Dynamical Systems

A dynamical system is a system in which the state at a given point in time is expressed in terms of its past state. For example, if we have $x_{n}$, it can be expressed as $x_{n+1} = f (x_{n} )$ for some map $f$, or the state of $x$ can be described by a differential equation such as $\dot{x} = g(x)$ for some function $g$. A system where deterministic values are obtained is called a dynamic system, while a non-deterministic system is called a stochastic process.¹

Dynamical systems represent a branch of mathematics that provides a mathematical approach to such dynamical systems, including mathematical modeling and system analysis. In contrast to its low awareness in Korea, it is a major field with diverse applications in physics, chemistry, biology, business, and more, being actively used not only in abstract exploration of spacetime but also in practical problem-solving.

General Dynamical Systems

• Rigorous Definition of Dynamical Systems
• 🔒(26/01/22)What are Governing Equations?

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
  • Basins of Attracting Sets
  • Natural Invariant Measure ⚫
  • Strange Attractors ⚫
  • Attractors that are Strange but not Chaotic ⚫
• Hyperbolicity of Fixed Points
• Hyperbolicity of Limit Cycles

Maps

• Dynamical Systems Expressed as Maps and Fixed Points
  • Orbits of Map Systems
• One-dimensional Maps
  • Sink-Source Test for One-dimensional Maps
  • Lyapunov Exponent for One-dimensional Maps
  • Schwarzian Derivative
  • Chaos in One-dimensional Maps ⚫
    • Li-Yorke Theorem
    • Sharkovskii Theorem
• Multidimensional Maps
  • Multidimensional Linear Maps
  • Multidimensional Nonlinear Maps
  • Lyapunov Exponents for Multidimensional Maps and Their Numerical Computation ⚫
  • Chaos in Multidimensional 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 as Differential Equations ● ◐ ○
• Conservation Laws in Systems Expressed as Differential Equations
• Omega Limit Sets in Systems Expressed as Differential Equations
  • Normal Forms of Vector Fields
• Two-dimensional Systems
  • Bendixson’s Criterion
  • Non-existence of Periodic Orbits in Two-dimensional Autonomous Systems
  • Poincaré-Bendixson Theorem: Chaos Does Not Occur in Two-dimensional Autonomous Systems ⚫
  • Nullclines
• Liouville’s Theorem in Dynamical Systems
• LaSalle’s Invariance Principle
• Definition of Lyapunov Spectrum
  • Variational Equations
  • Lyapunov Spectrum of Linear Systems
• Lyapunov Spectrum of Systems Expressed as Differential Equations and Its Numerical Computation
• Trajectories Without Fixed Points Have at Least One Zero Lyapunov Exponent
• Chaos in Systems Expressed as Differential Equations ⚫

Bifurcation Theory

• Bifurcation
  • Topological Equivalence Between Dynamical Systems
• Bifurcation Diagram
• 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
• von Koch Curve
• Fractals
  • Hausdorff Dimension
  • Similarity Dimension
  • Box-counting Dimension
  • Correlation Dimension
• Riddled Basins
• Mandelbrot Set and Julia Set

Mathematical Modeling

Named Systems

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

Population Growth

• Malthus Growth Model: Ideal Population Growth 🟢
• Logistic Growth Model: Limits to Population Growth 🟢
  • Allee Effect in Mathematical Biology 🟢
• Gompertz Growth Model: Time-dependent Growth Decline 🟢
• Bass Diffusion Model: Innovation and Imitation
• Lotka-Volterra Predator-Prey Model 🟢
  • Holling Type Functional Response 🟢
  • Food Chain System ⚫🟢
• Lotka-Volterra Competition Model 🟢
• May-Leonard Competition Model 🟢
• Lanchester’s Law
• Salvo Combat Model
• Leslie Age Structure Model 🟢
• von Foerster Equation 🟢
• Population Balance Equation 🟢

Disease Transmission

• Epidemiological Compartment Model 🟢
• What is the Basic Reproduction Number in Epidemic Models? 🟢
• SIR Model: The Most Basic Transmission Model 🟢
• SIS Model: Reinfection and Endemic Diseases 🟢
• SEIR Model: Incubation Period and Latent Period 🟢
• SIRV Model: Vaccines and Breakthrough Infection 🟢
• SIRD Model: Death and Mortality Rate 🟢
• STI Model: Disease Transmission Between Two Populations 🟢
• Cross-species Transmission Model: Disease Transmission Between Three Populations 🟢
• AIDS Transmission Model 🟢

Coupling

• Coupled Dynamical Systems
• Metapopulation Model 🟢
• Eulerian Movement Model 🟢
• Lagrangian Movement Model 🟢
• Slow-fast Systems

Non-smooth Systems

• Non-smooth Systems
  • Piecewise Smooth Systems PWS
  • Differential Inclusion $\dot{x} \in F(x)$
• DC-DC Buck Converter ⚫
• Atopic Dermatitis System 🟢
• Oscillating Impact Model ⚫
• Memristor Hindmarsh-Rose Neuron Model ⚫🟢
• Lozi Map ⚫

Simulation

Cellular Automata

• Dynamical Model Simulation

Agent-Based Simulation

• Getting Started with Agent-Based Simulation: Representing with Scatter Plots
• Reproduction in Agent-Based Model Simulation
• Mortality in Agent-Based Model Simulation

Lattice Model Simulation

• Getting Started with Lattice Model Simulation: Representing with Heatmaps
• Diffusion in Lattice Model Simulation

Main 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)