Dynamical Systems Introduction

Dynamical systems is a area of mathematics and science that studies how the state of systems change over time, in this module we will lay down the foundations to understanding dynamical systems as we talk about phase space and the simplest types of motion, transients and periodic motion, setting us up to approach the topic of nonlinear dynamical systems in the next module.

Within science and mathematics, dynamics is the study of how things change with respect to time, as opposed to describing things simply in terms of their static properties the patterns we observe all around us in how the state of things change overtime is an alternative ways through which we can describe the phenomena we see in our world.
A state space also called phase space is a model used within dynamic systems to capture this change in a system’s state overtime. A state space of a dynamical system is a two or possibly three-dimensional graph in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the state space. Now we can model the change in a system’s state in two ways, as continuous or discrete.
Firstly as continues where the time interval between our measurements is negligibly small making it appear as one long continuum and this is done through the language of calculus. Calculus and differential equations have formed a key part of the language of modern science since the days of Newton and Leibniz. Differential equations are great for few elements they give us lots of information but they also become very complicated very quickly.

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[manuelmartinelandong]

Thank you for this video. Regards from Oyala, Equatorial Guinea.

[quaidcarlobulloch]

Wow great video and great topic, make more!

[cjcrowley]

Great video, great explanation! For the most part this is correct, but there where a few things that were a bit off. For example, a dynamical system need not be dissipative. I cringed when you said "if it were not for the gravitational force between earth and the sun, the motion would cease to exist. There where a few other slight inaccuracies peppered throughout.

[izzatiazmi]

This video really2 help me out with my undergraduate project...thank you so much...

[mobiledevices]

I thought using gravity and earth revolving around the sun as a example for dissipating system was a poor if not incorrect example; Orbiting via gravity doesn't "deplete" "use" any finite input / energy, if you have infinite input, unchanging amount of input, it's not an input, it's a constant in the system. There are also no discernible differences between pre-and-post states exterior to the system. Further more, some modern science understanding has gravity as negative energy, (zero-energy hypothesis).

[Etrehumain]

Is there a difference between attractor and equilibrium ?