Dynamics

A system in which the state at a given time is represented by its past state is called a dynamical system. For example, if we have a variable $x_n$ that can be expressed by some map $f$ as $$x_{n+1} = f(x_n),$$ or if the state of $x$ can be represented by the differential equation $$\dot{x} = g(x)$$ for some function $g$, then such cases fall under this definition. In this context, systems that yield deterministic values are called dynamic systems, while non-deterministic systems are referred to as stochastic processes.¹

Dynamical systems is a branch of mathematics that takes a mathematical approach to these systems, encompassing mathematical modeling and system analysis. Despite its relatively low recognition domestically, it is a broad field with diverse applications in physics, chemistry, biology, business, and more, playing an active role in both abstract explorations of space-time and practical problem-solving.

General Dynamical Systems

• A Rigorous Definition of a Dynamical System

Sets and Spaces

• Orbits and Phase Portraits
  • Sensitive Dependence on Initial Conditions
• Invariant Sets
  • Stability and Instability of Invariant Manifolds
  • Chaos in Invariant Sets ⚫
• Attractors
  • The Basin of an Attracting Set
  • Natural Invariant Measure ⚫
  • 🔒(25/03/22) Strange Attractors ⚫
  • Strange Nonchaotic Attractors ⚫
• Hyperbolicity of Fixed Points
• Hyperbolicity of Limit Cycles

Maps

• Dynamical Systems Expressed as Maps and Fixed Points
  • Orbits in Map Systems
• One-Dimensional Maps
  • Criteria for Sinks and Sources in 1D Maps
  • Lyapunov Exponents in 1D Maps
  • Schwarzian Derivative
  • Chaos in 1D Maps ⚫
    • Re-Yorke Theorem
    • Sharkovsky’s Theorem
• Multi-Dimensional Maps
  • Linear Maps in Multiple Dimensions
  • Nonlinear Maps in Multiple Dimensions
  • Lyapunov Exponents and Their Numerical Calculation in Multi-Dimensional Maps ⚫
  • Chaos in Multi-Dimensional Maps ⚫
    • Conjugacy of Maps in Chaos Theory ⚫
  • Poincaré Recurrence Theorem

Differential Equations

• Dynamical Systems Expressed as Differential Equations and Equilibrium Points
  • Linearization of Nonlinear Systems
• Flows and Time-$T$ Maps
  • Poincaré Maps
• Orbits and Limit Cycles
  • Homoclinic and Heteroclinic Orbits
  • Lyapunov Stability and Orbital Stability
  • Lyapunov Functions
  • Quasiperiodic Orbits
• Classification of Fixed Points in Systems Expressed by Differential Equations ● ◐ ○
• Conserved Quantities in Systems Expressed by Differential Equations
• Omega Limit Sets in Systems Expressed by Differential Equations
  • Normal Forms of Vector Fields
• Two-Dimensional Systems
  • Bendixson’s Criterion
  • Absence of Periodic Orbits in 2D Autonomous Systems
  • Poincaré-Bendixson Theorem: No Chaos Occurs in 2D Autonomous Systems ⚫
  • Nullclines
• Liouville’s Theorem in Dynamical Systems
• LaSalle’s Invariance Principle
• Definition of the Lyapunov Spectrum
  • Variational Equations
  • Lyapunov Spectrum of Linear Systems
• Lyapunov Spectrum of Systems Expressed by Differential Equations and Its Numerical Calculation
• Trajectories Not Containing Fixed Points Possess at Least One Zero Lyapunov Exponent
• Chaos in Systems Expressed by Differential Equations ⚫

Bifurcation Theory

• Bifurcation
  • Topological Equivalence Between Dynamical Systems
• Bifurcation Diagrams
• Pitchfork Bifurcation $\dot{x} = rx \mp x^3$
• Transcritical Bifurcation $\dot{x} = rx - x^2$
• Saddle-Node Bifurcation $\dot{x} = r + x^2$
  • Hysteresis
• Homoclinic Bifurcation
• Heteroclinic Bifurcation
• Infinite-Period Bifurcation
• Hopf Bifurcation
• Period-Doubling Bifurcation
  • Feigenbaum Universality
• Neimark-Sacker Bifurcation
• Chaotic Transition ⚫

Fractals

• Self-Similar Sets
• Koch Curve
• Fractals
  • Hausdorff Dimension
  • Similarity Dimension
  • Box-Counting Dimension
  • Correlation Dimension
• Riddled Basins
• Mandelbrot and Julia Sets

Mathematical Modeling

Named Systems

• Mass Action Law in Mathematics
• Lorenz Attractor ⚫
• Rössler Attractor ⚫
• Duffing Oscillator
• Double Pendulum ⚫
• Logistic Family ⚫🟢
• Hénon Map ⚫
• Ikeda Map ⚫

Population Growth

• Malthusian Growth Model: Ideal Population Growth 🟢
• Logistic Growth Model: Limits to Population Growth 🟢
  • Allee Effect in Mathematical Biology 🟢
• Gompertz Growth Model: Growth Delay Over Time 🟢
• Bass Diffusion Model: Innovation and Imitation
• Lotka–Volterra Predator–Prey Model 🟢
  • Holling-Type Functional Response 🟢
  • Food Chain Systems ⚫🟢
• Lotka–Volterra Competition Model 🟢
• May–Leonard Competition Model 🟢
• Lanchester’s Laws
• Simultaneous-Fire Battle Model
• Leslie Age-Structured Model 🟢
• von Foerster Equation 🟢
• Population Balance Equation 🟢

Disease Spread

• Compartmental Models in Epidemiology 🟢
• What Is the Basic Reproduction Number in Epidemic Models? 🟢
• SIR Model: The Most Basic Epidemic Model 🟢
• SIS Model: Reinfection and Chronic Diseases 🟢
• SEIR Model: Incubation and Latency 🟢
• SIRV Model: Vaccination and Breakthrough Infections 🟢
• SIRD Model: Deaths and Fatality Rates 🟢
• STD Model: Disease Transmission Between Two Groups 🟢
• Cross-Species Transmission Model: Disease Spread Among Three Groups 🟢
• AIDS Transmission Model 🟢

Coupling

• Coupled Dynamical Systems
• Metapopulation Models 🟢
• Eulerian Movement Model 🟢
• Lagrangian Movement Model 🟢
• Slow–Fast Systems

Nonsmooth Systems

• Nonsmooth Systems
  • Piecewise Smooth Systems (PWS)
  • Differential Inclusion $\dot{x} \in F(x)$
• DC–DC Buck Converter ⚫
• Atopic Dermatitis System 🟢
• Impact Oscillator ⚫
• Memristor Hindmarsh–Rose Neuron Model ⚫🟢
• 🔒(25/03/23) Lozi Map ⚫

Simulation

Cellular Automata

• Dynamical Model Simulation

Agent-Based Simulation

• First Steps in Agent-Based Simulation: Visualizing with a Scatter Plot
• Reproduction in Agent-Based Model Simulation
• Mortality in Agent-Based Model Simulation

Lattice Model Simulation

• First Steps in Lattice Model Simulation: Visualizing with a Heatmap
• Diffusion in Lattice Model Simulation

Major References

• Allen. (2006). An Introduction to Mathematical Biology
• Ottar N. Bjørnstad. (2018). Epidemics Models and Data using R
• Capasso. (1993). Mathematical Structures of Epidemic Systems
• Guckenheimer. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
• Kuznetsov. (1998). Elements of Applied Bifurcation Theory(2nd Edition)
• Strogatz. (2015). Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering(2nd Edition)
• Yorke. (1996). CHAOS: An Introduction to Dynamical Systems
• Wiggins. (2003). Introduction to Applied Nonlinear Dynamical Systems and Chaos Second Edition(2nd Edition)