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.
Mark | Subcategory |
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⚫ | Chaos |
🟢 | Bio |
General Dynamics
Sets and Spaces
- Orbits and Phase Portraits
- Invariant Sets
- Attractors
- 🔒(24/11/22) Hyperbolicity of Fixed Points
- 🔒(24/11/26) Hyperbolicity of Limit Cycles
Maps
- Dynamical Systems Represented by Maps and Fixed Points
- One-Dimensional Maps
- Multidimensional Maps
Differential Equations
- Dynamical Systems Represented by Differential Equations and Equilibrium Points
- Flows and Time-T Maps
- Orbits and Limit Cycles
- 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
- Two-Dimensional Systems
- 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
- Bifurcation Diagram
- Pitchfork Bifurcation $\dot{x} = rx \mp x^{3}$
- 🔒(24/10/13) Transcritical Bifurcation $\dot{x} = rx - x^{2}$
- 🔒(24/10/17) Saddle-Node Bifurcation $\dot{x} = r + x^{2}$
- 🔒(24/10/21) Hysteresis
- 🔒(24/11/06) Homoclinic Bifurcation
- 🔒(24/11/10) Heteroclinic Bifurcation
- 🔒(24/11/14) Infinite Period Bifurcation
- 🔒(24/11/18) Hopf Bifurcation
- 🔒(24/11/30) 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 🟢
- 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
- DC-DC Buck Converter ⚫
- Atopic Dermatitis System 🟢
- Vibrating Impact Model ⚫
- 🔒(24/12/16) Memristor Hindmarsh-Rose Neuron Model ⚫🟢
Simulation
Cellular Automata
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)
Yorke. (1996). CHAOS: An Introduction to Dynamical Systems, p. 2. ↩︎
All posts
- Representing Dynamical Systems and Fixed Points with Maps
- Lorenz Attractor
- Identification of Sinks and Sources in One-Dimensional Maps
- Map System's Orbit
- Lyapunov Exponents of One-Dimensional Maps
- Chaos in One-Dimensional Maps
- Proof of the Lee-Yang Theorem
- Sharkovsky's Theorem
- Logistic Family
- Conjugate Maps in Chaos Theory
- Schwarzschild Derivative
- Bifurcation Diagram
- Natural Invariant Measure
- Chaotic Transition
- Multidimensional Linear Maps
- Multidimensional Nonlinear Maps
- Lyapunov Numbers and Their Numerical Calculation Methods for Multidimensional Maps
- Multidimensional Map Chaos
- Attractors in Chaos
- Dynamical Systems Described by Differential Equations and Equilibrium Points
- Autonomous Systems: Flow and Time-T Maps
- Autonomous Systems: Orbits and Limit Cycles
- Linearization of Nonlinear Systems
- Lyapunov Stability and Orbit Stability
- The Van der Pol Oscillator
- Classification of Fixed Points in Autonomous Systems
- Lyapunov Function
- Invariant Sets in Dynamics
- Stability of Invariant Manifolds
- Bendixson's Criterion
- Absence of Periodic Orbits in Two-Dimensional Autonomous Systems
- Proof of Poincaré bendixson Theorem
- Conservation Quantities of Autonomous Systems
- Proof of Liouville's Theorem in Dynamics
- Poincaré Recurrence Theorem Proof
- Omega Limit Sets of Autonomous Systems
- Attractors in Dynamical Systems
- Attracting Set's Basin
- Proof of the LaSalle Invariance Principle
- Malthus Growth Model: Ideal Population Growth
- Dynamical Model Simulation
- First Steps in Agent-Based Simulation: Representing with Scatter Plots
- Agent-based Model Simulation of Reproduction
- Agent-based Model Simulation of Mortality
- Logistic Growth Model: The Limits of Population Growth
- First Steps in Lattice Model Simulation: Representing with Heatmaps
- Diffusion in Lattice Model Simulations
- Allee Effect in Mathematical Biology
- Gompertz Growth Model: Time-dependent Growth Deceleration
- Bass Diffusion Model: Innovation and Imitation
- Lotka-Volterra Predator-Prey Model
- Lotka-Volterra Competition Model
- May-Leonard Competition Model
- Lanchester's laws
- Simultaneous Firing Combat Model
- Dynamics Compartment Model
- What is the Basic Reproduction Number in Epidemic Spread Models?
- SIR Model: The Most Basic Diffusion Model
- SIS Model: Reinfection and Chronic Disease
- Sexually Transmitted Diseases Model: Disease Transmission between Two Populations
- Inter-Species Transmission Model: Disease Spread among Three Populations
- Poincaré Map
- AIDS Transmission Model
- Law of Mass Action in Mathematics
- Rigorous Definition of Dynamical Systems
- Orbits and Phase Portraits in Dynamics
- Topological Equivalence between Dynamical Systems
- Coupled Dynamic Systems
- Leslie age structure Model
- SEIR Model: Latent Period and Incubation Period
- SIRV Model: Vaccines and Breakthrough Infections
- SIRD Model: Death and Fatality Rate
- Metapopulation Model
- Euler's Motion Model
- Lagrangian Motion Model
- 폰 푀르스터 방정식
- Population Balance Equation
- Rössler Attractor
- Smooth Systems in Each Segment of Dynamics
- Definition of Differential Inclusion
- Non-Smooth Systems in Dynamics
- Slow-Fast Systems
- DC-DC Buck Converter as a Dynamical System
- Dynamics in Atopic dermatitis Systems
- Bifurcation in Dynamics
- Normal Form of Vector Field in Dynamics
- Vibro-impact Model as a Dynamical System
- Pitchfork Bifurcation
- Transcritical Bifurcation
- Saddle-Node Bifurcation
- Hysteresis Phenomena in Dynamics
- Tipping Points in Dynamics
- Nullclines in Mathematical Analysis
- Homoclinic Orbit and Heteroclinic Orbit