Dynamics

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

Dynamics is a branch of mathematics that provides a mathematical approach to these dynamical systems, including mathematical modeling and system analysis. Despite its low recognition in Korea, it is a significant field widely applied in physics, chemistry, biology, business, and more. It is actively used not only in abstract exploration of space and time but also in practical problem-solving.

General Dynamics

• Precise Definition of Dynamical Systems

Sets and Spaces

• Orbits and Phase Portraits
• Invariant Sets
  • Stability and Instability of Invariant Manifolds
• Attractors
  • Basins of Attracting Sets
  • Chaos of Attractors ⚫
    • Natural Invariant Measure ⚫
• Hyperbolicity of Fixed Points
• Hyperbolicity of Limit Cycles

Maps

• Dynamical Systems Represented by Maps and Fixed Points
  • Orbits of Map Systems
• One-Dimensional Maps
  • Criteria for Sinks and Sources in One-Dimensional Maps
  • Lyapunov Exponents of One-Dimensional Maps
  • Schwarzian Derivative
  • Chaos in One-Dimensional Maps ⚫
    • Li-Yorke Theorem
    • Sharkovskii’s Theorem
• Multidimensional Maps
  • Multidimensional Linear Maps
  • Multidimensional Nonlinear Maps
  • Lyapunov Exponents of Multidimensional Maps and Their Numerical Computation ⚫
  • Chaos in Multidimensional Maps ⚫
    • Conjugacy of Maps in Chaos Theory ⚫
  • Poincaré Recurrence Theorem

Differential Equations

• Dynamical Systems Represented by Differential Equations and Equilibrium Points
  • Linearization of Nonlinear Systems
• Flows and Time-T Maps
  • Poincaré Map
• Orbits and Limit Cycles
  • Homoclinic and Heteroclinic Orbits
  • Lyapunov Stability and Orbit Stability
  • Lyapunov Functions
• Classification of Fixed Points in Systems Represented by Differential Equations ● ◐ ○
• Conservation Laws in Systems Represented by Differential Equations
• Omega Limit Sets in Systems Represented by Differential Equations
  • Normal Forms of Vector Fields
• Two-Dimensional Systems
  • Bendixson’s Criterion
  • Absence 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 Dynamics
• LaSalle’s Invariance Principle
• 🔒(24/12/24) Definition of Lyapunov Spectrum
  • 🔒(24/12/20) Variational Equations
  • 🔒(24/12/28) Lyapunov Spectrum of Linear Systems
• 🔒(25/01/03) Lyapunov Spectrum of Systems Represented by Differential Equations and Their Numerical Computation
• 🔒(25/01/15) A trajectory without a fixed point has at least one zero Lyapunov exponent

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
  • 🔒(24/12/08) Feigenbaum Universality
• 🔒(24/12/04) Neimark-Sacker Bifurcation
• Chaotic Transition ⚫

Mathematical Modeling

• Law of Mass Action in Mathematics
• Lorenz Attractor ⚫
• Rössler Attractor ⚫
• Logistic Family ⚫🟢
• Duffing Oscillator
• 🔒(24/12/12) Double Pendulum ⚫

Population Growth

• Malthusian Growth Model: Ideal Population Growth 🟢
• Logistic Growth Model: Limits of 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 🟢
  • 🔒(25/01/07) Holling type functional response 🟢
  • 🔒(25/01/11) Food Chain System ⚫🟢
• Lotka-Volterra Competition Model 🟢
• May-Leonard Competition Model 🟢
• Lanchester’s Laws
• Salvo Combat 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 Spread Models? 🟢
• SIR Model: The Most Basic Spread Model 🟢
• SIS Model: Reinfection and Endemic Diseases 🟢
• SEIR Model: Incubation and Latent Periods 🟢
• SIRV Model: Vaccination and Breakthrough Infections 🟢
• SIRD Model: Death and Fatality Rate 🟢
• STD Model: Disease Transmission Between Two Groups 🟢
• Inter-species Transmission Model: Disease Transmission Among Three Groups 🟢
• AIDS Transmission Model 🟢

Coupling

• Coupled Dynamic Systems
• Metapopulation Model 🟢
• 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 🟢
• Vibrating Impact Model ⚫
• 🔒(24/12/16) Memristor Hindmarsh-Rose Neuron Model ⚫🟢

Simulation

Cellular Automata

• Dynamical Model Simulation

Agent-Based Simulation

• First Steps in Agent-Based Simulation: Representing with Scatter Plots
• Reproduction in Agent-Based Model Simulation
• Death in Agent-Based Model Simulation

Lattice Model Simulation

• First Steps in Lattice Model Simulation: Representing with Heatmaps
• Diffusion in Lattice Model Simulation

Key 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)