CS-E5755 - Nonlinear Dynamics and Chaos D, 13.01.2021-12.04.2021
This course space end date is set to 12.04.2021 Search Courses: CS-E5755
Topic outline
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The Book FilePDF document
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This covers Chapters 1 and 2 of Strogatz.
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Most of Chapter 3 in Strogatz.
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Strogatz: the end of Chapter 3, parts of Chapter 4.
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For those interested. This is "bonus material" - not required in the exam.
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Strogatz: Ch 5 and part of Ch6.
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Strogatz: Chapter 6.
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Strogatz, Ch 7.
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On the 5th lecture I was puzzled by the claim in the book that trajectories leave nodes tangential to the slow direction, because if y is the slow direction and close to the node, say at (0, 0), x(t) = exp(A*t) and y(t) = exp(B*t), where A > B, then of course x grows faster in time than y , which contradicts the claim. Indeed as my fiddling around in the notebook here shows the trajectory's direction is more aligned with the x-axis than y-axis. So, the statement "trajectories are tangential to the slow eigendirection" has to be interpreted as "trajectories are more tangential to the slow eigendirection close to the node than farther away from it" - because asymptotically trajectories will be aligned with the fast eigendirection.
The above notion on trajectories not being strictly tangential to the slow direction applies also for stable nodes. Reversing time will make the system follow the same trajectories but in reverse direction, turning unstable nodes stable for the reverse time, and this applies to all systems we look at. Reversibility of a system is an additional property which states that after the time reversal the system looks completely identical to the non-reversed system.
In your sketches you can draw trajectories aligned with the slow eigendirection close to nodes just to show the qualitative behaviour, but strictly this is not quite so, and Strogatz's statement here is, in fact, wrong.
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Strogatz, Ch 8.
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Strogatz: Part III Chaos, Ch. 9 Lorenz Equations.
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Strogatz, Ch. 10.
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The last one. Strogatz, Chapters 11 and 12.
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