of the updating system. In addition, we theoretically show the desirable properties of the modified chaotic system and evaluate the improved CPSO-VQOs called CPSO-TSV and CPSO-SSV.
The particle swarm optimization method (PSO) is a population-based optimization technique. Since, in the PSO, the exploration ability is important to find a desirable solution, various kinds of methods have been investigated to improve it. In this paper, we propose a PSO with a new chaotic system derived from the steepest descent method for a virtual quartic objective function with perturbations having global minima at the personal and global bests obtained by particles so far, where elements of each particle's position are updated by the proposed chaotic system or the standard update formula. Thus, the proposed PSO can search for solutions around the personal and global bests intensively without being trapped at any local minimum due to the chaoticness. Moreover, we show approximately the sufficient condition of parameter values of the proposed system under which the system is chaotic. Through computational experiments, we verify the performance of the proposed PSO by applying it to some global optimization problems.
The particle swarm optimization method (PSO) is one of population-based optimization techniques for global optimization, where a number of candidate solutions called particles simultaneously move toward the tentative solutions found by particles so far, which are called the personal and global bests, respectively. Since, in the PSO, the exploration ability is important to find a desirable solution, various kinds of methods have been investigated to improve it. In this paper, we propose a PSO with a new chaotic system derived from the steepest descent method for a virtual quartic objective function with perturbations having its global minima at the personal and global bests, where elements of each particle’s position are updated by the proposed chaotic system or the standard update formula. Thus, the proposed PSO can search for solutions around the personal and global bests intensively without being trapped at any local minimum due to the chaoticness. Moreover, we show approximately the sufficient condition of parameter values of the proposed system under which the system is chaotic. Through computational experiments, we verify the performance of the proposed PSO by applying it to some global optimization problems.
Ever since the pioneering work of Hodgkin and Huxley, biological neuron models have consisted of ODEs representing the evolution of the transmembrane voltage and the dynamics of ionic conductances. It is only recently that discrete dynamical systems – also known as maps – have begun to receive attention as valid phenomenological neuron models. The present review tries to provide a coherent perspective of map-based biological neuron models, describing their dynamical properties; stressing the similarities and differences, both among them and in relation to continuous-time models; exploring their behavior in networks; and examining their wide-ranging possibilities of application in computational neuroscience.
In this paper, we investigate the dynamical behaviors of neural excitable system without periodic external current (proposed by Chialvo [Generic excitable dynamics on a two-dimensional map. Chaos, Solitons & Fractals 1995;5(3–4):461–79] and with periodic external current as system’s parameters vary. The existence and stability of three fixed points, bifurcation of fixed points, the conditions of existences of fold bifurcation, flip bifurcation and Hopf bifurcation are derived by using bifurcation theory and center manifold theorem. The chaotic existence in the sense of Marotto’s definition of chaos is proved. We then give the numerical simulated results (using bifurcation diagrams, computations of Maximum Lyapunov exponent and phase portraits), which not only show the consistence with the analytic results but also display new and interesting dynamical behaviors, including the complete period-doubling and inverse period-doubling bifurcation, symmetry period-doubling bifurcations of period-3 orbit, simultaneous occurrence of two different routes (invariant cycle and period-doubling bifurcations) to chaos for a given bifurcation parameter, sudden disappearance of chaos at one critical point, a great abundance of period windows (period 2 to 10, 12, 19, 20 orbits, and so on) in transient chaotic regions with interior crises, strange chaotic attractors and strange non-chaotic attractor. In particular, the parameter k plays a important role in the system, which can leave the chaotic behavior or the quasi-periodic behavior to period-1 orbit as k varies, and it can be considered as an control strategy of chaos by adjusting the parameter k. Combining the existing results in [Generic excitable dynamics on a two-dimensional map. Chaos, Solitons & Fractals 1995;5(3–4):461–79] with the new results reported in this paper, a more complete description of the system is now obtained.
Science, engineering, medicine, biology, and many other areas deal with signals acquired in the form of time series from different dynamical systems for the purpose of analysis, diagnosis, and control of the systems. The signals are often mixed with noise. Separating the noise from the signal may be very difficult if both the signal and the noise are broadband. The problem becomes inherently difficult when the signal is chaotic because its power spectrum is indistinguishable from broadband noise. This paper describes how to measure and analyze chaos using Lyapunov metrics. The principle of characterizing strange attractors by the divergence and folding of trajectories is studied. A practical approach to evaluating the largest local and global Lyapunov exponents by rescaling and renormalization leads to calculating the m Lyapunov exponents for m-dimensional strange attractors either modeled explicitly (analytically) or reconstructed from experimental time-series data. Several practical algorithms for calculating Lyapunov exponents are summarized. Extensions of the Lyapunov exponent approach to studying chaos are also described briefly as they are capable of dealing with the multiscale nature of chaotic signals. The extensions include the Lyapunov fractal dimension, the Kolmogorov--Sinai and Re´nyi entropies, as well as the Re´nyi fractal dimension spectrum and the Mandelbrot fractal singularity spectrum.
Recent technological advancements have enabled us to collect large volumes of geophysical noisy measurements that need to be combined with the model forecasts, which capture all of the known properties of the underlying system. This problem is best formulated in a stochastic optimization framework, which when solved recursively is known as Filtering. Due to the large dimensions of geophysical models, optimal filtering algorithms cannot be implemented within the constraints of available computation resources. As a result, most applications use suboptimal reduced rank algorithms. Successful implementation of reduced rank filters depends on the dynamical properties of the underlying system. Here, the focus is on geophysical systems with chaotic behavior defined as extreme sensitivity of the dynamics to perturbations in the state or parameters of the system. In particular, uncertainties in a chaotic system experience growth and instability along a particular set of directions in the state space that are continually subject to large and abrupt state-dependent changes. Therefore, any successful reduced rank filter has to continually identify the important direction of uncertainty in order to properly estimate the true state of the system. In this thesis, we introduce two efficient reduced rank filtering algorithms for chaotic system, scalable to large geophysical applications. Firstly, a geometric approach is taken to identify the growing directions of uncertainty, which translate to the leading singular vectors of the state transition matrix over the forecast period, so long as the linear approximation of the dynamics is valid.
The authors propose a correlation dimension as a new feature
vector of a speaker verification system among the persons whose voices
are much the same as each other. Since the speech production mechanism
is nonlinear, the ambiguity of speech utterance can be regarded as a
chaotic property. For speakers whose voices appear to be nearly
identical, using their vowel sound /i/ which is more chaotic than /a/,
/e/, /o/, or /u/ (in Korean), the authors carry out speaker verification
experiments and analyze the efficiency of speaker verification with CHMM
(continuous hidden Markov model), formant and correlation
dimension
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