Nonlinear Dynamics and Complexity: Mathematical Modelling of Real-World Problems

Preface
Contents
About the Editor
1 Compartmental Poisson Stability in Non-autonomous Differential Equations
1.1 Introduction
1.2 Preliminaries
1.3 Poisson Stable Solutions of Differential Equations
1.3.1 Linear System of Differential Equations
1.3.2 Quasilinear Differential Equations
1.3.3 A Case with MPPS Coefficients
1.4 Duffing Equations with Modulo Periodic Poisson Stable Coefficients
1.5 Numerical Example
1.6 Conclusions
References
2 Analysis of the Temporal Evolution of Plumeria Bud by Measuring Its Complex Impedance: Detection of the Fractal Element with Complex Conjugated Power-Law Exponents
2.1 Introduction and Formulation of the Problem
2.2 Experimental Details
2.3 Fitting Function and the Proposed Algorithm
2.4 Analysis of the Measured Data and Discussion of the Obtained Results
A.1 Mathematical Appendix
Hypothesis H1(bR < 1)
References
3 About the Simulations of Maxwell Equations: Some Applications
3.1 Motivation for Developing a Radiative Transfer Model (RTM)
3.2 Model Inputs
3.3 Calculating the Fluxes
3.4 Dust Aerosols
3.4.1 Second- and Higher-Order Moments
3.4.2 Present and Future Work
3.5 Propagation of Electromagnetic Waves in a Self-similar Fractal Structure
3.5.1 Metamaterials
References
4 A Computational Probabilistic Calibration of the Pielou's Model to Study the Growth of Breast Tumors: A Comparative Study
4.1 Introduction
4.2 Materials and Methods
4.2.1 Data
4.2.2 Assigning a PDF to the Data Using PME
4.2.3 Pielou's Model
4.2.4 Probabilistic Calibration (PC)
4.3 Results
4.4 Discussion
4.5 Conclusion
References
5 Invariant Manifolds in the Second-Order Maxwell–Bloch Equations
5.1 Introduction
5.1.1 The Semiclassical Laser Equations
5.1.2 Infinite-Dimensional Singular Perturbation Theory
5.1.3 Reduction
5.1.4 Adiabatic Elimination for Class-A Lasers
5.1.5 Geometric Singular Perturbation Theory (GSPT)
5.1.6 Main Results
5.2 Existence and Uniqueness
5.2.1 Notation
5.2.2 A Priori Estimates
5.2.3 Existence of a Smooth Flow
5.2.4 Geometry in the Singular Limit
5.3 Existence of the Invariant Manifold
5.3.1 The Main Theorem
5.3.2 The Modified Equations
5.3.3 A Priori Estimates
5.3.4 The Cone Property
5.3.5 The Graph Transform
5.4 Smoothness of the Invariant Manifold
5.4.1 Notation
5.4.2 C1 and Ck Smoothness
5.5 Conclusion
References
6 Computing Chaotic Eigenvectors in Narrow Energy Windows
6.1 Introduction
6.2 System Under Study
6.2.1 Classical Study
6.2.2 Quantum Study
6.2.2.1 Semiclassical Estimation of the Number of States
6.3 Method
6.3.1 Quantization of Periodic Orbits
6.3.2 Construction of Tube Wave Functions
6.3.3 Construction of Scar Wave Functions
6.3.4 Development of an Efficient and Small Basis Set
6.4 Results and Discussion
6.4.1 Calculated Eigenenergies and Eigenstates
6.4.2 Basis Sets of Scar Functions
6.4.3 Computational Errors
6.4.4 Reconstruction of Individual Eigenstates
6.4.4.1 A1 Eigenstates
6.4.4.2 A2 Eigenstates
6.4.4.3 B1 Eigenstates
6.4.4.4 B2 Eigenstates
6.4.5 Scar Intensities of the Eigenstates
6.5 Conclusions and Outlook
References
7 Composition of Fuzzy Numbers with Chaotic Maps
7.1 Introduction
7.2 Mathematical Preliminaries
7.2.1 Fuzzy Numbers
7.2.1.1 Gaussian Fuzzy Number
7.2.2 Chaotic Systems
7.2.2.1 Logistic Map
7.2.2.2 Sine Map
7.2.2.3 Chebyshev 1-D Chaotic Map
7.3 Fuzzy Chaotic Maps
7.3.1 Gaussian Fuzzy Number to Chaotic Maps
7.3.1.1 Gaussian Fuzzy Logistic Map
7.3.1.2 Gaussian Fuzzy Sine Map
7.3.1.3 Gaussian Fuzzy Chebyshev 1-D Chaotic Map
7.4 Conclusion
References
8 Dynamical Analysis of a Three-Dimensional Non-autonomous Chaotic Circuit Based on a Physical Memristor
8.1 Introduction
8.2 Mathematical Model
8.3 Numerical Results
8.3.1 The Dynamics Related to the Amplitude V0 of the AC Voltage
8.3.1.1 System's Behavior for V0=1V
8.3.1.2 System's Behavior for V0=1.11V
8.3.1.3 System's Behavior for f=2.0Hz
8.3.1.4 System's Behavior for C=0.07F
8.3.1.5 System's Behavior for C=0.6F
8.3.2 The Dynamics Related to the Amplitude of the Linear Resistance R
8.4 Conclusion
References
9 On the Probabilistic Extension of the Classical Epidemiological Compartmental Model
9.1 Introduction
9.2 SIR Deterministic Epidemic Model
9.2.1 The Reproduction Number R0
9.3 The SEIR Deterministic Epidemic Model
9.3.1 The Reproduction Number via the Next Generation Method
9.3.2 The Reproduction Number in the SARS-Cov2 Case
9.4 A Probabilistic SEIR Model: pSEIR
9.5 Numerical Simulations of the Model
9.6 Conclusions
Supplementary Material
Code to Simulate the SEIR and PSEIR Models
References
10 Repellers for the Laguerre Iteration Function
10.1 Introduction
10.2 Laguerre's Iterative Method
10.3 Laguerre Iteration Function Revisited
10.4 Dynamic Systems Approach
10.5 Parameter Space
10.6 Roots of Unity Under Laguerre's Iteration with ν=n
10.7 Symmetry Effects
10.8 Conclusions
References
11 Geometric Parametrisation of Lagrangian Descriptors for 1 Degree-of-Freedom Systems
11.1 Introduction
11.2 Revisiting Lagrangian Descriptors on the Pendulum
11.2.1 Some Properties of (E)
11.2.2 Rate of Divergence of "026A30C d (E)/d E "026A30C
11.3 Geometric Lagrangian Descriptors for Duffing's Oscillator
11.4 Geometric Lagrangian Descriptors for the Fish-Tail Separatrix
11.5 Conclusion
References
12 Exact Solutions of Two PDEs which Govern the 3D Inverse Problem of Dynamics
12.1 Introduction
12.2 The 3D Inverse Problem of Dynamics: Basic Facts
12.3 Potentials of One Argument. Case I
12.3.1 Conditions on the Orbital Functions
12.3.2 Suitable Pairs of Orbital Functions and Examples
12.4 Potentials of One Argument. Case II
12.4.1 Conditions on the Orbital Functions
12.4.2 Suitable Pairs of Orbital Functions and Examples
12.4.2.1 Exceptional Cases
12.5 Potentials of Two Arguments
12.5.1 Conditions on the Orbital Functions
12.5.2 Pertinent Examples
12.5.3 Special Case (B=0)
12.6 Families of Straight Lines
12.7 Conclusions
References
13 Random Vibration of One-Dimensional Acoustic BlackHole Beam
13.1 Introduction
13.2 Theoretical Model
13.2.1 Gaussian Expansion Method (GEM)
13.2.2 Dynamic Equations
13.3 Dynamic Response of Stochastic Loading
13.3.1 Random Response of ABH Beam Under Concentrated Load
13.3.2 Base Excitation
13.3.2.1 Harmonic Excitation
13.3.2.2 White Noise Excitation
13.4 Numerical Examples
13.4.1 Description
13.4.2 Concentrated Load
13.4.3 Base Excitation
13.4.3.1 Harmonic Excitation
13.4.3.2 White Noise Excitation
13.5 Conclusions
Appendix
References
14 A Pandemic Three-Sided Coin
14.1 Introduction
14.2 Model: A Biased Three-Sided Coin
14.3 Methods: Entropy Decomposition into Information Components Under Regular and Random Transition Times
14.3.1 Entropy Decomposition into Conditional Entropies
14.3.2 Random Transition Times Under Self-Quarantine
14.4 Results: Exacerbating Uncertainty with Vaccination and Self-Quarantine Restrictions
14.5 Conclusion
References
15 Target Tracking Algorithm Based on YOLOv3 and Feature Point Matching
15.1 Introduction
15.2 Principle of the Algorithm
15.2.1 Detection Method Based on YOLOv3
15.2.2 Multi-Target Tracking Algorithm: DeepSort
15.2.3 New Algorithm for Generating ID: Feature Point Matching
15.3 Algorithm Implementation
15.3.1 Improved DeepSort Target Tracking Algorithm
15.3.2 Evaluation Indexes
15.4 Result and the Analysis of the Application Study
15.4.1 Results of Target Detection
15.4.2 Algorithm Performance Comparison Between the Improved Method and DeepSort
15.4.3 Same MOT Framework Structure for Two Kinds of Targets
15.5 Conclusion
References
16 New Fractional Derivative for Fuzzy Functions and Its Applications on Time Scale
16.1 Introduction
16.2 Preliminaries
16.2.1 Time Scale Calculus
16.2.2 Fuzzy Calculus
16.3 New Concept of Fractional Differentiability for Fuzzy Functions on Time Scales
16.4 Solving Fuzzy Fractional Differential Equations on Time Scales
References
17 Analytical and Experimental Study of the Nonlinear Potentials in a Hindmarsh –Rose Neuron System
17.1 Introduction
17.2 Methodology
17.3 Semi-Analytical Periodic Firings
17.4 Electronic Simulations
17.5 Conclusion
References
18 Analyzing and Understanding Vortex in Typical Complicated Flows with Dynamical System Approach
18.1 Introduction
18.2 Description of Vortex Using LAVD
18.3 Descriptions of Mass Transport and Mixture Using Network
18.4 Numerical Examples
18.4.1 LCSs in Secondary Vortex Street
18.4.2 Dynamics in Double-Gyre System Using Network
18.4.2.1 Steady Flows
18.4.2.2 Periodic Flows
18.5 Concluding Remarks
References
19 Dynamical Analysis of a Prabhakar Fractional Chaotic Autonomous System
19.1 Introduction
19.2 Prabhakar Fractional Calculus
19.3 Stability Theorem for the System with the FRPD
19.4 Existence and Uniqueness of the Solution
19.5 Numerical Method of PFCAS
19.6 Controlling Chaos of PFCAS
19.7 Synchronizing Chaos of PFCAS
19.8 Conclusion
References
20 Mathematical Modeling of HBV Infection with DNA-Containing Capsids and Therapy
20.1 Introduction
20.2 Well-Posedness of HBV Viral Infection Problem
20.2.1 Positivity of Solutions
20.2.2 Boundedness of Solutions
20.3 Analysis of HBV Dynamics Model
20.3.1 Basic Reproduction Number
20.3.2 Equilibrium States of the System
20.3.3 Local Stability of the Equilibria
20.3.3.1 Stability of the Disease-Free Equilibrium
20.3.3.2 Stability of the Endemic Equilibrium
20.4 Numerical Simulations
20.4.1 Disease-Free Equilibrium
20.4.2 Endemic Equilibrium
20.4.3 Effect of Therapy
20.5 Conclusion
References
21 Pricing Options Under Time-Fractional Model Using Adomian Decomposition
21.1 Introduction and Preliminaries
21.2 The Fractional Black and Scholes Model
21.2.1 Pricing American Put Option Under Fractional Black and Scholes Model
21.2.2 Simulations and Numerical Results
21.3 The Fractional Heston Model
21.3.1 Closed-Form Solution of European Option Under Fractional Heston Model
21.3.2 Pricing American Put Option Under Fractional Heston Model
21.3.3 Simulations and Numerical Results
21.4 Conclusion
References
22 A Novel High-Efficiency Piezoelectric Energy Harvester Designed to Harvest Energy from Random Excitation
22.1 Introduction
22.2 Modeling
22.3 Results
22.4 Conclusion
References
23 Global Stability Analysis of Two-Strain SEIR Epidemic Model with Quarantine Strategy
23.1 Introduction
23.2 Mathematical Formulation
23.2.1 Mathematical Model
23.2.2 Positivity and Boundedness of Solutions
23.3 Analysis of the Model
23.3.1 The Basic Reproduction Number Calculation
23.3.2 Steady States
23.3.3 Global Stability
23.3.3.1 Global Stability of the Disease-Free Equilibrium
23.3.3.2 Global Stability of the Strain 1 Endemic Equilibrium
23.3.3.3 Global Stability of the Strain 2 Endemic Equilibrium
23.3.3.4 Global Stability of the Total Endemic Equilibrium
23.4 Numerical Simulations and Discussions
23.5 Conclusion
References
Index