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# Logistic map

About: Logistic map is a research topic. Over the lifetime, 2297 publications have been published within this topic receiving 38345 citations.

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TL;DR: This book discusses Chaos, Fractals, and Dynamics, and the Importance of Being Nonlinear in a Dynamical View of the World, which aims to clarify the role of Chaos in the world the authors live in.

Abstract: Preface 1. Overview 1.0 Chaos, Fractals, and Dynamics 1.1 Capsule History of Dynamics 1.2 The Importance of Being Nonlinear 1.3 A Dynamical View of the World PART I. ONE-DIMENSIONAL FLOWS 2. Flows on the Line 2.0 Introduction 2.1 A Geometric Way of Thinking 2.2 Fixed Points and Stability 2.3 Population Growth 2.4 Linear Stability Analysis 2.5 Existence and Uniqueness 2.6 Impossibility of Oscillations 2.7 Potentials 2.8 Solving Equations on the Computer Exercises 3. Bifurcations 3.0 Introduction 3.1 Saddle-Node Bifurcation 3.2 Transcritical Bifurcation 3.3 Laser Threshold 3.4 Pitchfork Bifurcation 3.5 Overdamped Bead on a Rotating Hoop 3.6 Imperfect Bifurcations and Catastrophes 3.7 Insect Outbreak Exercises 4. Flows on the Circle 4.0 Introduction 4.1 Examples and Definitions 4.2 Uniform Oscillator 4.3 Nonuniform Oscillator 4.4 Overdamped Pendulum 4.5 Fireflies 4.6 Superconducting Josephson Junctions Exercises PART II. TWO-DIMENSIONAL FLOWS 5. Linear Systems 5.0 Introduction 5.1 Definitions and Examples 5.2 Classification of Linear Systems 5.3 Love Affairs Exercises 6. Phase Plane 6.0 Introduction 6.1 Phase Portraits 6.2 Existence, Uniqueness, and Topological Consequences 6.3 Fixed Points and Linearization 6.4 Rabbits versus Sheep 6.5 Conservative Systems 6.6 Reversible Systems 6.7 Pendulum 6.8 Index Theory Exercises 7. Limit Cycles 7.0 Introduction 7.1 Examples 7.2 Ruling Out Closed Orbits 7.3 Poincare-Bendixson Theorem 7.4 Lienard Systems 7.5 Relaxation Oscillators 7.6 Weakly Nonlinear Oscillators Exercises 8. Bifurcations Revisited 8.0 Introduction 8.1 Saddle-Node, Transcritical, and Pitchfork Bifurcations 8.2 Hopf Bifurcations 8.3 Oscillating Chemical Reactions 8.4 Global Bifurcations of Cycles 8.5 Hysteresis in the Driven Pendulum and Josephson Junction 8.6 Coupled Oscillators and Quasiperiodicity 8.7 Poincare Maps Exercises PART III. CHAOS 9. Lorenz Equations 9.0 Introduction 9.1 A Chaotic Waterwheel 9.2 Simple Properties of the Lorenz Equations 9.3 Chaos on a Strange Attractor 9.4 Lorenz Map 9.5 Exploring Parameter Space 9.6 Using Chaos to Send Secret Messages Exercises 10. One-Dimensional Maps 10.0 Introduction 10.1 Fixed Points and Cobwebs 10.2 Logistic Map: Numerics 10.3 Logistic Map: Analysis 10.4 Periodic Windows 10.5 Liapunov Exponent 10.6 Universality and Experiments 10.7 Renormalization Exercises 11. Fractals 11.0 Introduction 11.1 Countable and Uncountable Sets 11.2 Cantor Set 11.3 Dimension of Self-Similar Fractals 11.4 Box Dimension 11.5 Pointwise and Correlation Dimensions Exercises 12. Strange Attractors 12.0 Introductions 12.1 The Simplest Examples 12.2 Henon Map 12.3 Rossler System 12.4 Chemical Chaos and Attractor Reconstruction 12.5 Forced Double-Well Oscillator Exercises Answers to Selected Exercises References Author Index Subject Index

2,949 citations

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01 Jan 1984

TL;DR: In this article, the authors present a model for the detection of deterministic chaos in the Lorenz model, which is based on the idea of the Bernoulli shift and the Kicked Quantum Rotator.

Abstract: Preface.Color Plates.1 Introduction.2 Experiments and Simple Models.2.1 Experimental Detection of Deterministic Chaos.2.2 The Periodically Kicked Rotator.3 Piecewise Linear Maps and Deterministic Chaos.3.1 The Bernoulli Shift.3.2 Characterization of Chaotic Motion.3.3 Deterministic Diffusion.4 Universal Behavior of Quadratic Maps.4.1 Parameter Dependence of the Iterates.4.2 Pitchfork Bifurcation and the Doubling Transformation.4.3 Self-Similarity, Universal Power Spectrum, and the Influence of External Noise.4.4 Behavior of the Logistic Map for r ≤ r.4.5 Parallels between Period Doubling and Phase Transitions.4.6 Experimental Support for the Bifurcation Route.5 The Intermittency Route to Chaos.5.1 Mechanisms for Intermittency.5.2 Renormalization-Group Treatment of Intermittency.5.3 Intermittency and 1/f-Noise.5.4 Experimental Observation of the Intermittency Route.6 Strange Attractors in Dissipative Dynamical Systems.6.1 Introduction and Definition of Strange Attractors.6.2 The Kolmogorov Entropy.6.3 Characterization of the Attractor by a Measured Signal.6.4 Pictures of Strange Attractors and Fractal Boundaries.7 The Transition from Quasiperiodicity to Chaos.7.1 Strange Attractors and the Onset of Turbulence.7.2 Universal Properties of the Transition from Quasiperiodicity to Chaos.7.3 Experiments and Circle Maps.7.4 Routes to Chaos.8 Regular and Irregular Motion in Conservative Systems.8.1 Coexistence of Regular and Irregular Motion.8.2 Strongly Irregular Motion and Ergodicity.9 Chaos in Quantum Systems?9.1 The Quantum Cat Map.9.2 A Quantum Particle in a Stadium.9.3 The Kicked Quantum Rotator.10 Controlling Chaos.10.1 Stabilization of Unstable Orbits.10.2 The OGY Method.10.3 Time-Delayed Feedback Control.10.4 Parametric Resonance from Unstable Periodic Orbits.11 Synchronization of Chaotic Systems.11.1 Identical Systems with Symmetric Coupling.11.2 Master-Slave Configurations.11.3 Generalized Synchronization.11.4 Phase Synchronization of Chaotic Systems.12 Spatiotemporal Chaos.12.1 Models for Space-Time Chaos.12.2 Characterization of Space-Time Chaos.12.3 Nonlinear Nonequilibrium Space-Time Dynamics.Outlook.Appendix.A Derivation of the Lorenz Model.B Stability Analysis and the Onset of Convection and Turbulence in the Lorenz Model.C The Schwarzian Derivative.D Renormalization of the One-Dimensional Ising Model.E Decimation and Path Integrals for External Noise.F Shannon's Measure of Information.F.1 Information Capacity of a Store.F.2 Information Gain.G Period Doubling for the Conservative H-enon Map.H Unstable Periodic Orbits.Remarks and References.Index.

1,693 citations

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TL;DR: Applying measures of complexity based on vertical structures in recurrence plots and applying them to the logistic map as well as to heart-rate-variability data is able to detect and quantify the laminar phases before a life-threatening cardiac arrhythmia occurs thereby facilitating a prediction of such an event.

Abstract: The knowledge of transitions between regular, laminar or chaotic behaviors is essential to understand the underlying mechanisms behind complex systems. While several linear approaches are often insufficient to describe such processes, there are several nonlinear methods that, however, require rather long time observations. To overcome these difficulties, we propose measures of complexity based on vertical structures in recurrence plots and apply them to the logistic map as well as to heart-rate-variability data. For the logistic map these measures enable us not only to detect transitions between chaotic and periodic states, but also to identify laminar states, i.e., chaos-chaos transitions. The traditional recurrence quantification analysis fails to detect the latter transitions. Applying our measures to the heart-rate-variability data, we are able to detect and quantify the laminar phases before a life-threatening cardiac arrhythmia occurs thereby facilitating a prediction of such an event. Our findings could be of importance for the therapy of malignant cardiac arrhythmias.

890 citations

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TL;DR: This work proposes an image cryptosystem employing the Arnold cat map for bit-level permutation and the logistic map for diffusion, demonstrating the superior security and high efficiency of this algorithm.

Abstract: In recent years, a variety of chaos-based digital image encryption algorithms have been suggested. Most of these algorithms implement permutations and diffusions at the pixel level by considering the pixel as the smallest (atomic) element of an image. In fact, a permutation at the bit level not only changes the position of the pixel but also alters its value. Here we propose an image cryptosystem employing the Arnold cat map for bit-level permutation and the logistic map for diffusion. Simulations have been carried out and analyzed in detail, demonstrating the superior security and high efficiency of our cryptosystem.

596 citations

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TL;DR: In the proposed image encryption, this spatiotemporal chaotic system has more outstanding cryptography features in dynamics than the logistic map or the system of coupled map lattices does, and the strategy of bit-level pixel permutation is employed.

Abstract: We propose a new image encryption algorithm based on the spatiotemporal chaos of the mixed linear–nonlinear coupled map lattices. This spatiotemporal chaotic system has more outstanding cryptography features in dynamics than the logistic map or the system of coupled map lattices does. In the proposed image encryption, we employ the strategy of bit-level pixel permutation which enables the lower bit planes and higher bit planes of pixels permute mutually without any extra storage space. Simulations have been carried out and the results demonstrate the superior security and high efficiency of the proposed algorithm.

517 citations