Period adding bifurcation in a logistic map with memory

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.

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.

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

IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 1995

A method of controlling spatiotemporal chaos in a chain of the logistic maps, based on the Ott, Grebogi, Yorke (OGY) approach, is proposed in the paper. Forms of spatiotemporal disturbances of a system parameter are determined analytically to stabilize desired spatiotemporal states. The results are illustrated by numerical experiments. Transitions of the system under control from the regime of spatiotemporal chaos to the regular homogeneous ones with temporal period one, two, four and also to more complicated regular regimes with different spatial and temporal periods have been demonstrated.

In this paper a two dimensional non linear map is taken whose period doubling dynamical behavior has been analyzed. The bifurcation points have been calculated numerically and have been observed that the map follows a universal behavior that has been proposed by Feigenbaum. With the help of experimental bifurcation points the accumulation point where chaos starts has been calculated.

Open Systems & Information Dynamics (OSID), 2001

A hierarchy of universalities in families of 1-D maps is discussed. Breakdown of universalities in families of 3-D maps is shown on selected examples of such families.

International Journal of Bifurcation and Chaos, 2002

Cycling behavior involving steady-states and periodic solutions is known to be a generic feature of continuous dynamical systems with symmetry. Using Chua's circuit equations and Lorenz equations, Dellnitz et al. [1995] showed that "cycling chaos", in which solution trajectories cycle around symmetrically related chaotic sets, can also be found generically in coupled cell systems of differential equations with symmetry. In this work, we use numerical simulations to demonstrate that cycling chaos also occurs in discrete dynamical systems modeled by one-dimensional maps. Using the cubic map f (x, λ) = λx - x3 and the standard logistic map, we show that coupled iterated maps can exhibit cycles connecting fixed points with fixed points and periodic orbits with periodic orbits, where the period can be arbitrarily high. As in the case of coupled cell systems of differential equations, we show that cycling behavior can also be a feature of the global dynamics of coupled itera...

Pramana-journal of Physics, 1996

By introducing a periodic perturbation in the control parameter of the logistic map we have investigated the period locking properties of the map. The map then gets locked onto the periodicity of the perturbation for a wide range of values of the parameter and hence can lead to a control of the chaotic regime. This parametrically perturbed map exhibits many other interesting features like the presence of bubble structures, repeated reappearance of periodic cycles beyond the chaotic regime. dependence of the escape parameter on the seed value and also on the initial phase of the perturbation etc. . Keywords_ Logistic map; periodic perturbation; bubble structures; fractal basin of escape.