# State of Brandon's sub-projects and to-do list
Let's keep a list of the various ideas we have about directions to take this project. Please add to it and check things off as you go.

## January 2020

### General
- [ ] Now that my goals and what needs to be done is more straightforward, I should implement the Things app. When I first tried to use it I don't feel like I understood the next steps in my project well enough to implement them.
- [ ] To this end, I need to reorganize my code and do a rewrite implementing a style guide.
- [ ] Find good Julia style guides. I have found some, and plenty for Python (PEP 8). This is important both to reduce headaches later on, but so it can be available on github and it is good practice for both jobs and postdocs to have code that is written with standard formats

- [ ] Apply to NoLineal2020. Abstracts can be submitted February 15. I think my work on LD escapes would be a good fit based off Ana María Mancho being a speaker.

### Jobs!
- [ ] Finish research statement. Needs to have a final draft before semester starts (1/21/2020) at *latest*.
- [ ] Need to make plots of the Lagrangian descriptors to replace temporary images. The quality of the images here is important.
- [ ] Need to finish text for research project 2.
- [ ] Edits from RG on draft?
- [ ] Finish rewriting CV in latex

- [ ] Make git account public.
- [ ] Make new website (Lower prioriy ).
- [ ] Update linkedin- necessary evil.

- [ ] Continue working on Python and R data-sciency stuff. Been working on this in my spare time since the summer and have gotten pretty far. Has a transferred a lot to general coding ability- and a lot of things in Julia I didn't know from my MATLAB experience came from Python.

### Escape Paper

#### Mathematics
- [x] Clean up and organize LD plots
- [x] Implement GPU accerlation using Julia. Should expect big speed up since problem is massively parallelizable.
* https://juliadiffeq.org/2019/05/09/GPU.html

#### Writing

- [ ] Make a storyboard

## List of jobs

#### Academic
A running list of all jobs to apply to
- [ ] BU https://www.mathjobs.org/jobs/jobs/15125
- [ ] Rabadan Lab https://pa334.peopleadmin.com/postings/4539
- [ ] Dimitris Giannakis https://www.mathjobs.org/jobs/jobs/15325 Start processign Jan 15th
- [ ] Buhler https://www.mathjobs.org/jobs/jobs/14619
- [ ] Wiggins https://champsproject.com/jobs/
- [ ] Lorenzo Polvani https://pa334.peopleadmin.com/postings/4873
- [ ] Casey Diekman https://njit.csod.com/ats/careersite/JobDetails.aspx?site=1&id=1453
- [ ] Microsoft (still a post doc..) https://careers.microsoft.com/us/en/job/704686/Post-Doc-Researcher-Machine-Learning?jobsource=indeed&utm_source=indeed&utm_medium=indeed&utm_campaign=indeed-feed
- [ ] Flatiron?
- [ ] CCM https://simonsfoundation.wd1.myworkdayjobs.com/en-US/simonsfoundationcareers/job/162-Fifth-Avenue/Flatiron-Research-Fellow--CCM_R0000448
- [ ] CCP https://simonsfoundation.wd1.myworkdayjobs.com/en-US/simonsfoundationcareers/job/162-Fifth-Avenue/Flatiron-Research-Fellow_R0000058-1
- [ ] Software (prob not good fit/ have 70% of skills required at moment) https://simonsfoundation.wd1.myworkdayjobs.com/en-US/simonsfoundationcareers/job/162-Fifth-Avenue/Software-Engineer_R0000570
- [ ] https://www.rfcuny.org/careers/postings?pvnID=RC-2001-003469
#### Outside the Walls
- [ ] https://nytimes.wd5.myworkdayjobs.com/en-US/NYT/job/New-York-NY/Data-Scientist_REQ-006691-1
- [ ] https://jobs.jpmorganchase.com/ShowJob/Id/256048/CIB%20AI%20Research%20%20%20Research%20Scientist
- [ ] https://jobs.jpmorganchase.com/ShowJob/Id/278268/CIB%20Global%20Research%20%20%20Applied%20AI%20%20%20Machine%20Learning%20Senior%20Associate
- [ ] https://jobs.jpmorganchase.com/ShowJob/Id/272754/CIB%20%20%20QR%20%20%20Quantitative%20Research%20Analytics%20%20%20Associate%20%20%20New%20York,%20NY#4
- [ ] https://www.citadel.com/careers/details/quantitative-researcher-full-time-us/
- [ ] https://externaltalent-mskcc.icims.com/jobs/38102/job?utm_sohttps://social.icims.com/viewjob/po157852852308555723#.XhZvCInyhKA.mailto

- [ ]https://www.moodys.jobs/TGnewUI/Search/Home/Home?partnerid=25546&siteid=5212#jobDetails=1528018_5212

- [ ] https://careers.google.com/jobs/results/74889671231840966/
- [ ] https://www.amazon.jobs/en/jobs/986874

- [ ] https://careers.google.com/jobs/results/74889671231840966/
- [ ] https://uscareers-goldmansachs.icims.com/jobs/57307/job?utm_source=indeed_integration&iis=Job+Board&iisn=Indeed&indeed-apply-token=73a2d2b2a8d6d5c0a62696875eaebd669103652d3f0c2cd5445d3e66b1592b0f&mode=job&iis=Job+Boards&iisn=Indeed&mobile=false&width=730&height=500&bga=true&needsRedirect=false&jan1offset=-300&jun1offset=-240
- [ ] https://careers.bloomberg.com/job/detail/80592?utm_medium=Bayard&utm_source=IndeedOrganic
- [ ] https://jobs.americanexpress.com/jobs/20000650?lang=en-us#.Xh3XF2aW-2Y.mailto
- [ ] https://jobs.colgate.com/colgate/job/Piscataway-Data-Scientist-NJ-08854/558028700/?feedId=168900&utm_source=Indeed&utm_campaign=Colgate_Indeed&sponsored=ppc
- [ ] https://www.amazon.jobs/en/jobs/993096/applied-scientist
- [ ] https://www.amazon.jobs/en/jobs/905238/data-scientist
- [ ] https://www.capitalonecareers.com/job/-/-/1732/14871581
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## Proposal cleanup
- [x] Make all fixes suggested by committee, including putting proper labels on Poincare sections and adding additional description of numerical procedure used to generate them.
- [x] Need to redo images for setup of leapfrogging, need to make a way to both be clear which ones are positive and which ones are negative and how they are paired. Does color represent sign or pairs? Boatto suggested using different shades of blue for negative and different shades of red for positive. First pair would be light red and light blue and second pair could be dark red and dark blue. Other idea is to make the $+$ and $-$ signs much larger and clearer.

## Stability of periodic orbit via exact formulation of Floquet problem
- [x] Write down 1 parameter family of exact jacobi-elliptic function parameterized floquet problems. **Obsolete by explicit polar formulation**
- [x] Do proper writeup of set of simulations showing that initial condition with $E=\frac{1}{8}$ eigenvalue = 1.
- [x] Investigate Panos's suggestion about Vakhitov-Kolokolov condition proof. **Seems only applicable to saddle-node, our Energy-Frequency ratio is monotonic.**
## Numerical investigations
- [x] Reimplement in Julia. Determine if faster, more reliable, more readable and MORE ORGANIZED.

- [x] Get Jupyter working- may need to reinstall OSX on my computer, wasted a lot of time over the weekend debugging a bad python installation, probably just do it again from scratch. Julia works find but the python extension which I need for graphing and using Jupyter aren't working properly. **I cheated by doing everything on my desktop, now that the semester is over I can do a fresh install to get julia working properly on it (mainly the plotting wasn't working right, probably because of a bad python install and it can't run matplotlib)**
- [x] Reorganize my papers! It has become a bit of a mess this semester. I read a lot, but I need to organize and notate which ones to revist and why.

## Harmonic Balance/Hill's Determinant
- [x] ~~For explicit system--what to do with the fourier expansion of the Jacobi functions? They are more unwiedly than expected since they depend on the Elliptic Nome, q and the elliptic integral K evaluated at the energy H. Perhaps I can taylor expand these in terms of H. (first attempt failed but I realized I used h instead of H=1/2h as the expansion parameter)~~-**Obsolete by polar explicit form**

- [x] ~~Can't write the explicit form into Hill's equation because the coefficents are not continous when trying to write it out as a second order ODE. The original matrix ceofficent is continous, but when I turn it into a second order ODE~~ **obsolete by polar explicit form**

- [x] ~~Harmonic balance with the Fourier-Taylor expansion in theta and H that I used for the Casas algorithm (this can be written as Hill's equation). Have some results- but something doesn't seem right-~~ **works now, was a misalignment in what order I took**
- [ ] Investigate Kronig-Penney Model, anything inspirational there? Essentially step functions, but maybe I can use it to make some bounds/get a new idea?
- [x] Properly explain why we need only odd terms and rewrite Mathematica code to only use the odd terms.
- [x] Clearly explain why the $A(\theta)$ is $\pi$ periodic and the periodic solution is $2\pi$-periodic (which only exits for $E=1/8$.

## Closely related questions
- [ ] At what energy level is escape to infinity possible for solutions starting close by leapfrogging orbit. Relation to destruction of KAM tori. To what extent is this answered in Whitchurch?

## More distantly related questions (in order of priority)
- [ ] Send thank yous/follow ups to Kevin Mitchell, Panos Kevrekidis, Cristina Stoica, Stephanella Boatto, Vered Rom-Kedar, Gareth Roberts.
- [x] Collect and organize references of papers in Papers. Found a lot of great papers over break but I need to organize them both in the computer and in my mind so we can discuss them!!
- [ ] Investigate Meiss's visualization methods for 2 DoF system, such as the Poincare cube to see layers of PSS.
- [ ] Use Naik paper to visualize unstable periodic orbit as NHIM
- [ ] Finish write-up on different canonical reductions to get leap-frogging equations.
- [x] Set up the Subharmonic Melnikov integral using marsden/holmes/guckenheimer-**postponed for the forseeable future**
- [x] Is the CONLEY-ZEHNDER INDEX only another way to understand floquet for a 2x2 system? Understand enough of Deng & Offin to find out. Does this provide a stability test? Try to understand the paper well enough to visualize the index so I can properly decide whether the direction is worth pursing.-**Talked to GE ROBERTS. He was very impressed that we had gone down this line of thought, but Offin himself is still trying to work out the theory for a 2x2 Floquet system.**
- [ ] The reductions for leapfrogging can be repeated on a sphere. Boatto was curious what would happen and wanted me to send her some plots. Need to hurry up with the more important things so I have time to investigate this.
- [x] ~~Panos suggested that I also investigate the change in Chaotic behavior using SALI and that it would be a good technique to get in my reptoire before I graduate.~~
- [ ] Continue to read Marsden space design/ better undestand CRTBP since it is the standard example used for many techniques.

## Paper Check list (in order)

### By Tuesday 6/18
- [x] Finish writing floquet section.
- [x] I think the Floquet theory section may need to be reordered. It feels like it goes linearized equations, floquet theory, comment again on our system. The flow feels awkward.

- [x] Include numerics showing trace=-2 at h=1/8. The orbit you find is anti-periodic so you can do harmonic balance with only odd harmonics. **Part of this has been pushed forward to weekend as I redo my numerics for section 4.2, this has been done in Matlab, but redoing in Julia with BigFloat**
- [x] The mondoromy matrix before the change of variables W=BZ has tr=2 , after (the one I do the expansion on) has $tr(M)=2$. This makes sense since at $h=0$ we have $C_0=\left(\begin{smallmatrix} 0 & -2 \\ 0 & 0 \end{smallmatrix} \right)$ so $M=e^{\pi C_0}=\left(\begin{smallmatrix} 1 & -2 \pi \\ 0 & 1 \end{smallmatrix} \right)$. Compared to the numerics I did in the paper for the original system, the Floquet multipers start at (1,0) and move along the unit circle to (-1,0).

- [x] Rewrite the Harmonic Balance section to now just reference earlier floquet analysis. Write with only odd harmonics.
- [x] Rewrote mathematica code so to only use odd harmonics and gives correct values of $h$.

- [x] Show in addition the first few truncated Hill determinants. What are their quotients and remainders when you divide through by (h-1/8)?
- [x] It appears that the remainder increase exponentially even though the root of the polynomial goes to 1/8th.

- [x] Fix period and t(X) expressions

- [x] Print out and proof read again on tuesday morning.

- [x] Add things based off Tuesday conversation.

### By Thursday 6/20

- [x] Write conclusion draft

- [x] Rewrite intro

### By Tuesday 6/25
- [x] Rewrite intro paragraph.

- [x] Include information about Hill's equation as a second order ODE.
- [x] Future work.
- [ ] Not sure how much detail to go into here. For example, when do I first mention disintegration?
- [x] Plot trace/ periodic
- [x] Make plot showing leapfrogging trajectories and initial configuration.
- [x] what is the best way to draw label figures in MATLAB?

### Later
- [x] Make plot showing leapfrogging trajectories and initial configuration. Need to decide the best way to do this, the earlier method I used was too tedius to make good figures.

- [x] Research Journals

- [ ] Can I expand out $C$ as a Fourier series rather than a Fourier-Taylor. i.e. get pure harmonic terms rather than new low-harmonics appearing as expand in $h$. I multipled the system $\dot{W}=C W$ through by the common denomitor to solve a system like $\rm{Denominator} \dot{W}=\rm{NumeratorMatrix} W$ and am able to do harmonic balance by expanding both sides out. I wasn't able to realistically Fourier expand $C$, but the numerator and denominator both seem to work fine.

- [x] Do numerics for tr=2 and recreate plots and figures
- [x] Write section 4.2 (Numerical solution of the Floquet problem)

- [x] Write abstract.

- [x] clean up including bibliography (most info is there now, but need to make sure it is displaying right).
- [x] **Spend an hour a day just rereading paper to fix phrasing/typos/grammar
**
- [x] fix font in figure labels
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