Nonlinear Dynamics: Introduction to Nonlinear Dynamics

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After a quick overview of the field and its history, we review the basic background that students need in order to succeed in this course. We then dig deeper into the dynamics of maps—discrete-time dynamical systems—encountering and unpacking the notions of state space, trajectories, attractors and basins of attraction, stability and instability, bifurcations, and the Feigenbaum number. We then move to the study of flows, where we revisit many of the same notions in the context of continuous-time dynamical systems. Since chaotic systems cannot, by definition, be solved in closed form, we spend some time thinking about how to solve them numerically, and learning what challenges arise in that process. We then learn about techniques and tools for applying all of this theory to real-world data and close with a number of interesting applications: control of chaos, prediction of chaotic systems, chaos in the solar system, and uses of chaos in music and dance.

In each unit of this course, students will begin with paper-and-pencil exercises regarding the corresponding topics, and then write computer programs that operationalize the associated mathematical algorithms. This will not require expert programming skill, but you should be comfortable translating basic mathematical ideas into code. Any computer language that supports simple plotting—points on labelled axes—will suffice for these exercises. We will not ask you to turn in your code, but simply report and analyze the results that your code produces.

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This channel is just awesome; I'm a starter in nonlinear dynamics and this is pure gold! Thank you for sharing

pedromanuelcastorena

Started a class on Nonlinear Dynamics at UC Santa Cruz today, I thought the first class was quite interesting. I was really surprised to hear you mention UC Santa Cruz in your video! I did not know that they had what seems like a somewhat large role in the study of this field. I will have to ask my professor about the Eudaemons!

cjhuffpuff

I am so excited for these YouTube videos!

pasdechat

This sounds very nice and perfect for between my physics semesters.
I just can't find the course on your website?

Joe

This might be a stupid question but is it possible that NLD&C takes place at an atomic or even quantum level? Meaning that the building blocks of life are pre-determined?

nickpitts

How do we know that these plots look the way the actually do? What I mean is that I assume these plots are generated through computer simulation, which requires us to select an initial starting value. Since the system is so chaotic, these initial values are very important. Since computers are only able to represent numbers with finite precision, that means all the initial conditions we choose will be rational numbers (is that correct?), which is a meager set of all real numbers. For example, how do we know that initial condition with transcendental numbers will not produce a completely different looking plot?

xichenjiang

I am having trouble understanding the difference between nonlinear systems that are not chaotic and those that are. They both have the nonlinear characteristic that a change in the input is not proportional to the change in the output, but isn't this exactly what sensitivity to initial conditions is? I guess then my question moreso pertains to: what is the difference between nonlinear characteristics and sensitivity to initial conditions? They both represent the idea that with small changes, big effects can occur....

luisanarodriguez