• ES Español
• EN English

3.6. Mathematical Modeling and Simulation (MMS)

This area covers mathematical tools for describing and simulating computational and physical systems, spanning ordinary differential equations, dynamical systems, control theory, and stochastic differential equations, with direct applications to simulation engineering, robotics, and AI systems.

Knowledge Area (KA)	CS Core	KA Core
3.6.1 Ordinary Differential Equations	Elective
3.6.2 Dynamical Systems and Chaos	Elective
3.6.3 Mathematical Control Theory	Elective
3.6.4 Stochastic Differential Equations	Elective

3.6.1. MMS/Ordinary Differential Equations ↑ Back to top

Theory and solution methods for ODEs: first and higher order equations, linear systems, qualitative analysis, and Laplace transform methods used in system modeling.
Topics:
Core

• First-order ODEs: separable, linear, exact equations, and integrating factors
• Higher-order linear ODEs: characteristic equations, variation of parameters, and undetermined coefficients
• Systems of linear ODEs: matrix exponential, eigenvalue methods, and phase portraits
• Existence and uniqueness theorems (Picard-Lindelöf) and continuation of solutions
• Laplace transform methods for solving ODEs with discontinuous forcing

Learning Outcomes:
Core:

1. Classify ODEs by order, linearity, and type, and select an appropriate solution method [Familiarity]
2. Solve linear systems of ODEs using eigenvalue methods and sketch their phase portraits [Usage]
3. Apply the Laplace transform to solve ODEs with piecewise and impulsive forcing functions [Assessment]

3.6.2. MMS/Dynamical Systems and Chaos ↑ Back to top

Phase plane analysis, stability, bifurcations, limit cycles, and chaos in continuous and discrete systems, relevant to network dynamics, biological computing models, and agent systems.
Topics:
Core

• Phase portraits of 2D systems: equilibria, stability, and limit cycles
• Lyapunov stability theory and the linearization (Hartman-Grobman) theorem
• Bifurcation theory: saddle-node, pitchfork, Hopf, and global bifurcations
• Chaos: Lorenz system, sensitivity to initial conditions, and Lyapunov exponents

Learning Outcomes:
Core:

1. Sketch phase portraits of 2D autonomous systems and classify equilibria by stability [Familiarity]
2. Apply Lyapunov's method and the Hartman-Grobman theorem to assess stability of equilibria [Usage]
3. Identify bifurcation types in parameterized systems and trace the evolution of invariant sets [Assessment]

3.6.3. MMS/Mathematical Control Theory ↑ Back to top

State-space models, controllability and observability, feedback design, Lyapunov stability, and optimal control, with applications to robotics, embedded systems, and autonomous agents.
Topics:
Core

• State-space representation of linear systems: transfer functions and realizations
• Controllability and observability: rank conditions and Kalman decomposition
• Feedback stabilization: pole placement, LQR design, and the separation principle
• Lyapunov stability for nonlinear systems and input-output stability
• Optimal control: the Pontryagin Maximum Principle and Hamilton-Jacobi-Bellman equation

Learning Outcomes:
Core:

1. Determine the controllability and observability of a linear system from its state-space matrices [Familiarity]
2. Design a state-feedback controller using pole placement or LQR optimization [Usage]
3. Apply the Pontryagin Maximum Principle to solve a basic optimal control problem [Assessment]

3.6.4. MMS/Stochastic Differential Equations ↑ Back to top

Itô calculus, SDEs driven by Brownian motion, the Fokker-Planck equation, and numerical simulation of SDEs, foundational for probabilistic ML and diffusion models.
Topics:
Core

• Itô integral, quadratic variation, and Itô's formula
• Existence and uniqueness of solutions to SDEs under Lipschitz conditions
• Fokker-Planck equation: derivation and connection to SDEs
• Feynman-Kac formula and connections to parabolic PDEs
• Euler-Maruyama and Milstein schemes for numerical simulation of SDEs

Learning Outcomes:
Core:

1. State Itô's formula and apply it to compute the differential of a function of a stochastic process [Familiarity]
2. Derive the Fokker-Planck equation from a given SDE and interpret it as a probability flow [Usage]
3. Implement the Euler-Maruyama scheme to simulate trajectories of a stochastic differential equation [Assessment]