Recent Trends in Chaotic, Nonlinear and Complex 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