On the bifurcation in a “modulated” logistic map

We comment on a paper by Luang [On the bifurcation in a ''modulated'' logistic map, Physics Letters A 194(1994) 57]. The numerical evidence given in that paper, for a peculiar type of bifurcation, is shown to be incorrect. The causes of such anomalous results are explained. An accurate bifurcation diagram for the map is also given.

Logistic Map耦合格子模型模拟环带

本论文采用Logistic Map耦合格子模型对高聚物中特有的环带球晶进行了模拟，所得到的模拟结果与实验结果吻合较好。同时，研究结果能够对实验制备环带球晶样品提供可靠的理论指导。 首先，我们对Logistic Map耦合格子模型及模型中的两个模拟参量μ和ε进行分析，同时结合实验中各种实验条件对聚合物结晶行为的影响，认为Logistic Map的动力学特征与聚合物结晶行为非常相似，并且参量μ与实验中的结晶温度相关，即随温度的升高而减小，而参量ε与实验中影响扩散的因素有关，即随温度的升高而增大、随分子量的增大而减小，并且随样品厚度的增大而增大。我们对模型的整个参数空间进行计算，得到了可以形成环带球晶形貌的参数范围，通过进一步研究发现环带图案的带宽随参量μ的增大而变窄，随参量ε的增大而变宽。上述研究结果与实验中带宽随实验条件的变化规律是一致的。 在得到环带图案的基础上，我们又进一步计算得到了靶状和螺旋状形貌的参量μ和ε的具体取值范围。通过改变μ和ε的参数取值，模拟了环带球晶形貌由靶状过渡到螺旋状的过程，即靶状图案的环带由外层向内层逐渐断裂成较短的条带结构，所有的条带结构呈现出以空间某处为中心团聚在一起的形貌；随后，这种“团聚”的形貌逐渐消失了，空间中小的条带结构的排列呈无序状态。随着参数的进一步变化，短的条带结构变成较长的带状结构，并且这些带状结构的边缘逐渐发生卷曲，最终形成了螺旋状图案。我们还考察了系统初值和耦合方式对上述图案的影响，结果发现，形成环带球晶的参数范围对系统初值没有明显的依赖性，然而靶状和螺旋状图案的分布受初值的影响较大。此外，发现只有采用交替耦合、并考虑长程耦合作用的Logistic Map耦合格子模型才可以得到环带球晶图案。 为了更好地与实验结果进行对比，我们利用Logistic Map耦合格子模型对二维空间中的几种受限体系进行了模拟。（一）对温度梯度场中的环带球晶进行模拟，发现环带球晶在低温处较易成核，向高温处生长，并且，高温处环带的带宽比低温处宽。（二）对格子宽度受限情况进行了模拟，发现随着受限方向的宽度越来越窄，球晶尺寸逐渐变小，相邻两个环带球晶碰撞产生的界线变短，三个相邻环带球晶所形成的界线的交汇点减少。（三）研究了受限边界上的成核作用对狭长格子中环带球晶的影响，结果发现，随着受限边界上成核点密度的不断增加，其形貌转变分为三个不同阶段：①当成核密度稍有增大时，环带球晶数量增加，直径变小；②继续增大边界成核密度，使得大量晶层从受限边界向格子内生长，导致环带球晶的数量减少，直径也减小；③当成核点增加到一定程度时，整个空间中只有极少数由格子内部成核生长且直径非常小的环带球晶，而占主导地位的是由成核点垂直于受限边界生长出的穿透晶层。这些模拟结果均与实验结果相符合。 我们将Logistic Map耦合映象格子模型发展到三维空间格子中，得到了与环带球晶形貌一致的图案，并且其带宽随模拟参量μ的增大而变窄，随ε的增大而变宽。这一规律性结果与二维正方格子的模拟结果是一致的。这一部分的研究结果还表明，边界条件和格子尺寸对模拟结果有显著的影响，周期性边界条件导致在小体积立方格子中只能得到靶状图案；而当格子尺寸很大时，可以得到螺旋状环带球晶的图案。最后，通过调节垂直于薄膜平面方向上的格子数来研究薄膜厚度对环带图案带宽的影响，发现环带的带宽随厚度的增加而变宽，这与实验中环带球晶的带宽随样品厚度的增加而变大的结论是一致的。

The study of banded spherulite patterns by coupled logistic map lattice model

Banded spherulite patterns are simulated in two dimensions by means of a coupled logistic map lattice model. Both target pattern and spiral pattern which have been proved to be existent experimentally in banded spherulite are obtained by choosing suitable parameters in the model. The simulation results also indicate that the band spacing is decreased with the increase of parameter mu in the logistic map and increased with the increase of the coupling parameter epsilon, which is quite similar to the results in some experiments. Moreover, the relationship between the parameters and the corresponding patterns is obtained, and the target patterns and spiral patterns are distinguished for a given group of initial values, which may guide the study of banded spherulite.

Simulation of Banded Spherulite Patterns by Coupled Logistic Map Lattice Model in Three Dimensions

Banded spherulite patterns are simulated in three dimensions by means of a Coupled Logistic map lattice model. The patterns obtained by numerical calculation are consistent with those in experiments. The simulation results also indicate that the hand spacing is decreased with the increase of parameter mu in the Logistic map and increased with the increase of the coupling parameter e for cube lattices, and increased with the increase of the thickness of the lattice for polymer film, which is quite similar to the results in some experiments. Spiral pattern in three dimensions is also shown in this paper, which helps us understand the form of banded spherulite in polymers.

Investigations on the Dynamics of Combination Maps and the Analysis of Bifurcation in a Discontinuous Logistic Map

This thesis is a study of discrete nonlinear systems represented by one dimensional mappings.As one dimensional interative maps represent Poincarre sections of higher dimensional flows,they offer a convenient means to understand the dynamical evolution of many physical systems.It highlighting the basic ideas of deterministic chaos.Qualitative and quantitative measures for the detection and characterization of chaos in nonlinear systems are discussed.Some simple mathematical models exhibiting chaos are presented.The bifurcation scenario and the possible routes to chaos are explained.It present the results of the numerical computational of the Lyapunov exponents (λ) of one dimensional maps.This thesis focuses on the results obtained by our investigations on combinations maps,scaling behaviour of the Lyapunov characteristic exponents of one dimensional maps and the nature of bifurcations in a discontinous logistic map.It gives a review of the major routes to chaos in dissipative systems,namely, Period-doubling ,Intermittency and Crises.This study gives a theoretical understanding of the route to chaos in discontinous systems.A detailed analysis of the dynamics of a discontinous logistic map is carried out, both analytically and numerically ,to understand the route it follows to chaos.The present analysis deals only with the case of the discontinuity parameter applied to the right half of the interval of mapping.A detailed analysis for the n –furcations of various periodicities can be made and a more general theory for the map with discontinuities applied at different positions can be on a similar footing

Existence of multiple attractors and the nature of bifurcations in a discontinuous logistic map

We present the analytical investigations on a logistic map with a discontinuity at the centre. An explanation for the bifurcation phenomenon in discontinuous systems is presented. We establish that whenever the elements of an n-cycle (n > 1) approach the discontinuities of the nth iterate of the map, a bifurcation other than the usual period-doubling one takes place. The periods of the cycles decrease in an arithmetic progression, as the control parameter is varied. The system also shows the presence of multiple attractors. Our results are verified by numerical experiments as well.

Dynamics of the logistic map under discrete parametric perturbation

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.

Bifurcation structure and Lyapunov exponents of a modulated logistic map

We have studied the bifurcation structure of the logistic map with a time dependant control parameter. By introducing a specific nonlinear variation for the parameter, we show that the bifurcation structure is modified qualitatively as well as quantitatively from the first bifurcation onwards. We have also computed the two Lyapunov exponents of the system and find that the modulated logistic map is less chaotic compared to the logistic map.

An analogue of the logistic map in two dimensions

This is a sequel to our earlier work on the modulated logistic map. Here, we first show that the map comes under the universality class of Feigenbaum. We then give evidence for the fact that our model can generate strange attractors in the unit square for an uncountable number of parameter values in the range μ∞<μ<1. Numerical plots of the attractor for several values of μ are given and the self-similar structure is explicity shown in one case. The fractal and information dimensions of the attractors for many values of μ are shown to be greater than one and the variation in their structure is analysed using the two Lyapunov exponents of the system. Our results suggest that the map can be considered as an analogue of the logistic map in two dimensions and may be useful in describing certain higher dimensional chaotic phenomena.

Universal behaviour in a ``modulated'' logistic map

We study the period-doubling bifurcations to chaos in a logistic map with a nonlinearly modulated parameter and show that the bifurcation structure is modified significantly. Using the renormalisation method due to Derrida et al. we establish the universal behaviour of the system at the onset of chaos.

Noise-induced attractor annihilation in the delayed feedback logistic map

We study dynamics of the bistable logistic map with delayed feedback, under the influence of white Gaussian noise and periodic modulation applied to the variable. This system may serve as a model to describe population dynamics under finite resources in noisy environment with seasonal fluctuations. While a very small amount of noise has no effect on the global structure of the coexisting attractors in phase space, an intermediate noise totally eliminates one of the attractors. Slow periodic modulation enhances the attractor annihilation.

Critical lattice size limit for synchronized chaotic state in one- and two-dimensional diffusively coupled map lattices

We consider diffusively coupled map lattices with P neighbors (where P is arbitrary) and study the stability of the synchronized state. We show that there exists a critical lattice size beyond which the synchronized state is unstable. This generalizes earlier results for nearest neighbor coupling. We confirm the analytical results by performing numerical simulations on coupled map lattices with logistic map at each node. The above analysis is also extended to two-dimensional P-neighbor diffusively coupled map lattices.

Chaos in modulated logistic systems

This paper is a review of the work done on the dynamics of modulated logistic systems. Three different problems are treated, viz, the modulated logistic map, the parametrically perturbed logistic map and the combination map obtained by combining two maps of the quadratic family. Many of the interesting features displayed by these systems are discussed.

Evidence for the existence of Sarkovskii ordering in a two-dimensional map

We analyse numerically the bifurcation structure of a two-dimensional noninvertible map and show that different periodic cycles are arranged in it exactly in the same order as in the case of the logistic map. We also show that this map satisfies the general criteria for the existence of Sarkovskii ordering, which supports our numerical result. Incidently, this is the first report of the existence of Sarkovskii ordering in a two-dimensional map.