Mathematical Modelling and Computing in Physics, Chemistry and Biology: Fundamentals and Applications

Contents
1 Basic Characteristics of Dynamical Systems
	1.1 Cornerstone
	1.2 Concise Characterization of Complex Dynamical Systems
	1.3 Relevance and Emergence of Natural Systems
	1.4 Unusual Behavior of Complex Nonlinear Dynamical Systems
	1.5 Entropy of Complex Processes in Dynamical Systems
	1.6 Instantaneous States of Dynamical Systems
	References
2 Basic Ingredients of Nonlinear Systems Dynamics
	2.1 Introduction
	2.2 Nonlinear Phenomena and Nonlinear Equations
		2.2.1 Preliminaries
		2.2.2 Paradigmatic Nonlinear Models
		2.2.3 Nonlinear Oscillators Solvable in Elementary Functions
	2.3 Models of Complex Nonlinear Dynamic Systems
		2.3.1 Basic Forms of Systems Oscillations
		2.3.2 Systems Under Periodic Pulsed Excitation
		2.3.3 Regular Periodic Pulses in Linear Systems
		2.3.4 Oscillators Under the Periodic Impulsive Loading
		2.3.5 Periodic Impulses with a Temporal ‘Dipole’ Shift
		2.3.6 Fractional Order Differential Models
		2.3.7 Artificial Intelligence Models
	2.4 Mixed-Mode Oscillations
	2.5 Stability and Bifurcation of Dynamic States
	2.6 Chaotic Oscillations
	References
3 Oscillations in Physical Systems
	3.1 Lorenz System and Its Properties
	3.2 Logistic Equation and Its Applications
	3.3 Predator–Prey Systems
	3.4 Systems with the Double Scroll Attractors
	3.5 Fractal Systems
	3.6 Applications of Van der Pol Equation
	3.7 The Rössler Attractor
	3.8 Dynamic Analysis of the Nonlinear Energy Harvesting System
	3.9 Duffing’s Forced Oscillator
	References
4 Oscillatory Chemical Systems
	4.1 Preliminary
	4.2 Oscillations in Chemical Systems
	4.3 Autocatalysis, Activation and Inhibition
	4.4 Enzyme Kinetics
	4.5 BZ Oscillating Reactions
	4.6 Limit Cycle Oscillations in the BZ Reaction
	4.7 Numerical Simulations of Kinetic Bistability of Reactions
	4.8 Mathematical Modeling of Electrochemical Reactors
	4.9 MMOs in Electrochemical Reactors
	References
5 Oscillations in Biological Processes
	5.1 Motivation, Brief History and Background
	5.2 Feedback Control Mechanisms
	5.3 The Hodgkin–Huxley Neuron
		5.3.1 Basic Mathematical Model
		5.3.2 Periodic Neuron Firing
	5.4 Reduced Model of a Single Neuron Activity
	5.5 Nonlinear Human Cardiovascular System
		5.5.1 Concise Characterization of a Cardiovascular System
		5.5.2 Thermodynamic Model of the Cardiovascular System Dynamics
		5.5.3 MMOs as an Indicator of Illness in the Cardiovascular System
	References
6 Energy Flow Analysis of Nonlinear Dynamical Systems
	6.1 Introduction and Short Historical References
	6.2 New Standards for the Energy Avenue in Non-sinusoidal States
		6.2.1 Preliminary
		6.2.2 Hysteresis Loops on Energy Phase Plane
		6.2.3 Quantitative Measures of the Energy Hysteresis Loop
		6.2.4 Estimates for One-Period Energy Loops
		6.2.5 Analysis for Energy Aspects in Dynamical Systems with Switches
	6.3 The Energy Approach to Electrochemical Corrosion Studies of Nano-copper Coatings
		6.3.1 Introduction
		6.3.2 The One-Period Energy Approach
		6.3.3 Experiments
		6.3.4 Results and Discussion
	6.4 Effective Harvesting of Braking Energy in Electric Cars
		6.4.1 Introduction
		6.4.2 Energy Losses in Sub-systems of Electric Cars
		6.4.3 Energy Regeneration in Sub-systems of Electric Cars
		6.4.4 Modification of the Car Brake Sub-system
		6.4.5 Regenerative Damping
		6.4.6 Supercapacitor Characterization
		6.4.7 Simulations
		6.4.8 Summary and Conclusions
	6.5 Electromechanical System for Charging Batteries of Electric Cars
		6.5.1 Introduction
		6.5.2 Principle of Batteries Charging with the Faraday Disk Generator
		6.5.3 Battery Property and Modelling
		6.5.4 Electro-magnetic-mechanical Model of Faraday Disk Generators
		6.5.5 Structure of the Charging System
		6.5.6 State-Space Equations
		6.5.7 Computer Simulations
		6.5.8 Discussion and Summary
	6.6 Modeling of Energy Processes in Wheel-Rail Contacts Operating Under Influence of Periodic Discontinuous Forces
		6.6.1 Introduction
		6.6.2 The Problem Description
		6.6.3 Dynamic Model of Wheel-Rail Contacts
		6.6.4 Exact Periodic Solutions
		6.6.5 Application of One-Period Energy Approach
		6.6.6 Sleeper Nonlinear Characteristics
		6.6.7 Discussion and Summary
	References
7 Artificial Intelligence in the Service of Dynamical Systems Studies
	7.1 Background of AI Modeling
	7.2 Common Types of AI Algorithms
	7.3 Artificial Neural Network
		7.3.1 Preliminary
		7.3.2 Architectures of Artificial Neural Networks
		7.3.3 Hopfield Artificial Neural Network
		7.3.4 Equilibrium States in the Hopfield Network
	7.4 Learning Neural Networks
	7.5 Fuzzy Logic Models
		7.5.1 Concise Overview
		7.5.2 Fuzzy Systems
		7.5.3 Building Fuzzy Models
		7.5.4 Defuzzification
	7.6 Real-Life Artificial Intelligence Applications
	References