October 21, 2022

"Dynamism," the quality of being characterized by vigorous activity and progress, is the basis of the function of the entire world. A dynamical system is a mathematical model that describes the behavior of a man-made or natural system. It generally models any phenomenon or process with quantities that change over time, e.g., fish growing in a pond, water flowing in a pipe, fuel combustion in an engine. Dynamic systems and models serve many important purposes, most importantly, prediction where they are used to calculate the future conditions of the situations they model. For instance, you wouldn't be able to tell if it would rain tomorrow or next if dynamical systems and models did not exist.

Dynamical systems are a core foundation of every evolving situation in the real world. From engineering and space travel, environmental observation and artificial intelligence, to modern medicine and societal processes, dynamical systems are relevant in basically every area of life. Any situation or system that is subject to change with time can be modeled as a dynamical system.

Without these systems, predicting, diagnosing, or troubleshooting real-world situations would practically be impossible. Therefore, it is important to understand the dynamical systems perspective so we can better make the world around us better. For example, modeling dynamical systems can help leaders and companies innovate on their hardware products faster and with less risk. It can also help governments develop and implement policy improvements that save lives and improve humanity by leaps and bounds - e.g., in 2020, understanding how COVID spreads allowed governments to come up with better and more informed policies on how to contain the spread of the virus.

The dynamical systems theory is the unifying framework used in the study of all complex dynamical systems. It has roots in mathematics, astronomy, physics, meteorology, and biology. It describes the behavior of each dynamical system using a differential equation, generally represented as:

$$f(x) = dx/dt$$

This describes the rate of change of a quantity x with respect to time t - a simple summary of the core behavior of a dynamical system.

The dynamic systems theory has found extensive use and vast application in basically every technology-based industry in the world. Across mechanical, aerospace, civil, chemical and biomedical engineering, dynamical systems theory plays an important role in modeling and simulation of heavy-machinery industrial processes, rocket take off and landing, autonomous vehicle operation, etc. to optimize costs, enhance safety and improve efficiency.

For example, in fluid mechanics where internal states of every fluid changes over time, one can accurately predict the parameters and behavior of all particles as the fluid flows through different conditions (temperatures, pressures, velocities), the effect the container has on the fluid, and the effect of dormancy on both the fluid and the container.

Mathematical models of real-world situations enable better awareness, understanding and intervention across several different fields of study. For instance, modeling bacterial growth curves over time was the basis of the creation of modern day antibiotic medications. Other examples of dynamical systems include mathematical representations of the following situations:

- Pendulum swings
- Investment maturity
- Economic decline
- Global warming

The state space is the encompassing set of all possible states of the system at any specified point in time. Determined by parameters called 'state variables', state space equations show abstractions of what can be expected from the system at any time.

To define a dynamical system, parameters or initial conditions must be identified. Depending on the type of situation being modeled, some of the initial conditions or parameters may remain constant, such as the size of a water tank.****

State variables directly describe events of the state space. State variables are measurable or visible quantities that provide information about the current state of a system. Hence, all possible state variables combine to form the state space.

State variables are highly subject to change in most real-world systems, such as the temperature of a heating tank or the velocity of a missile. However, they can sometimes be programmed to remain constant over time while still producing visible effects on the system.

Some of the state variables that would be modeled and simulated for a rocket during take off, flight and re-entry include:

- Acceleration
- Velocity
- Altitude
- Fuel consumption rate
- Interior pressure
- Rocket-body temperature
- Exhaust velocity of the gasses
- External forces (gravity and drag)

Rockets are some of the most extremely advanced aeronautical and mechanical systems in the entire world, and as a result, these state variables are typically highly dynamic and subject to remarkable changes.

The term "phase space" is often used interchangeably with "state space", however, it is slightly different and mostly used to describe continuous dynamical systems. Phase space describes the identical underlying space in which all possible states of a system are represented, just with a different set of coordinates. For example in classical mechanics, it consists of a set of all possible position coordinates and momenta of a system.

The dynamical system approach to prediction and diagnosis is so vast that there are several categories for modeling different types of situations.

Types of dynamical systems include:

- Discrete and continuous - classified according to time evolution
- Deterministic and stochastic - classified according to the randomness or behavioral factor

The common thread across is that all these types of dynamical systems will change and evolve over time.

Discrete dynamical systems are models of situations where the evolution occurs in discrete or uniform timestamps. This type of dynamical system usually consists of an array of state variables (x1, …, xn), where the change must occur with each time step, n. For example, if the rate of change is 2 seconds, 2 minutes, 1 years, or 2 years, the change occurs in the specified time range based on its rules and the changes in between the time steps are either non-existent or ignored.

Discrete dynamical systems are a primary characteristic of man-made systems where the time of change is programmed, such as factory processes. This is because computers are programmed in binary (0s and 1s), therefore, the time steps need to be well-defined for it to be implemented in an electronic or digital system.

Discrete time systems are easier to model, simpler to study, and more convenient to deploy to existing computers and programmable logic controllers (PLCs).

Examples of situations that can be modeled with discrete dynamical systems include:

- Factory processes (e.g. batch packaging)
- Time-stamped industrial processes (repetitive manufacturing)
- The biological bacteria doubling process (binary fission in a specified time, depending on the type of bacteria)
- Investment profit accumulation (every 30 days, a specified amount tops your capital)

Continuous dynamical systems are models of situations where the evolution occurs in continuous or wavy timestamps. It still occurs over a specified rule but there is no definite time stamp for changes to happen. They will either constantly occur, even in invisible bits. Continuous dynamical systems are abundant in biological and natural processes.

Due to the absence of vacant time steps, continuous dynamical systems are often harder to model and simulate. However, they are often estimated with relative precision as discrete systems with variable time steps.

Examples of situations that can be modeled with continuous dynamical systems include:

- Vehicle wheels in motion
- Water flowing out of a tap and into a tank
- Environmental weather and climate progression
- Continuous industrial processes (dispensation, mixing, combustion, etc.)

A deterministic dynamical system is one that allows no room for a variety of outputs. It's coined from the word determinism, which means no "free will". It is usually a discrete type of system where the variables and inputs in a given process must produce a unique and unchanging set of outputs, with very little to no randomness allowed. Ideally, it is 100% randomness free.

A dynamical system uses a deterministic model if the preset state can be determined uniquely from the past states. It has a very high predictability attribute and it's easy to determine what the future states will be. Deterministic systems are simple to study, easy to understand, but not very useful for modeling intricately complex systems.

A real-world example of a deterministic system is the use of formulae in mathematics and physics. To convert temperatures in degrees Celsius to Fahrenheit, multiply by 1.8 and add 32. For as long as the calculations are performed accurately, the answers for any given set of initial values will always be constant. There is no possibility of randomness except human errors that can be easily identified and rectified. Any form of mathematical conversion using an established formula is always deterministic. The predictability is high and reliability is unwavering.

The stochastic dynamical system is the direct opposite of the deterministic system. Practically, it is described as a system that can be inherently affected by "noise" or interference. Stochastic systems can handle uncertainties. Their outputs aren't set in stone and are prone to inconsistencies caused by inherent randomness.

Stochastic models are a class of non-linear dynamical systems where the signals are constantly varying while they accumulate sensory information over time. Stochastic systems are suitable for modeling complex processes where the predictability function is not 100% reliable.

Stochastic dynamical systems can either be discrete or continuous according to time variation.

A real-world example of a stochastic model system can be found in the financial technology industry. The systems used for trading stocks, cryptocurrencies, and other non-fixed rate financial investments are built to accommodate the very high levels of volatility. Stocks have no 100% perfect predictability model, if any at all. They can go from 10% to -5% in one second, and back up again in an hour. The randomness can only be managed by a designated stochastically modeled system.

Dynamical analysis includes the broad range of activities used in studying and observing a dynamical system. It's the technical process of extracting the features of a dynamical system, predicting or determining future states of the system, diagnosing current or possible problems, and fixing or troubleshooting anomalies within the system. Dynamical analysis is an essential process because it's used in the accurate determination of all possible inputs and outputs of a system or invention. In the real-world, dynamic system analysis can be used to perform extremely important predictability functions, such as prevention of natural disasters, system failure, etc.

A few common terms thrown around a lot in the dynamical system study:****

The Chaos theory is an extremely important real-world idea that's often misunderstood and wrongly described. It is generally misconceived to represent random and wild behavior of systems. However, chaos represents randomness or unpredictable results arising from normal equations because of the complexity of the systems involved. The system was expected to be stable, but under normal conditions, the output goes off course. It describes a deterministic system with massive randomness. It can also be defined as the point at which stability moves to instability and non-linearity begins.

The applications of chaotic dynamical systems in modern technology are limitless and wide, from fluid mechanics and structural dynamics to modeling many big data systems.

An attractor is the stable set of physical properties or states towards which a dynamical system evolves over time. Long-term behavior of the system is dependent on the attractor. It is a set of positions in the state space, to which all state variables from a certain vicinity move toward with time.

For example, in rocket propulsion, an attractor is the atmosphere which the rocket relies on for its performance through flight and re-entry. A solid understanding of the attractor of a system is essential to prediction and diagnosis.

Perturbation is a small change in the phase space of a dynamical system, mostly in a physical system that is disturbed from the outside when at equilibrium. It's likened to a small nagging disturbance that occurs out of nowhere and might actively interrupt the smooth flow of activity in the system, or causes the system to re-adjust to accommodate it.

Practically, the perturbation theory is applied in devising methods and techniques of keeping systems actively running under the influence of possibly constant disturbances.

The dynamic system approach presents a myriad of advantages to engineers and process developers. Some of the benefits that come with modeling processes into dynamical systems include:

- Getting insights into real-world situations
- Error correction in industrial and technological processes before implementation
- Forecasting for disaster prevention and preparation (such as storm forecasting and shelter provisions)
- Predicting outcomes of physical processes

While their benefits cannot be over-emphasized, modeling dynamical systems isn't always the easiest of processes. A few challenges with the process:

- It can be complex to create a virtual representation of a system
- It can be time consuming to generate all the differential equations that go into modeling a system with a high level of precision
- It can be compute intensive to simulate the system at the level of fidelity required to get good insights
- It's often difficult to completely account for all the randomness possibilities

Dynamical systems are a foundational part of every technological process in the world today. It has found versatile use in several industries including automotive, aerospace and defense, industrial, manufacturing, and energy. Being able to model dynamical systems faster and more accurately is a competitive advantage for many companies today.

Collimator presents a unique solution to the challenges faced by engineers trying to model dynamical systems:

- Graphical user interfaces to simplify modeling of dynamical systems - especially those with feedback
- Reusable function blocks at a useful level of abstraction such as engines, drivetrains, vehicle dynamics, etc.
- High performance compute so you can run your simulations at a high enough level of fidelity for it to be useful
- System identification and AI tools to automatically define system level behavior in case you do not have the corresponding differential equations or when they are too complex to define
- Ability to stream data in and data out via API directly to your database
- Ability to integrate and co-simulate with other 3rd party tools for further analysis, e.g., in 3D simulation environments such as Carla and AirSim

Find out more about how Collimator can help you model and simulate your own dynamical systems by booking a meeting with our application engineering team.