Dynamical Systems: a window to understand every changes

Dynamical Systems: a window to understand every changes

I am trying to write something for our university students (specifically for STEM students) with my very little knowledge about Dynamical Systems. STEM students should start learning this specific field; this will motivate them to carry on mathematical modelling related stuff in future. Whatever definition, examples, history, applications I am going to write below, are not actually sufficient to entirely describe this field. Researchers who have expertise in this field can describe better than me. Nevertheless, the following is based on my understanding:

Let’s first try to look at the definition of dynamical systems. Mathematically, a dynamical system is a framework, which is used to analyse how a system changes over time. It basically deals with variables that change according to specific rules with time; such rules are usually in the form of differential equations, difference equations, or maps. If we want to know how epidemics / pandemics happen, how cancer-tumors grow and spread, how bacteria acquire drug resistance, how the numbers of predators and prey fluctuate, how weather patterns form and change, how satellites maintain stable orbits, how stock markets fluctuate, how rumours / misinformation / disinformation go viral, how robots walk stably, and many more, dynamical systems will help us in such scenarios to understand the inside dynamics.

If we try to know the behind history of this field, then we will understand that the early ideas date back to Newton's time, but the modern dynamical systems theory has actually evolved from the works of Leibniz, Euler, Lagrange, Laplace, Hamilton, Verhulst, Poincaré, Lyapunov, Lorenz, Lotka, Volterra, and many others. Some of the well-known terms / subfields of dynamical systems could be: hamiltonian dynamics, stability theory, ergodic theory, bifurcation theory, control theory and chaos theory. Mathematical models such as Lorenz system, Lotka-Volterra model, Logistic growth model, Van der Pol oscillator, Duffing equation etc. are some of the very popular examples from this field.

When to talk about the applications of dynamical systems, I always say one short sentence: ''it’s vast.'' We can think about many areas where dynamical systems play crucial roles to describe the underlying facts. Therefore, applications are everywhere, we can’t write all those down. However, applications can be observed in physics (planetary motions, electromagnetic systems, thermodynamics); medicine and biology (cardiac dynamics, epidemiology, neuroscience, population dynamics); chemistry (chemical reactions, oscillating reactions); engineering (control systems, robotics, electric circuits); economics (stock markets, optimization); ecology and environment (climate change, weather forecast, species interactions). Ideas of dynamical systems are also being used in network science, data science, machine learning and artificial intelligence.

So lastly, I must say that dynamical systems is a branch of mathematics, which we can’t ignore; it is actually everywhere. It is indeed a very powerful tool for understanding the changing events in the real world around us. Through the concepts of dynamical systems, we can create real models, predict the future, control and analyse complex systems. That is why, our students can dive into this field and feel its beauty through their studies and research.

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(written by Md. Azmir Ibne Islam)

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