Dynamical Systems Modeling and Analysis

Dynamical Systems Modeling and Analysis is a core methodology within Systems Science that uses mathematical formalisms, such as differential or difference equations, to represent how a system's state evolves over time. The primary goal is to understand and predict the system's behavior by identifying key features like equilibrium points (stable states), periodic orbits (cycles), and bifurcations (sudden changes in behavior), as well as determining whether the system is stable, oscillatory, or chaotic. This approach allows researchers to simulate complex phenomena, from population dynamics and climate change to neural networks and economic cycles, revealing the underlying rules that govern their temporal evolution.

1. Introduction to Dynamical Systems
   1. Fundamental Concepts
      1. Definition of a Dynamical System
      2. State of a System
      3. State Variables
      4. State Vector Representation
      5. State Space
         1. Geometric Interpretation
         2. Phase Space Terminology
         3. Dimensionality of State Space
      6. Evolution Rules
         1. Deterministic Evolution
         2. Time Evolution Operators
         3. Stochastic Evolution
      7. Trajectories and Orbits
         1. Definition of Trajectories
         2. Forward Orbits
         3. Backward Orbits
         4. Periodic Orbits
         5. Time Series Representation
      8. System Parameters
         1. Role of Parameters
         2. Parameter Space
         3. Parameter Variation Effects
   2. Classification of Dynamical Systems
      1. Time Structure Classification
         1. Continuous-Time Systems
            1. Differential Equations as Flows
            2. Flow Maps
         2. Discrete-Time Systems
            1. Iterated Maps
            2. Recurrence Relations
      2. Linearity Classification
         1. Linear Systems
            1. Definition of Linearity
            2. Superposition Principle
            3. Matrix Representation
         2. Nonlinear Systems
            1. Sources of Nonlinearity
            2. Qualitative Differences
      3. Time Dependence Classification
         1. Autonomous Systems
            1. Time-Independent Dynamics
            2. Phase Space Analysis
         2. Non-autonomous Systems
            1. Time-Dependent Parameters
            2. Periodically Forced Systems
      4. Energy Classification
         1. Conservative Systems
            1. Conservation Laws
            2. Hamiltonian Structure
         2. Dissipative Systems
            1. Energy Dissipation
            2. Attracting Sets