using-simulation-foundations

Differential equations suit continuous dynamics like physics or ecology; state-space modeling handles linear systems and control problems; stochastic methods capture randomness and uncertainty. This Skill routes you to the right approach based on your problem structure, whether you're building physics, AI, or economic simulations.

Theory-driven approaches use differential equations and stability analysis when your system has clear governing laws; empirical tuning works when rules are unknown or complex. This Skill clarifies the trade-off and helps you decide which strategy fits your simulation's requirements and constraints.

Numerical methods approximate solutions to differential equations when closed-form answers don't exist. This Skill guides you through continuous versus discrete modeling approaches and routes you to specific numerical techniques for physics, AI, and economic simulations with stability guarantees.

Continuous systems use differential equations and stability analysis to capture how variables evolve over time. This Skill routes you to differential-equations-for-games and stability-analysis resources with workflows for modeling ecosystem dynamics, control systems, and chaotic behavior.

Stability analysis determines whether a system converges to equilibrium or diverges; chaos describes sensitive dependence on initial conditions. This Skill helps you distinguish these behaviors and select the appropriate mathematical framework—state-space, stochastic, or nonlinear dynamics—for your simulation.