Chaotic Map with No Fixed Points: Entropy, Implementation and Control
TL;DR: A map without equilibrium has been proposed and studied and shown its chaotic behavior using tools such as return map, bifurcation diagram and Lyapunov exponents’ diagram.
Abstract: A map without equilibrium has been proposed and studied in this paper. The proposed map has no fixed point and exhibits chaos. We have investigated its dynamics and shown its chaotic behavior using tools such as return map, bifurcation diagram and Lyapunov exponents’ diagram. Entropy of this new map has been calculated. Using an open micro-controller platform, the map is implemented, and experimental observation is presented. In addition, two control schemes have been proposed to stabilize and synchronize the chaotic map.
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TL;DR: In this paper, the authors studied the dynamics of a new fractional-order map with no fixed points and introduced a control scheme to stabilize the states of the fractional map and ensure their convergence to zero asymptotically.
Abstract: This paper studies the dynamics of a new fractional-order map with no fixed points. Through phase plots, bifurcation diagrams, largest Lyapunov exponent, it is shown that the proposed fractional map exhibit chaotic and periodic behavior. New Hidden chaotic attractors are observed, and transient state is found to exist. Complexity of the new map is also analyzed by employing approximate entropy. Results, show that the fractional map without fixed point have high complexity for certain fractional order. In addition, a control scheme is introduced. The controllers stabilize the states of the fractional map and ensure their convergence to zero asymptotically. Numerical results are used to verify the findings.
37 citations
TL;DR: In this paper, a new fractional order discrete-time system with no equilibrium is proposed, and the complex dynamical behaviors of such a system are discussed numerically by means of a bifurcation diagram, the largest Lyapunov exponents, a phase portrait, and a 0-1 test.
Abstract: Chaotic systems with no equilibrium are a very important topic in nonlinear dynamics. In this paper, a new fractional order discrete-time system with no equilibrium is proposed, and the complex dynamical behaviors of such a system are discussed numerically by means of a bifurcation diagram, the largest Lyapunov exponents, a phase portrait, and a 0–1 test. In addition, a one-dimensional controller is proposed. The asymptotic convergence of the proposed controller is established by means of the stability theory of linear fractional order discrete-time systems. Next, a synchronization control scheme for two different fractional order discrete-time systems with hidden attractors is reported, where the master system is a two-dimensional system that has been reported in the literature. Numerical results are presented to confirm the results.
34 citations
TL;DR: It is shown that nonlinear integration techniques significantly change the behavior of discrete models of nonlinear systems, increasing the values of Lyapunov exponents and spectral entropy.
Abstract: In this paper, we consider nonlinear integration techniques, based on direct Pade approximation of the differential equation solution, and their application to conservative chaotic initial value problems. The properties of discrete maps obtained by nonlinear integration are studied, including phase space volume dynamics, bifurcation diagrams, spectral entropy, and the Lyapunov spectrum. We also plot 2D dynamical maps to enlighten the features introduced by nonlinear integration techniques. The comparative study of classical integration methods and Pade approximation methods is given. It is shown that nonlinear integration techniques significantly change the behavior of discrete models of nonlinear systems, increasing the values of Lyapunov exponents and spectral entropy. This property reduces the applicability of numerical methods based on Pade approximation to the chaotic system simulation but it is still useful for construction of pseudo-random number generators that are resistive to chaos degradation or discrete maps with highly nonlinear properties.
23 citations
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References
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TL;DR: This is an interpretive review of first-order difference equations, which can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations.
Abstract: First-order difference equations arise in many contexts in the biological, economic and social sciences. Such equations, even though simple and deterministic, can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations. There are consequently many fascinating problems, some concerned with delicate mathematical aspects of the fine structure of the trajectories, and some concerned with the practical implications and applications. This is an interpretive review of them.
6,118 citations
"Chaotic Map with No Fixed Points: E..." refers background or methods in this paper
...It has previously been observed that simple first-order nonlinear maps can generate complex dynamical behavior including chaos [8]....
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...Chaotic maps such as Hénon map [9], Logistic map [8], Lozi map [10], and zigzag map [11] are found....
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TL;DR: Analysis of a recently developed family of formulas and statistics, approximate entropy (ApEn), suggests that ApEn can classify complex systems, given at least 1000 data values in diverse settings that include both deterministic chaotic and stochastic processes.
Abstract: Techniques to determine changing system complexity from data are evaluated. Convergence of a frequently used correlation dimension algorithm to a finite value does not necessarily imply an underlying deterministic model or chaos. Analysis of a recently developed family of formulas and statistics, approximate entropy (ApEn), suggests that ApEn can classify complex systems, given at least 1000 data values in diverse settings that include both deterministic chaotic and stochastic processes. The capability to discern changing complexity from such a relatively small amount of data holds promise for applications of ApEn in a variety of contexts.
5,055 citations
"Chaotic Map with No Fixed Points: E..." refers background in this paper
...It is noted that there is no reported threshold to be achieve in the ApEn in order to exhibit chaos [28,29]....
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...In particular, approximate entropy (ApEn) [28,29] is useful to study chaotic systems [19,30]....
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TL;DR: A review of the main mathematical ideas and their concrete implementation in analyzing experiments can be found in this paper, where the main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions).
Abstract: Physical and numerical experiments show that deterministic noise, or chaos, is ubiquitous. While a good understanding of the onset of chaos has been achieved, using as a mathematical tool the geometric theory of differentiable dynamical systems, moderately excited chaotic systems require new tools, which are provided by the ergodic theory of dynamical systems. This theory has reached a stage where fruitful contact and exchange with physical experiments has become widespread. The present review is an account of the main mathematical ideas and their concrete implementation in analyzing experiments. The main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions). The relations between these quantities, as well as their experimental determination, are discussed. The systematic investigation of these quantities provides us for the first time with a reasonable understanding of dynamical systems, excited well beyond the quasiperiodic regimes. This is another step towards understanding highly turbulent fluids.
4,619 citations
TL;DR: Work in different scientific fields is now suggesting the existence of generic early-warning signals that may indicate for a wide class of systems if a critical threshold is approaching.
Abstract: Complex dynamical systems, ranging from ecosystems to financial markets and the climate, can have tipping points at which a sudden shift to a contrasting dynamical regime may occur. Although predicting such critical points before they are reached is extremely difficult, work in different scientific fields is now suggesting the existence of generic early-warning signals that may indicate for a wide class of systems if a critical threshold is approaching.
3,450 citations
TL;DR: In this article, the same properties can be observed in a simple mapping of the plane defined by: \({x i + 1}} = {y_i} + 1 - ax_i^2,{y i+ 1} = b{x_i}\).
Abstract: Lorenz (1963) has investigated a system of three first-order differential equations, whose solutions tend toward a “strange attractor”. We show that the same properties can be observed in a simple mapping of the plane defined by: \({x_{i + 1}} = {y_i} + 1 - ax_i^2,{y_{i + 1}} = b{x_i}\). Numerical experiments are carried out for a =1.4, b = 0.3. Depending on the initial point (x 0,y 0), the sequence of points obtained by iteration of the mapping either diverges to infinity or tends to a strange attractor, which appears to be the product of a onedimensional manifold.by a Cantor set.
2,507 citations
"Chaotic Map with No Fixed Points: E..." refers methods in this paper
...Chaotic maps such as Hénon map [9], Logistic map [8], Lozi map [10], and zigzag map [11] are found....
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