Recent Trends in Chaotic, Nonlinear and Complex Dynamics

Description

In recent years, enormous progress has been made on nonlinear dynamics particularly on chaos and complex phenomena. This unique volume presents the advances made in theory, analysis, numerical simulation and experimental realization, promising novel practical applications on various topics of current interest on chaos and related fields of nonlinear dynamics.Particularly, the focus is on the following topics: synchronization vs. chaotic phenomena, chaos and its control in engineering dynamical systems, fractal-based dynamics, uncertainty and unpredictability measures vs. chaos, Hamiltonian systems and systems with time delay, local/global stability, bifurcations and their control, applications of machine learning to chaos, nonlinear vibrations of lumped mass mechanical/mechatronic systems (rigid body and coupled oscillator dynamics) governed by ODEs and continuous structural members (beams, plates, shells) vibrations governed by PDEs, patterns formation, chaos in micro- and nano-mechanical systems, chaotic reduced-order models, energy absorption/harvesting from chaotic, chaos vs. resonance phenomena, chaos exhibited by discontinuous systems, chaos in lab experiments.The present volume forms an invaluable source on recent trends in chaotic and complex dynamics for any researcher and newcomers to the field of nonlinear dynamics. Contents Preface 1. New Trends and Recent Advances — An Introduction 1.1. Introduction Part I. Fundamental Results in Nonlinear Dynamics Part II. Stochastic Dynamics, Fractal Structure Analysis and Numerical Errors Part III. Recent Trends in Applications of Chaotic and Complex Systems 1.2. Fundamental Results in Nonlinear Dynamics 1.3. Stochastic Dynamics, Fractal Structure Analysis and Numerical Errors 1.4. Recent Trends in Applications of Chaotic and Complex Systems 1.5. Conclusions References Part I. Fundamental Results in Nonlinear Dynamics 2. Enhanced Vibrational Resonance by an Amplitude-Modulated Force 2.1. Introduction 2.2. The Duffing Oscillator 2.3. Single-Well Duffing Oscillator with One High-Frequency Force 2.3.1. Theoretical description of VR 2.3.2. Analysis of VR 2.4. Single-Well System with a Two High-Frequency Force 2.4.1. Equations for slow and fast variables 2.4.2. Enhanced Q(ω) by the amplitude-modulated force 2.4.3. Nonsmooth variation of Q(ω) 2.4.4. Hysteresis and jump phenomenon 2.5. Double-Well Duffing Oscillator 2.5.1. System with the force F2(t) = f cos ωt + g cos(Ω + ω)t 2.5.2. Vibrational resonance with two high-frequency forces 2.6. Conclusion Acknowledgments References 3. Generation of Self-excited and Hidden Multiscroll Attractors in Multistable Systems 3.1. Multiscroll Attractor 3.2. Hidden Attractors 3.3. Emergence of a Hidden Double-Scroll Attractor 3.4. Generalization of Hidden Multiscroll Attractors 3.5. Extension for 2D and 3D Grid Scroll Hidden Attractors Acknowledgment References 4. Dynamics of the Dipole-Segment with Equal Masses and Arbitrary Rotation 4.1. Introduction 4.2. Potential of the Dipole-Segment Problem 4.3. Equilibria for Two Equal Masses m1 = m2(μ = 1/2) 4.4. Families of Symmetric Periodic Orbits for Values of κ 4.4.1. The grid-search method 4.5. Families Evolution in Function of the Rotation Parameter κ 4.5.1. Case k = 1 4.5.2. Families evolution in Zone Zi 4.5.3. Families evolution in Zone Zd 4.6. Conclusions Acknowledgments References 5. The Interaction of Two Vortices Near a Boundary in Rotating Stratified Incompressible Flows 5.1. Introduction 5.2. A Brief Review of Quasi-2D Flow and Vortex Models 5.2.1. Two-dimensional incompressible flows and vortex models 5.2.1.1. Two-dimensional Euler equations 5.2.1.2. Two-dimensional vortices 5.2.2. Quasi-two-dimensional fluid motion in stratified rotating fluids 5.2.2.1. The quasi-geostrophic model for a rapidly rotating, continuously stratified incompressible fluid 5.2.2.2. The two-layer quasi-geostrophic model for the ocean 5.2.2.3. Point vortices in the two-layer quasi-geostrophic model 5.2.2.4. Numerical model for point vortices in the two-layer quasi-geostrophic dynamics 5.3. Interaction of Two Vortices Along a Wall/Coast in the Two-Layer Model 5.3.1. Reference simulation 5.3.2. Sensitivity study: Influence of the physical parameters 5.3.2.1. Varying the layer thicknesses and vortex intensities 5.3.2.2. Inclusion of the planetary and topographic beta effects 5.3.2.3. Influence of the coastline orientation in both layers 5.3.2.4. Influence of a neighboring cyclone 5.3.2.5. Influence of an oscillating, sheared, mean flow 5.4. Discussion, Conclusion 5.4.1. Discussion 5.4.2. Conclusions Appendix: Mathematical Solution of the Point-Vortex Motion in the Two-Layer Model Acknowledgments References 6. Modal Asynchronicity in Pre-stressed Continuous Simply-Supported Beams with Transversal and Rotational Oscillators 6.1. Introduction 6.2. Mathematical Model 6.2.1. Beam 6.2.2. Oscillators 6.2.3. Modal analysis 6.3. Results 6.3.1. Parametric analysis 6.3.2. Sensitivity analysis 6.3.3. Considerations on a symmetric case 6.4. Concluding Remarks References 7. Weak Decay of Autocorrelations in Periodic Barrier Billiards 7.1. Introduction 7.2. Dynamical Properties of the Collision Map of Periodic Barrier Billiard Model 7.2.1. The dynamics as a statistical process 7.2.2. Continuous spectral measure of function f 7.3. Concluding Remarks Acknowledgments References Part II. Stochastic Dynamics, Fractal Structure Analysis and Numerical Errors 8. Stochastic Sensitivity Analysis of Noise-Induced Phenomena in Discrete Systems 8.1. Introduction 8.2. Stochastic Sensitivity 8.2.1. Stochastic sensitivity of equilibria 8.2.2. Stochastic sensitivity of k-cycles 8.2.3. Stochastic sensitivity of closed invariant curves 8.2.4. Stochastic sensitivity of chaotic attractors 8.3. Confidence Domains 8.3.1. Confidence ellipsoid around the equilibrium 8.3.2. Confidence domains for k-cycle 8.3.3. Confidence domains for closed invariant curves 8.4. Examples Acknowledgment References 9. The Role of Noise in Chaotic Intermittency 9.1. Introduction 9.2. Noise Effect: Classical Approach 9.2.1. Fokker–Planck approach 9.2.1.1. Type I intermittency 9.2.1.2. Type II and type III intermittencies 9.3. Renormalization Group and Scaling Theory 9.4. Extended Intermittency Theory to a More General Reinjection 9.5. Effect of Noise on the RPD 9.5.1. NRPD in type II intermittency 9.5.2. NRPD in type III intermittency 9.6. Characteristic Relations 9.7. Experimental Confirmation 9.8. Conclusions and Future Researches Acknowledgments References 10. Fractal Structures in a Binary Schwarzschild Black Hole System 10.1. Introduction 10.2. Basic Equations 10.3. The Scattering Map 10.4. Escape Basins 10.5. Basin Entropies 10.6. Conclusion Acknowledgments References 11. Characterizing Fractal Basin of Attraction in Planar Switched Systems 11.1. Introduction 11.2. Switched Systems and Auxiliary Dynamical Systems 11.3. Basic Concepts and Preliminaries 11.3.1. Stable manifold and unstable manifold 11.3.2. Wada basin boundary and basin cell 11.3.3. Prime ends and chains of regions 11.3.4. Alternate definitions of prime ends, extensions and the classification 11.4. Characterizing Fractal (or Wada) Basin Boundaries 11.4.1. Methods from manifolds of dynamical systems 11.4.2. Methods from limit sets and prime ends in topology 11.5. Conclusions Acknowledgment References 12. Shadowing, Errors, and Exact Orbits of Quadratic Maps 12.1. Introduction 12.2. The There-and-Back Game with the Hénon Map 12.3. Exact Coordinates of Orbital Points 12.4. Orbital Inheritance: Solving High Degree Equations 12.5. Preperiodic Points as Access to Exact Orbital Points 12.6. Conclusions and Outlook Appendix Acknowledgments References Part III. Recent Trends in Applications of Chaotic and Complex Systems 13. Electromechanical Models of Micro- and Nanoresonators 13.1. Introduction 13.2. Monolayer Nanoresonator 13.3. Differential Resonator 13.4. Parametrical Resonator 13.5. Parametrical Nanoresonator with Magnetic Exciting of Oscillation 13.6. Self-oscillation Regime of Nanoresonator 13.7. “Pull-In” Effect for Micro-Rod with Tensile Load Acknowledgments References 14. The Retina as a Dynamical System 14.1. Introduction 14.2. Brief Overview of the Retina 14.2.1. Structure 14.2.2. Modelling 14.3. The Multiscale Dynamics of the Retina: Retinal Waves 14.3.1. Context 14.3.2. The dynamics of Starburst Amacrine Cells 14.3.2.1. Fast potassium channels 14.3.2.2. Slow after-hyperpolarization current 14.3.2.3. Acetylcholine coupling 14.3.3. Retinal waves propagation 14.3.3.1. Toward a generic mechanism controlling waves 14.3.3.2. Chain of SACs 14.3.3.3. Two-dimensional dynamics 14.3.4. Consequences 14.3.4.1. Explaining variability with a unique mechanism 14.3.4.2. Dynamical response to stimuli 14.4. Response to Stimuli 14.4.1. Convolution kernel 14.4.2. Nonlinearities 14.4.3. Network 14.4.4. Waves of activity induced by a stimulus 14.4.5. Consequences 14.5. The Structure of Correlations 14.5.1. The spontaneous activity of Ganglion cells 14.5.2. An example: the discrete time integrate and fire model 14.5.3. Response to stimuli 14.5.4. Consequences 14.6. Discussion and Perspectives 14.6.1. Functional retinal circuits 14.6.2. Changing physiological parameters Acknowledgments References 15. Collective Sustained Oscillations in Complex Systems 15.1. Biological Rhythms: An Introduction 15.1.1. Self-sustained oscillation in simple nonlinear systems 15.1.2. Self-sustained oscillations in complex systems 15.2. The Dominant Phase-Advanced Driving Method 15.3. The Functional-Weight Approach 15.4. Self-Sustained Oscillation in Excitable Networks 15.4.1. The Bär–Eiswirth network 15.4.2. Winfree loop and wave propagations 15.4.3. Minimum Winfree loop and self-sustained oscillations 15.4.4. Recent progresses 15.5. Sustained Oscillation in Gene-Regulatory Networks 15.5.1. Modeling gene-regulatory networks 15.5.2. Oscillating skeletons and cores 15.6. Concluding Remarks References 16. Stability of Anti-bunched Buses and Local Unidirectional Kuramoto Oscillators 16.1. Introduction 16.2. Preliminaries 16.3. Fourier Expansion for the Bus Loop System 16.4. Local Unidirectional Kuramoto Oscillators 16.5. Partial Synchronization of local Unidirectional Kuramoto oscillators 16.6. Anti-bus Bunching 16.6.1. Direct calculation 16.6.2. Alternative condition 16.7. Stable Staggered Solutions of the Local Unidirectional Kuramoto Oscillators 16.8. Creating Stability for a Staggered Configuration in the Bus Loop System Supplementary Material Acknowledgments References 17. Feedback Delay as a Control Tool: The Complex Ginzburg–Landau Equation with Local and Nonlocal Delayed Perturbations 17.1. Introduction 17.2. Existence and Uniqueness of Solutions: Problems (17.3) and (17.6) 17.3. Turbulence Control by Feedback Delay Perturbations 17.3.1. The pseudo-linearization principle 17.3.2. Applications of the abstract result to the complex Ginzburg–Landau equation 17.3.3. Study of the eigenvalues of the linearized problem 17.4. Hopf Bifurcation and Delay Terms 17.4.1. Hopf bifurcation for the Stuart–Landau equation with a time delay feedback 17.4.2. Hopf bifurcation for the complex Ginzburg–Landau equation on the whole space and with delayed time feedback 17.4.3. Hopf bifurcation for the delayed CGLE in a bounded domain 17.4.3.1. An abstract Hopf bifurcation theorem for semilinear functional equations 17.4.3.2. Application to the delayed CGLE on a bounded domain 17.4.3.3. Some comments on the associated transversality assumption Acknowledgments References 18. Nonlinear Analysis and Control of Quasi-passive Dynamic Walker Based on Entrainment Effect 18.1. A Brief Overview of Passive Dynamic Walking 18.1.1. The most natural gait of walking robots 18.1.2. Equation of motion 18.1.3. Collision equations 18.1.4. Typical gait of passive dynamic walking 18.2. Entrained Limit-Cycle Walking 18.2.1. Entrainment-based control 18.2.2. Typical gait of entrained limit-cycle walking 18.2.3. Evaluation of entrainability 18.3. Summary References 19. Physics-Inspired Swarm Optimization: The General Algorithmic Search 19.1. Introduction 19.1.1. Particle swarm optimization 19.1.2. General algorithmic search 19.2. The GAS Algorithm 19.3. Benchmark 19.4. Results 19.5. Conclusion Appendix: 2D Test Functions Acknowledgments References