Dynamics

A system where a state at a certain point in time is expressed in terms of its past states is called a dynamical system. For example, if there is an $x_{n}$, it can be represented as $x_{n+1} = f(x_{n})$ for some map $f$, or the state of $x$ can be represented by a differential equation $\dot{x} = g(x)$ for some function $g$. Systems where deterministic values are obtained are called dynamical systems, while nondeterministic systems are referred to as stochastic processes¹.

Dynamics is the mathematical approach to such dynamical systems, including mathematical modeling and analysis of systems, and is a major branch of mathematics that is widely applied in physics, chemistry, biology, business, etc., despite its low recognition in some regions. It is used actively both in abstract investigations of space-time and in practical problem-solving.

General Dynamics

• Rigorous Definition of Dynamical Systems

Sets and Spaces

• Orbits and Phase Portraits
• Attractors
  • Basin of Attracting Set
  • Chaos of Attractors ⚫
    • Natural Invariant Measure ⚫
• Invariant Sets
  • Stability of Invariant Manifolds

Maps

• Dynamical Systems Represented by Maps and Fixed Points
  • Orbits in Map Systems
• 1-Dimensional Maps
  • Sink and Source Judgement in 1-Dimensional Maps
  • Lyapunov Exponent of 1-Dimensional Maps
  • Schwarzian Derivative
  • Chaos in 1-Dimensional Maps ⚫
    • Li-Yorke Theorem
    • Sharkovsky’s Theorem
• Multi-Dimensional Maps
  • Multi-Dimensional Linear Maps
  • Multi-Dimensional Nonlinear Maps
  • Lyapunov Exponents of Multi-Dimensional Maps and Their Numerical Calculation Methods
  • Chaos in Multi-Dimensional Maps ⚫
    • Conjugacy in Map Chaos Theory ⚫
  • Poincaré Recurrence Theorem

Differential Equations

• Dynamical Systems Represented by Differential Equations and Equilibrium Points
  • Flow and Time-T Map in Autonomous Systems
  • Orbits and Limit Cycles in Autonomous Systems
  • Linearization of Nonlinear Systems
  • Lyapunov Stability and Orbit Stability
    • Lyapunov Function
  • Classification of Fixed Points in Autonomous Systems
  • Conserved Quantities in Autonomous Systems
  • Omega Limit Set in Autonomous Systems
  • 🔒(24/08/10) Normal Form of Vector Fields
• 2-Dimensional Systems
  • Bendixson Criterion
  • Absence of Periodic Orbits in 2-Dimensional Autonomous Systems
  • Poincaré-Bendixson Theorem: No Chaos in 2-Dimensional Autonomous Systems ⚫
• Liouville’s Theorem in Dynamics
• LaSalle’s Invariance Principle
• Poincaré Map

Bifurcation Theory

• 🔒(24/08/06) Bifurcation
  • Topological Equivalence Among Dynamical Systems
• Bifurcation Diagram
• 🔒(24/08/20) Pitchfork Bifurcation
• Chaotic Transition ⚫

Mathematical Modeling

• Law of Mass Action in Mathematics
• Lorenz Attractor ⚫
• 🔒(24/04/10) Rössler Attractor ⚫
• Logistic Family ⚫
• Duffing Oscillator

Population Growth

• Malthus Growth Model: Ideal Collective Growth 🟢
• Logistic Growth Model: Limits of Collective Growth 🟢
  • Allee Effect in Mathematical Biology 🟢
• Gompertz Growth Model: Growth Delay Over Time 🟢
• Bass Diffusion Model: Innovation and Imitation
• Lotka-Volterra Predator-Prey Model 🟢
• Lotka-Volterra Competition Model 🟢
• May-Leonard Competition Model 🟢
• Lanchester’s Laws
• Salvo Combat Model
• Leslie age structured Model 🟢
• Von Foerster Equation 🟢
• 🔒(24/03/17) Population Balance Equations 🟢

Disease Spread

• Epidemic Compartment Models 🟢

• Basic Reproduction Number in Epidemic Spread Models 🟢

• SIR Model: The Most Basic Spread Model 🟢

• SIS Model: Reinfection and Chronic Diseases 🟢

• SEIR Model: Incubation and Latency Periods 🟢

• SIRV Model: Vaccination and Breakthrough Infections 🟢

• SIRD Model: Death and Fatality Rates 🟢

• Sexually Transmitted Diseases Model: Disease Transmission Between Two Groups 🟢

• Inter-Species Transmission Model: Disease Transmission Among Three Groups 🟢

• AIDS Transmission Model 🟢

Coupling

• Coupled Dynamical Systems
• Meta-Population Model 🟢
• Eulerian Movement Model 🟢
• Lagrangian Movement Model 🟢
• 🔒(24/06/29) Slow-Fast Systems

Nonsmooth Systems

• 🔒(24/06/25) Nonsmooth Systems
  • 🔒(24/06/17) Piecewise Smooth Systems PWS
  • 🔒(24/06/21) Differential Inclusions $\dot{x} \in F(x)$
• 🔒(24/07/03) DC-DC Buck Converter ⚫
• 🔒(24/07/07) Atopic Dermatitis System 🟢
• 🔒(24/08/14) Vibrational Impact 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
• 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)