Feigenbaum Constant Research Articles

Dynamics of a discrete one‐predator two‐prey system with Michaelis–Menten‐type prey harvesting and prey refuge

In this paper, we propose and study a discretized one‐predator two‐prey system along with prey refuge and Michaelis–Menten‐type prey harvesting. The interaction among the species is considered as Holling type III functional response. Firstly, existence and local stability of all the fixed points are derived under certain parametric conditions. Furthermore, a special consideration is made to global asymptotic stability of the interior fixed point. Then, we have shown that the system undergoes different types of bifurcations including transcritical bifurcation, flip bifurcation, and Neimark–Sacker bifurcation by using center manifold theorem, bifurcation theory, and normal form method. Also, Feigenbaum's constant of the system is calculated. It is observed that both harvesting and refuge have a stabilizing effect on the system, and the stabilizing effect of harvesting dominates the stabilizing effect of refuge. Of most interest is the finding of coexisting attractors and multistability. In particular, optimal harvesting policy has been obtained by extension of Pontryagin's maximum principle to discrete system. Finally, some intriguing numerical simulations are provided to verify our analytic findings and rich dynamics of the three species system.

Nonlinear dynamical systems: Time reversibility versus sensitivity to the initial conditions

Time reversal of vast classes of phenomena has direct implications with predictability, causality and the second principle of thermodynamics. We propose a new method of investigating reversibility in time series data through the instance of a paradigmatic dissipative nonlinear dynamical system, namely the logistic map xt+1=1−axt2. The method splits the original data into two series, one symmetrical and the other antisymmetrical with respect to time. A close relation is revealed between time reversibility and the sensitivity to the initial conditions. Indeed, depending on the initial condition and the size of the time series, time reversal can enable the recovery, within a small error bar, of past information when the Lyapunov exponent is non-positive, notably at the Feigenbaum point (edge of chaos), where weak chaos is known to exist. Past information is gradually lost for increasingly large Lyapunov exponent (strong chaos), notably at a=2 where it attains a large value.

Quantification of Training in Human Thought using Chaos Theory: Degree of Decreasing Entropy as an Indicator

Many analytical methods used to quantify the technical training of human resources have been reported, but they are poor at assessing the training of human thought. Chaos theory and the degree of decreasing entropy have not yet been used in such analytics, although all phenomena—except for those in a completely fixed state (e.g., mathematics, historical facts, and true natural laws) or a random state (e.g., dialing a random digit)—are based on chaos theory. Chaos theory must therefore be considered in the quantification of the technical and intellectual training of human resources because human thinking is involved. This report compares a method for chaos theoretical quantification of human resource training to methods that do not involve chaos theory. The development of the ability to change to a fixed state from a chaotic state beyond the Feigenbaum point is the goal for the training in thinking; this process, which involves a decrease in entropy, is the most difficult, and important process in such training. In a chaotic state with an infinite number of solutions, humans can never show their own state using any method. In many reports, each goal in a fixed state is considered after thinking has been rearranged; however, a chaotic state in human thinking can never be expressed using indices such as productivity, efficiency, or job satisfaction. Indeed, many reported results are merely a part of human resource training, but the change of entropy in the fixed state is small (this corresponds to the “creativity” of artificial intelligence). By contrast, the change in entropy from a chaotic state to a fixed state is large, and this corresponds to human intuitive. Considered in terms of chaos theory, goal achievement for thinking must therefore be quantified in terms of the degree of decreasing entropy (i.e., the problem-solving speed, and degree of problem difficulty). This type of problem-solving speed is not about achieving ta goal but about creating a new idea. The concrete methods considered are the Schedule for the Evaluation of Individual Quality of Life-Direct Weighting method, the Kawakida Jiro method, and the Mandala chart. Based on the findings presented here, considering entropy change in the chaotic state should become an index for evaluating the creation of a new idea instead of the repetition of an existing idea, as this type of thinking is beyond the scope of artificial intelligence. Given that almost all natural phenomena, including human thinking, are based on chaos theory, using this theory can promote scientific development in all academic fields.

Period-doubling bifurcations and islets of stability in two-degree-of-freedom Hamiltonian systems.

In this paper, we show that the destruction of the main Kolmogorov-Arnold-Moser (KAM) islands in two-degree-of-freedom Hamiltonian systems occurs through a cascade of period-doubling bifurcations. We calculate the corresponding Feigenbaum constant and the accumulation point of the period-doubling sequence. By means of a systematic grid search on exit basin diagrams, we find the existence of numerous very small KAM islands ("islets") for values below and above the aforementioned accumulation point. We study the bifurcations involving the formation of islets and we classify them in three different types. Finally, we show that the same types of islets appear in generic two-degree-of-freedom Hamiltonian systems and in area-preserving maps.

Time evolution of nonadditive entropies: The logistic map

Due to the second principle of thermodynamics, the time dependence of entropy for all kinds of systems under all kinds of physical circumstances always thrives interest. The logistic map xt+1=1−axt2∈[−1,1](a∈[0,2]) is neither large, since it has only one degree of freedom, nor closed, since it is dissipative. It exhibits, nevertheless, a peculiar time evolution of its natural entropy, which is the additive Boltzmann–Gibbs-Shannon one, SBG=−∑i=1Wpilnpi, for all values of a for which the Lyapunov exponent is positive, and the nonadditive one Sq=1−∑i=1Wpiqq−1 with q=0.2445… at the edge of chaos, where the Lyapunov exponent vanishes, W being the number of windows of the phase space partition. We numerically show that, for increasing time, the phase-space-averaged entropy overshoots above its stationary-state value in all cases. However, when W→∞, the overshooting gradually disappears for the most chaotic case (a=2), whereas, in remarkable contrast, it appears to monotonically diverge at the Feigenbaum point (a=1.4011…). Consequently, the stationary-state entropy value is achieved from above, instead of from below, as it could have been a priori expected. These results raise the question whether the usual requirements – large, closed, and for generic initial conditions – for the second principle validity might be necessary but not sufficient.

Effective algorithms for solving functional equations with superposition on the example of the Feigenbaum equation

Purpose. New algorithms were consider for functional equations solving using the Feigenbaum equation as an example. This equation is of great interest in the theory of deterministic chaos and is a good illustrative example in the class of functional equations with superposition. Methods. The article proposes three new effective methods for solving functional equations — the method of successive approximations, the method of successive approximations using the fast Fourier transform and the numerical-analytical method using a small parameter. Results. Three new methods for solving functional equations were presented, considered on the example of the Feigenbaum equation. For each of them, the features of their application were investigated, as well as the complexity of the resulting algorithms was estimated. The methods previously used by researchers to solve functional equations are compared with those described in this article. In the description of the latter, the numerical-analytical method, several coefficients of expansions of the universal Feigenbaum constants were written out. Conclusion. The obtained algorithms, based on simple iteration methods, allow solving functional equations with superposition without the need to reverse the Jacobi matrix. This feature greatly simplifies the use of computer memory and gives a gain in the operating time of the algorithms in question, compared with previously used ones. Also, the latter, numerically-analytical method made it possible to obtain sequentially the coefficients of expansions of the universal Feigenbaum constants, which in fact can be an analytical representation of these constants.

Study of decay rate of materials using logistic map equation

The bifurcation diagram obtained till date is for population dynamics given by Robert May and analyzed and plotted for many relative graphs by Feigenbaum with emergence of Feigenbaum constant. Author here tried to apply the same logistic equation with negative growth rate, can be called as decay rate which can relate to decomposition processes in nature or materials. The universality of the bifurcation diagram and the Feigenbaum constant is also proved in this region. The decay rate hence obtained has a different values than that of existing growth rate values.

Fiat numero: trigintaduonion emanation theory and its relation to the fine-structure constant α, the Feigenbaum constant c∞, and π

Fiat numero: trigintaduonion emanation theory and its relation to the fine-structure constant α, the Feigenbaum constant c∞, and π

Numerical Analysis of Multiple Steady States, Limit Cycles, Period-Doubling, and Chaos in Enzymatic Reactions Involving Oxidation of L-tyrosine to Produce L-DOPA

We analyze the nonlinear dynamics of an isothermal system involving complex enzymatic reactions for L-DOPA (L-3,4-dihydroxyphenylalanine) production by numerical simulation. The mass action kinetics of the system forms a family of 9ordinary differential equations with 22 parameters. The multiple steady states are calculated by the chemical reaction network toolbox. Starting from one of the steady states, a limit point is guaranteed to be detected due to a change in the system parameters using the numerical continuation software MatCont. Other bifurcations are also obtained via the bifurcation continuations of the limit point, such as cusp bifurcations, Hopf bifurcations, limit cycles, zero Hopf bifurcations, generalized Hopf bifurcations, period-doubling, and so on. A transition of a period-doubling bifurcation to chaos occurs by numerical simulations. Positive values, 0.25~0.71, of Lyapunov exponents are obtained for the chaotic dynamics. Poincare maps and power spectrum densities are also plot. The Feigenbaum constant is computed to be 4.681~4.705.

Period multiplication in mode-locked figure-of-9 fiber lasers

Period multiplication is a bifurcation phenomenon, which depicts a typical roadmap from periodic steady-state to chaos. There has been particular interest in studying period multiplication properties in mode-locked lasers, but the mechanism behind is still not well-studied. Here, we experimentally demonstrate a figure-of-9 fiber laser which can generate period-multiplied pulses with long-term stability. Further theoretical insight is obtained by numerical simulations, which clearly shows that the ratio between the distances of each bifurcation point is approaching the Feigenbaum constant. We believe that the laser design presented here could be a stable and reliable ultrafast optical platform to study bifurcation and chaos, which also has potential applications in optical frequency comb and all-optical modulation.

Fractal Characteristics of Porosity of Electrospun Nanofiber Membranes

In this paper, the method of measuring the porosity of electrostatic nanofiber membrane by VC++ and Matlab is introduced. It is found that the ratio of the calculated porosity to the porosity measured by the mercury intrusion method accords with the famous Feigenbaum constant (α=2.5029078750957⋯). The porosity distribution of nanofiber membranes was studied by VC++ and Matlab based on the image obtained by using a scanning electron microscope. The porosity distribution calculated by using a computer is magnified by eα times which was named as relative porosity distribution. According to the relative porosity distribution, we use the algorithm proposed by Grassberger and Procaccia (briefly referred to as the G-P algorithm) to calculate the correlation fractal dimension. The correlation fractal dimension calculated from the relative porosity distribution series was between 1 and 2, consistent with geometric characteristics of coincidence samples. The fractal meaning of the Feigenbaum constant was verified again. In the end, we obtained the relationship between the associated fractal dimension and the filtration resistance by fitting in accordance with the secondary function relationship and reached the maximum correlation fractal dimension when the filtration resistance was 15–20 pa.

Transition to chaos through period doublings of a forced oscillating cylinder in steady current

Transition to chaos through a cascade of period doublings of the primary synchronization mode is discovered in steady approaching flow around a forced inline oscillating cylinder near a plane boundary at a Reynolds number of 175. The transition occurs well within the otherwise synchronized region (known as the Arnold tongue) in the frequency and amplitude space of the oscillating cylinder, creating two parameter strips of desynchronized flows within the Arnold tongue. Five orders of period doublings from mode to mode are revealed by progressively increasing the frequency resolution in the simulation. The ratio of frequency intervals of two successive period-doubling modes asymptotes towards the first Feigenbaum constant, reaching a value of 4.52 at mode of . Additional three-dimensional simulations demonstrate the existence of period doubling with a regular spanwise flow structure similar to regular mode B of steady flow around an isolated cylinder. Although transition to chaos through cascades of period doublings is primarily reported for the primary synchronization mode, it is also observed for other synchronization modes (Tang et al., J. Fluid Mech., vol. 832, 2017, pp. 146–169), where and are integers with a non-reducible , such as . The physical mechanisms responsible for the present period-doubling bifurcations and transition to chaos through cascades of period doublings are ascribed to the interaction of asymmetric vortex shedding from the cylinder (due to a geometric asymmetry) and the boundary layer developed on the plane boundary, through specifically designed numerical tests.

Analysis of population dynamics and chaos theory

Population growth is a topic of great interest to biologists, epidemiologists, ecologists, microbiologists and bioanalysts. Describing the dynamics of a population system through mathematical models is very useful in order to predict the behavior of the study population. Chaos theory supports studies of this type through the analysis of the logistic equation which allows observing this behavior under the variation of the constant k that represents the rate of increase in the number of times of the population values in a given time and the orbit diagram that summarizes the asymptotic behavior of all orbits in which we have values of k between zero and four. These models work with discrete time under measurement by iteration in observation and not continuously. The objective is to show the relationship of the logistic equation and the orbit diagram with the Feigenbaum constant in order to show the order that exists in the population dynamics.

INVESTIGATION OF DIFFERENCE EQUATIONS WITH A RATIONAL RIGHT-HAND SIDES

The aim of the present article is to investigate of some properties of solutions of nonli- near difference equations. A period doubling bifurcation in a discrete dynamical system leads to the appearance of deterministic chaos. We use permutable rational functions for study of some classes of one-dimensional mappings. Also n-dimensional generalizations of permutable polynomials may be obtained. We investigate polynomial and rational mappings with invariant measure and construct equivalent piecewise linear mappings. These mappings have countably many cycles. We applied the methods of symbolic dynamics to the theory of unimodal mappi- ngs. We use whole p-adic numbers for study the invariant set of some mapping in the theory of universal properties of one-parameter families. Feigenbaum constants play an important role in this theory.

Difference in the dynamics of logistic maps

We investigate the evolution of the difference in the conventional logistic map and observed complicated but interesting dynamics. Our results show that the bifurcation, the attractor and the asymptotic measure depend on initial values. The reverse bifurcation, the multiple bifurcation and other bifurcation properties obtained here are different from those of the original logistic map. Furthermore, we find that attractors form a structure which looks like a jigsaw puzzle, and the corresponding distributions related to the Feigenbaum constant are also analyzed.

Research on System Control Based on a Novel Theory

A system is combined by itself and the periphery, and the periphery controls the system. The Periphery Theory describes that a system input is dependent on a system gate of the periphery, and the gate is considered as a switch which will adjust the state of the system automatically. In this research, a new logistic model with switch is created to learn how the switch adjust the system. The switch is set to close/open the gate of the system to decrease/increase the input, and it is related to the final states of the system. By comparing this new model with the original one, the switch also improve the complexity of the system, which changes the distribution of final system states, making the positions of the transition and bifurcation points occur earlier. Moreover, a new Feigenbaum constant for the new logistic model is 4.68274650, and it is totally different from the original one. In addition, a quantitative relationship between the probability of system switch and the final system states shows the protection provided by the gate to the system.

Transition of Feigenbaum Point on a family of 1-dimensional Maps

Transition of Feigenbaum Point on a family of 1-dimensional Maps

Feigenbaum's constants in reverse bifurcation of fractional-order Rössler system

This paper demonstrates the existence of Feigenbaum's constants in reverse bifurcation for fractional-order Rössler system. First, the numerical algorithm of fractional-order Rössler system is presented. Then, the definition of Feigenbaum's constants in reverse bifurcation is provided. Third, in order to observe the effect of fractional-order to Feigenbaum's constants in reverse bifurcation, a series of bifurcation diagrams are computed. The Feigenbaum's constants in reverse bifurcation are measured and the error percentage in fractional-order Rössler system is presented. The simulation results show that Feigenbaum's constants exist in reverse bifurcation for fractional-order Rössler system. Especially, the Feigenbaum's constants still exist in the periodic windows. A summary on previous others’ works about Feigenbaum's constants is proposed. This paper draw a conclusion that the constants are universal in both period-doubling bifurcation and reverse bifurcation for both integer and fractional-order system.

Structure of the correlation function at the accumulation points of the logistic map

The correlation function of the trajectory exactly at the Feigenbaum point of the logistic map is investigated and checked by numerical experiments. Taking advantage of recent closed analytical results on the symbol-to-symbol correlation function of the generating partition, we are in position to justify the deep algorithmic structure of the correlation function apart from numerical constants. A generalization is given for arbitrary m · 2∞ Feigenbaum attractors.

Computation of the anharmonic orbits in two piecewise monotonic maps with a single discontinuity

In this paper, the bifurcation values for two typical piecewise monotonic maps with a single discontinuity are computed. The variation of the parameter of those maps leads to a sequence of border-collision and period-doubling bifurcations, generating a sequence of anharmonic orbits on the boundary of chaos. The border-collision and period-doubling bifurcation values are computed by the word-lifting technique and the Maple fsolve function or the Newton–Raphson method, respectively. The scaling factors which measure the convergent rates of the bifurcation values and the width of the stable periodic windows, respectively, are investigated. We found that these scaling factors depend on the parameters of the maps, implying that they are not universal. Moreover, if one side of the maps is linear, our numerical results suggest that those quantities converge increasingly. In particular, for the linear-quadratic case, they converge to one of the Feigenbaum constants \(\delta _F= 4.66920160\cdots \).