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- Detailed Program: Introduces dynamical systems, continuous and discrete models (1D, 2D, n-dimensional), local bifurcations, piecewise-linear systems, and applications in population dynamics, epidemic models, oligopoly theory, and financial markets.
- Mathematical Modelling: Explains the process of representing real systems mathematically, identifying measurable quantities, and analyzing system behavior through schematic, mathematical, and formal models.
- Dynamical Systems - General Definitions:
- Defines dynamical systems as systems that change over time, described by dynamic variables.
- Categorizes variables as states (measurable quantities) and parameters (fixed over time).
- Illustrates concepts like continuous and discrete time, phase curve (orbit), equilibrium, trapping sets, and invariant sets.
- Discusses stability: local (basin of attraction) and global attractor.
- Examples of 1D Models in Continuous Time:
- Malthusian growth model: Linear, homogeneous, with equilibrium points and stability analysis based on birth/mortality rates.
- Affine dynamic equation: Introduces constant immigration/emigration, analyzing equilibrium stability.
- Walrasian model: Price dynamics in a partial market with linear demand and supply functions.
- Logistic growth model: Nonlinear, incorporates population density effects, introducing carrying capacity and transcritical bifurcation.
- SIR epidemic model: Models Susceptibles, Infected, and Recovered populations, analyzing equilibria and their stability.
- Continuous-Time Dynamical Systems - 1D: Explores linear homogeneous and affine models, focusing on equilibrium conditions and local stability based on the sign of parameter 'a'. Introduces `x = f(x, α)` as `x` = `αx` for linear, and `x` = `αx + b` for affine.
- Nonlinear Continuous 1D & Local Bifurcations:
- Defines qualitative equivalence and local bifurcations.
- Describes Fold Bifurcation: creation/annihilation of equilibrium points as a parameter varies (e.g., `x` = `μ - x²`).
- Explains Transcritical Bifurcation: exchange of stability between two equilibrium points (e.g., `x` = `μx - x²`).
- Details Pitchfork Bifurcation: transition from one stable equilibrium to three (one unstable, two stable) (e.g., `x` = `μx - x³`).
- Discusses Hyperbolic and Non-Hyperbolic Equilibria based on the derivative `f'(x*)`.
- Continuous 2D Linear Systems:
- Introduces 2D systems, nullclines, and linear homogeneous systems (`x` = `Ax`).
- Analyzes equilibrium stability using the characteristic equation, trace, and determinant of the Jacobian matrix.
- Classifies equilibria based on eigenvalues: stable/unstable node, saddle, stable/unstable focus, centre.
- Continuous 2D Nonlinear & Hopf Bifurcation:
- Extends concepts to nonlinear 2D systems, using Taylor expansion and Jacobian matrix for local stability analysis.
- Introduces the Andronov-Hopf theorem: relates complex conjugate eigenvalues crossing the imaginary axis to the creation of limit cycles (periodic orbits). Describes supercritical (soft loss of stability) and subcritical (hard loss of stability) cases.
- Continuous ND Systems & Chaos:
- Discusses linear n-dimensional systems and stability conditions based on the real part of eigenvalues.
- Introduces concepts of chaos, deterministic chaos, sensitivity to initial conditions (butterfly effect), and strange attractors (Lorenz attractor).
- Discrete-Time Dynamical Systems - 1D:
- Defines discrete time and difference equations (`x(t+1) = T(x(t))`).
- Analyzes linear homogeneous iterated maps (`x(t+1) = ax(t)`) and their stability based on `|a|`.
- Discusses affine maps (`x(t+1) = ax(t) + b`) and their equilibrium `x* = b/(1-a)`.
- Covers the Cobweb model for price dynamics with static expectations.
- Examines financial market models with fundamentalists and chartists, leading to linear/nonlinear difference equations.
- Explores periodic cycles and their stability, including Fold, Transcritical, and Pitchfork bifurcations in discrete time.
- Highlights key differences from continuous-time systems, such as oscillatory behavior for negative 'a'.
- Introduces the logistic map (`x(t+1) = μx(t)(1-x(t))`) and its bifurcations (transcritical, flip), leading to cascades of period-doubling and chaos.
- Explains basin of attraction and different types of increasing/decreasing maps.
- Discrete 2D Maps:
- Extends discrete systems to two dimensions (`x'(t) = T(x(t))`).
- Analyzes stability using eigenvalues within the unit circle of the complex plane (`|λi| < 1`).
- Classifies equilibria based on eigenvalues (stable/unstable node, saddle, stable/unstable focus).
- Introduces Neimark-Sacker bifurcation: complex conjugate eigenvalues crossing the unit circle, leading to invariant closed curves.
- Piecewise-Linear Maps - Border-Collision Bifurcations:
- Defines piecewise-linear maps and their characteristics.
- Examines dynamics based on slopes (`aL`, `aR`) and intercepts (`bL`, `bR`).
- Introduces Border-Collision Bifurcations (BCB) as cycles collide with border points, leading to new stable cycles or chaos.
- Discusses period-adding and period-incrementing structures.
- Applications: Detailed models for financial markets (with various trader types) and Cournot oligopoly (static and dynamic versions, with different assumptions on rationality and costs).
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