1. Dynamical Systems Modeling
Andrew Pendergast

2. Dynamical Systems modeling
Dynamical Systems: Mathematical object to describe behavior that changes over time
Modeling a functional relationship such that time is a primary variable wherein a value or vector function is produced
Integral in physics, mathematics, chemistry, and other applied sciences
Overall mechanisms of prediction and modeling of complex behaviors

3. Overview
Dynamic Systems as functions modeling behavior that follow multiple partial differential equations including time
Predictions of output behavior for complex systems with multiple inputs
State Space Models: Discrete vs. Continuous Modeling
Difference Equations: Analysis of a given state within discretized units (subsequent elements)
Ordinary Differential Equations & Partial Differential Equations

4. Why dynamical systems modeling?
Method for predicting complex behavior with a series of functions while allowing for discretization through difference equations
Generalized methodology for predictive behavior while allowing for recursive definition and continual optimization
Functionally applied computer science: theoretically can be done by hand but tasked with the issues of efficiency

5. Example: Dynamical Modeling: Electrochemistry
Consider the case of electrochemical interactions:
Molecules of interest
Force conditions
Solvents and interfering agents
Thermodynamic equilibrium
Kinetic equilibrium
Temporal limitations on efficiency
Fundamentally difficult to determine exact contribution of each component

6. Mathematical Approach
Determining the ∆ values of variables such that the discretization of independent shifts can be quantified
Combinatorics approach of analyzing systems of high order can be approached as a task of multiple more simple systems
Fundamentally based on the determination of end result per unit time
∆Result/∆Time or other factor
Reliance on State Space Modeling

7. State space models
State Space: Value set that can be taken as parameters for a process: individual states can be considered as instances of given variables
Ordinary Differential Equations: Allows continuous, not practical for computer space
dx/dt = f(x, u), Y = g(x, u)
Difference Equations: method to discretize time or other parameters
Xk+1 = f(xk, uk), Yk = h(xk, uk)
Yk = h(xk, uk), Yk+1 = h(xk+1 , uk+1)

8. State space model of electrochemistry
Consider group of values: each component value combines to form an “array” of values corresponding to the end result
Example: { Ca2+ / Cl- , 740 torr., 5 torr/sec, H2O / OH-, Keq, 60 seconds }, { Ca2+ / Cl- , 680 torr., 7 torr/sec, Methanol, Keq2, 45 seconds }, { Ca2+ / Cl- , 800 torr., 3 torr/sec, Ethyl Acetate / OH-, Keq3, 80 seconds }
Might produce { 9V, 12V, 3V }
Overall goal: mapping sets of values efficiently to an observed end result
Computer science allows for compilation of all data and therefore easier modeling

9. Inferential Methods What makes a good model?
Predictability, reproducibility, internal consistency
Additional issue of weighting variables and forming relationship values that not only model current behavior but predict future behavior
Advantage of computer science approach: method for evaluating potential models rapidly and efficiently through repeated simulation

10. Inferential Methods: electrochemistry
Logical approach
Application of physical and chemical equations: determine PDE for each factor
Presents chemically and physically accurate but inefficient method for analysis
CS approach
Combination of deep learning systems and mathematical analysis of different potential combinations to determine the most efficient combination of factors
Requires discretization of individual factors, specifically with respect to time

11. Mathematical approach
Primarily analyzed through continuous PDE modeling: great for computations, poor for prediction and behavior in computer systems
Viewing phenomena as classical function behavior which includes Dynamical Systems

12. Computer science approach
Bridges the gap between observed Dynamical Systems and the process of modeling dynamical systems through discretization
PDE converted into difference equations from which models can be synthesized
Broken down into set of components and final values: viewing the total effect changes in terms of the discrete changes in initial values

13. Computer science approach: applied
Given an initial set of data
Model set: yi = (β0 + β1x1i … + βpxpi) + (εi); yj = (β0 + β1x1j … + βpxpj) + (εj); etc.
Combination of statisticsal analysis and numerical optimization to determine optimal parameters based on high volume simulations compared to observed data
Account for the fact that Dynamical Systems often can have significantly different end results based on similar initial parameters

14. Electrochemistry Revisited
Determination of effect on end reaction by alteration of individual components
Combination of difference equations for each component to determine overall effect with the individual changes
Approximate “linearization” of chemical behavior to predict optimal combination of conditions given a target output or required inputs

15. How are dynamical systems applied?
1. Gather data for the desired result through experimentation into database
2. Develop initial model based on physical or mathematical phenomena
3. Test and optimize model through high volume simulations
4. Application of machine learning systems to vary factors and select for most accurate models based on simulations of step 3
5. Allow for the addition of data points in the future to allow for recursive modeling

16. Further extrapolations
Weather systems: additional problem of spherical physics
Linguistic Synthesis Simulation
Biochemical pathway optimization

17. Simple Real example: Rössler Attractor
System of 3D PDE of theoretical attractor to study the behaviors of chaotic systems
Relies on combination of linear and nonlinear (Poincaré maps of state space fields, etc) methods to determine the behaviors of the system in discretized form

18. Resources