Period adding bifurcation in a logistic map with memory (original) (raw)
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Periodicity and chaos in a modulated logistic map
International Journal of Theoretical Physics, 1990
We study the onset of chaos in a logistic map whose parameter is modulated nonlinearly. The bifurcation pattern with respect to a parameter/z is obtained and the critical value of ~ is seen to be 0.89, where periodicity just ends. Further evidence for this regime is obtained from the analysis of the intermittency pattern. The stability in the different ranges of p, under repeated iteration is exhibited by the values of Lyapunov exponents. Beyond /~ = 0.89, the largest Lyapunov exponent becomes positive and the situation turns out to be unstable. Confirmation comes from a functional analysis of the stable and unstable manifolds which touch at ~ = 0.89.
Abstract and Applied Analysis
The focus of this research work is to obtain the chaotic behaviour and bifurcation in the real dynamics of a newly proposed family of functions fλ,ax=x+1−λxlnax;x>0, depending on two parameters in one dimension, where assume that λ is a continuous positive real parameter and a is a discrete positive real parameter. This proposed family of functions is different from the existing families of functions in previous works which exhibits chaotic behaviour. Further, the dynamical properties of this family are analyzed theoretically and numerically as well as graphically. The real fixed points of functions fλ,ax are theoretically simulated, and the real periodic points are numerically computed. The stability of these fixed points and periodic points is discussed. By varying parameter values, the plots of bifurcation diagrams for the real dynamics of fλ,ax are shown. The existence of chaos in the dynamics of fλ,ax is explored by looking period-doubling in the bifurcation diagram, and cha...
Bifurcation and basin in two coupled parametrically forced logistic maps
2011 IEEE International Symposium of Circuits and Systems (ISCAS), 2011
Two coupled logistic maps whose parameters are forced into periodic varying are investigated. From the investigation of bifurcation in this system, nonexistence of odd periodic orbit except fixed point and existence of many coexisting attractors, which consist of periodic orbits or chaotic orbits, are observed. Basins where boundary depends on the invariant manifold of saddle points are numerically analyzed by considering second order iteration and using superposition with Newton method, although the system has discontinuity.
Deterministic chaos in one-dimensional maps—the period doubling and intermittency routes
Pramana-journal of Physics, 1992
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Route to Chaos in Generalized Logistic Map
Acta Physica Polonica A, 2015
Motivated by a possibility to optimize modelling of the population evolution we postulate a generalization of the well-know logistic map. Generalized difference equation reads: x n+1 = rx p n (1 − x q n), (1) x ∈ [0, 1], (p, q) > 0, n = 0, 1, 2, ..., where the two new parameters p and q may assume any positive values. The standard logistic map thus corresponds to the case p = q = 1. For such a generalized equation we illustrate the character of the transition from regularity to chaos as a function of r for the whole spectrum of p and q parameters. As an example we consider the case for p = 1 and q = 2 both in the periodic and chaotic regime. We focus on the character of the corresponding bifurcation sequence and on the quantitative nature of the resulting attractor as well as its universal attribute (Feigenbaum constant).
Power law periodicity in the tangent bifurcations of the logistic map
Physica A-statistical Mechanics and Its Applications, 2001
Numerical studies were carried out for the average of the logistic map on the tangent bifurcations from chaos into periodic windows. A critical exponent of 1 2 is found on the average amplitude as one approaches the transition. Additionally, the averages oscillate with a period that decreases with the same exponent. This Power Law Periodicity is related to the reinjection mechanism of the map. The undulations appear at control parameter values much earlier than the values where the critical exponent of the bifurcation shows signiÿcant changes in the average amplitude.
Chaotic sub-dynamics in coupled logistic maps
Physica D: Nonlinear Phenomena, 2016
We study the dynamics of Laplacian-type coupling induced by logistic family f µ (x) = µx(1 − x), where µ ∈ [0, 4], on a periodic lattice, that is the dynamics of maps of the form F (x, y) = ((1 − ε)f µ (x) + εf µ (y), (1 − ε)f µ (y) + εf µ (x)) where ε > 0 determines strength of coupling. Our main objective is to analyze the structure of attractors in such systems and especially detect invariant regions with nontrivial dynamics outside the diagonal. In analytical way, we detect some regions of parameters for which a horseshoe is present; and using simulations global attractors and invariant sets are depicted.
Bifurcation and Synchronization in Coupled Parametrically Forced Logistic Maps
In this study, a parametrically forced logistic map that a parameter of the logistic map are forced into periodic varying is suggested. Unique bifurcations from period to chaos are observed in the map. Then, synchronization phenomena in globally coupled system of the parametrically forced logistic map are investigated. When the number of coupling is three, various synchronization phenomena are observed by choosing a coupling intensity. The synchronization phenomena fall into three general categories, which are asynchronous, self-switching of synchronization, synchronization of two among the three maps and synchronization of all the maps. Further more, relationship between sojourn time and the coupling intensity in the selfswitching of synchronization is investigated. The sojourn time increase exponentially with the coupling intensity.
Journal of Mathematical Sciences, 2009
This paper introduces a new 2D piecewise smooth discrete-time chaotic mapping with rarely observed phenomenon – the occurrence of the same chaotic attractor via different and distinguishable routes to chaos: period doubling and border-collision bifurcations as typical futures. This phenomenon is justified by the location of system equilibria of the proposed mapping, and the possible bifurcation types in smooth dissipative systems.
Logarithmic Periodicities in the Bifurcations of Type-I Intermittent Chaos
Physical Review Letters, 2004
The critical relations for statistical properties on saddle-node bifurcations are shown to display undulating fine structure, in addition to their known smooth dependence on the control parameter. A piecewise linear map with the type-I intermittency is studied and a log-periodic dependence is numerically obtained for the average time between laminar events, the Lyapunov exponent and attractor moments. The origin of the oscillations is built in the natural probabilistic measure of the map and can be traced back to the existence of logarithmically distributed discrete values of the control parameter giving Markov partition. Reinjection and noise effect dependences are discussed and indications are given on how the oscillations are potentially applicable to complement predictions made with the usual critical exponents, taken from data in critical phenomena. PACS numbers: 05.45.Pq, 05.45.-a, 64.60.Fr