What is dynamic systems modeling? A practical introduction

A dynamic system is anything whose condition changes over time because of its own rules and its surroundings: a room warming up, a battery draining, a bank losing deposits, a robot arm swinging. Dynamic systems modeling means writing those rules down so a computer can step the system forward and show you what happens.

The three ingredients

Every dynamic model has the same three parts.

• State. The variables that describe the system right now, such as a temperature, a stock level or a joint angle.
• Rules. Equations that say how the state changes. In continuous time these are usually differential equations. In discrete steps they are difference equations.
• Inputs. Outside influences, such as a heater switched on or a surge in demand.

A hot plate warming a room shows the idea. The state is the air temperature. The rule says the air warms at a rate proportional to the gap between the plate and the air, which you could write as dT/dt = k × (T_plate − T_air). The input is the plate being on or off. Step the rule forward in time and you get a curve that rises and levels off.

Common ways to build the model

There is no single right way. The main approaches differ in how they ask you to describe the rules.

• Equations in code. You write the differential equations directly in a language such as Python, MATLAB or Julia. This is flexible, but the structure of the system is hidden inside the code.
• Block diagrams. You connect blocks that pass signals to each other, as in Simulink. This suits control systems, where signal flow is the point.
• Stock-and-flow models. You draw quantities that accumulate and the flows between them. This suits business and policy questions.
• Acausal component models. You connect physical components such as resistors, springs and tanks, as in Modelica tools. The software works out the equations.
• Agent-based and discrete-event models. You describe individual actors or events rather than continuous quantities.
• Graph-based models. Each component is a node that holds its own state, and each relationship is an edge describing how one node affects another. The structure is the model.

When a graph fits

A graph is a good fit when the structure of the system matters. If many parts interact in similar ways, a graph shows you at a glance what depends on what. It also lets you swap one relationship without rewriting the rest, and it makes it natural to ask what happens if a link fails.

A typical workflow

1. Define the state. List the quantities that matter and give each a starting value.
2. Write the relationships. Describe how each quantity is affected by the others.
3. Choose the time span and step. Decide how long to run and how finely to step.
4. Run it. Watch how each state evolves.
5. Compare with data. Check the output against measurements, and adjust constants until they agree.
6. Refine. Add the pieces you left out, and change the structure if the data demands it.

Steps 2 and 5 are where most of the effort goes. If you already have measurements, some tools can fit a known equation's constants to the data, or even propose which relationships exist from the data alone.

Dynamic systems modeling in Konjugate

Konjugate is open-source software for building dynamic systems models as graphs. You build the graph visually, write relationships as equations or in C++ or Python, and run it on a native engine. It can fit constants to data and recover structure from a CSV of measurements.

The browser version at web.konjugate.com is a quick demo without all the features. For the full experience, download the desktop app from konjugate.com. To build a first model in about five minutes, follow "Getting started with Konjugate".