Nonlinear Dynamics And Chaos Homework Solutions

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This first course in nonlinear dynamics and chaos is aimed at upper-level undergraduate and graduate students. We will use analytical methods, concrete examples, and geometric intuition to develop the basic theory of dynamical systems, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles, and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors.

There is no required text. Instead readings will be assigned from theliterature and from several of the following books. All the books are onreserve in the library. Biophysics of Computation by C. Koch is an excellentbook that covers most of the material in the course. Foundations of Cellular Neurophysiologyby D. Johnston and S. Wu covers some of the material in a more elementaryfashion and P. Dayan and L.F. Abbott Theoretical Neuroscience provides a moremodern view of some topics. Several chapters from Methods in Neuronal Modeling(2nd ed.) edited by C. Koch and I. Segev will beused. An excellent and comprehensive book on membrane physiology is IonicChannels of Excitable Membranes, (3rd ed.) by B. Hille.This book covers ion channels in more depth than those above; it is requiredreading for anyone seriously interested in this subject. Additional referencesinclude the following: G.M. Shepherd, The Synaptic Organization of the Brain(4th ed.) is a good introduction to neural systems for persons with no previousexperience. S.H. Strogatz Nonlinear Dynamics andChaos cover aspects of nonlinear dynamics and network theory that will bediscussed in the course; some material from H.R. Wilson, Spikes Decisions andActions will also be used. The book Dynamical Systems in Neuroscience by E.M. Izhikevich provides a useful view of 2nd-order nonlinearsystems of the type used in neuroscience. A good overview of network theory isJ. Hertz, A Krogh, and R.G. Palmer, Introduction to the Theory of NeuralComputation. Finally, J.J.B. Jack, D. Noble, and R.W. Tsien,Electric Current Flow in Excitable Cells contains detailed discussions of olderwork, especially useful for cable theory.

Introduction toNonlinear Dynamics and Chaos Tu,Th 9:35-10:55 Howey S204 ChaosBook.org/predrag/courses/PHYS-4267-16/ August 23 1. IntroductionReading: Chapter 1; Chapter 2, sections 2.1-2.3 Optional reading: ChaosBook.org Brief history of chaos might amuse you. Online: Strogatz lecture 1 August 252. Flows on the lineReading: Chapter 2Online: Strogatz lecture 2 up to minute 34 Fun stuff:snowflakesProblem set #1: 2.1.5, 2.2.10, 2.3.3, 2.4.2, 2.4.9(click here)August 303. Experimental nonlinear dynamicsReading: Chapter 2Presentation: CRAB lab - Will Savoie, Andy Karsai and Dan GoldmanSeptember 14. Numerical solution of nonlinear ODEsDue in class: Homework #1(for homework solutions, go to t-square, links above)Reading: Chapter 2: Reading: Sections16.0-16.3 of Numerical Recipes by Press et al.Problem set #2: 2.5.3, 2.7.6, 2.8.3(for solutions, go to t-square)Note:There is a typo in 2.8.3 part (c): you are to plot ln(E) vs. ln(Delta t).Explain how the slope is related to the order of the method.September 65. Bifurcations in one-dimensional systems IReading: Chapter 3 Online: Strogatz lecture 2, starting at minute 34September 86. Bifurcations in one-dimensional systems IIDue in class: Homework #2(remember, always due next Thursday)Reading: Chapter 3 Online: Strogatz lecture 3Online: Strogatz lecture 4Problem set 3: 3.1.3, 3.2.6, 3.3.1, 3.4.8, 3.5.7September 13, guest lecturer Prof. F. Fentonstein7. Flows on a circleReading: Chapter 4 Fun stuff:firefliesand mosquitos synchronizationSeptember 15, guest lecturer Prof. F. Fentonstein8. Two-dimensional systemsReading: Chapter 5 Online: Strogatz lecture 5Problem set 4: 3.6.2, 3.7.6, 4.1.8, 4.3.7, 4.6.3 September 209. Phase plane analysisReading: Chapter 6 Online: Strogatz lecture 6September 2210. Conservative systemsReading: Chapter 6 Problem set 5: 5.1.10, 5.2.13, 6.1.3, 6.2.1, 6.3.1, 6.4.7Online: Strogatz lecture 7September 2711. Index theory. Limit CyclesReading: Chapter 7 Online: Strogatz lecture 8September 2912. No periodic orbits Perturbation theoryReading: Chapter 7 Problem set 6: 6.5.10, 6.6.3, 6.7.2, 6.8.5, 6.8.7 Online: Strogatz lecture 9Online: Strogatz lecture 10October 413. Nonlinear oscillators and averagingOptional reading: Bender and Orszag Chapter 7 Online: Strogatz lecture 11October 614. Bifurcations in two dimensionsReading: Chapter 8 Prof. Flavio Fenton: temporal Belousov-Zhabotinsky demonstrationProblem set 7: 7.1.6, 7.2.9, 7.3.9, 7.4.1. Due October 20.Chris says: use a computer to do plots - in the long run, it is faster, and it's what you need to learn anyhow.Online: Strogatz lecture 12October 10-11Fall breakOctober 13Mid-term exam9:35-10:55 Howey S204; see instructionsOctober 1815. Hopf bifurcationReading: Chapter 8Storgatz bores you Optional ChaosBook.orgreading (none of this will be on the final): Section 4.3 A linear diversion, Example 4.3 Linear stability of 2-dimensional flows,Example 4.4 In-out spirals,Section 5.1 Equilibria,Appendix C.2 Eigenvalues and eigenvectors. Online: Strogatz lecture 13 October 20 16. Global bifurcations of cycles Reading: Chapter 8 Due in class: Homework #7 Prof. Flavio Fenton: Belousov-Zhabotinsky spirals demonstration Problem set 8: 7.5.4, 7.6.3, 7.6.6, 7.6.17 Online: Strogatz lecture 14 October 25 17. Quasiperiodicity and Poincare maps Reading: Chapter 8 October 27 18. Floquet Theory Reading: Chapter 8 Optional reading: Nonlinear Dynamics 1: Geometry of Chaos, week 2 Problem set 9: 8.1.8, 8.1.11, 8.2.1, 8.2.9, 8.3.1 November 1 19. Lorenz equations Reading: Chapter 9, skip Sect 9.1 A Chaotic Waterwheel (Malkus and Howard were my friends, but this is not the why Lorenz work was so great!) Reading: Chapter 9, R. Grigoriev notes Optional reading: Nonlinear Dynamics 1: Geometry of Chaos, week 1 Online: Strogatz lecture 17 Numerical exploration of different dynamical regimes (: Maple! :) November 3 20. Lorenz equations Reading: Chapter 9 Problem set 10: 8.4.2, 8.4.12, 8.5.2, 8.6.1, 8.7.5 Bonus problem set 10+: Geometry of Chaos, homework 1 (email me the grade sent to you by the autograder) Online: Strogatz lecture 18 November 8 21. One-dimensional maps Reading: Chapter 10 Optional reading: Nonlinear Dynamics 1: Geometry of Chaos, week 7 Optional: Matlab simulations of the Rossler system: reduction to 2D and 1D maps and stretching of phase space volumes Online: Strogatz lecture 19 November 10 22. Universality Reading: Chapter 10, R. Grigoriev lecture notes Optional reading, ChaosBook.org bootleg chapters: Universality in transitions to chaos and Complex universality Problem set 11: assignment Online: Strogatz lecture 20 November 15 23. Charting the state space Reading: ChaosBook.org, Chapter 14 and Chapter 15, edited for PG-13 Optional viewing: Nonlinear Dynamics 1: Geometry of Chaos, week 7 November 17 24. Kneading theory Reading: Chapter 11, Sect. 11.2 (rest mostly useless 1980's flounderings) Fractals in nature, biology, and mathematics. Problem set 12: 10.1.11, 10.2.6 (see comments for 10.2.3), 10.3.11, 10.4.1 - due December 1 November 22 25. Learning how to count Reading: ChaosBook.org, Chapter 17 and Chapter 18, edited for PG-13 Optional viewing: Nonlinear Dynamics 1: Geometry of Chaos, week 9 Optional reading: Aligood, Sauer, and Yorke Problem set 13: assignment - due December 1 November 24 Holiday November 29 26. Learning how to measure Reading: ChaosBook.org, Chapter 19 and Chapter 20, edited for PG-13 Optional viewing: Nonlinear Dynamics 1: Geometry of Chaos, week 10 December 1 27. Ah, chaos! December 6 28. Wrapping it all up; overture Reading: ChaosBook.org, Chapters 21 to 23, edited for PG-13 Optional viewing: Nonlinear Dynamics 1: Geometry of Chaos, week 11 December 8 Final exam 2:50-5:40pm Howey S204; see instructions

Upon completion of this course, a student will be able to: Provide a qualitative bifurcation analysis of a simple one-dimensional, one parameter nonlinear differential equation; Understand and analyze basic types of linear and nonlinear oscillators; Linearize a two-dimensional non-linear system of differential equations at an equilibrium, and use this linearization to analyze the behavior of nearby solutions; Analyze dynamics of a two-dimensional nonlinear system of differential equations using a phase plane analysis.

Rafael worked with me to integrate some visualization and analysis capabilities specific to tracking halos in a cosmological simulation over time into AstroViz. He developed an additional module to AstroViz which can read in halo track information from an SQLite database of a standard format and visualize it in several interesting ways controlled by the user from the ParaView GUI. The final portion of the project integrated the ability to highlight tracks which experience collisions along their path as well as plotting capability for various physical quantities as a function of their collision history.Teaching: InstructorPastSummer App Space, Summer 2017.MIT MEET, Y2. Computer Science, Introduction to Java, Summer 2007.MIT MEET, Y3. Computer Science, Android Application Development, Summer 2013. MIT Global Startup Labs: University of the Philippines Diliman Program, Lead Technical Instructor, Summer 2013.Lecturer, UP Diliman Android Development Lecture Series, August, 2013.\\r\\nAssistantElectricity and Magnetism, ESG MIT, 2004.Mathematical Methods for Physicists I, Fall 2010 This course covered a broad swath of mathematical concepts: elementary calculus, linear algebra, vector calculus, Fourier series and transforms, ordinary differential equations, calculus of variations, tensor analysis, functions of a complex variable and more. We largely followed the excellent text by Prof. Mary Boas which I wholeheartedly recommend. Switching off with my co-assistant, the excellent Lucia Hosekova who has helpfully assisted the class previously, I prepared homework assignments, graded them, and gave a weekly review session. In addition, I gave a lecture on the gamma and beta functions and the Bromwich integral method of finding the inverse Laplace transform. Finally, I prepared the review sheet for the final written exam, helped give the review sessions, and prepared the final exam in consultation with both the professor, Professor George Lake and my co-assistant.\\r\\nMathematical Methods for Physicists II, Spring 2011This course took up where MMP I left off, including Sturm Liouville theory, a few more special functions, complex analysis, integral equations, group theory, nonlinear dynamics and chaos.Astrophysical Dynamics, Fall 2011Stars, planets, black holes, dark matter and more. This graduate course followed the canonical textbook in the field, Binney and Tremaine's Galactic Dynamics.Advanced Computational Science, Spring 2012Software Engineering.Quantum Mechanics I, Fall 2012 This course covered an introduction to non-relativistic single-particle quantum mechanics. In particular, the basic concepts of quantum mechanics, such as the quantisation of classical systems, the description of observables as operators on a Hilbert space, and the formulation of symmetries were discussed and illustrated with generic examples. We followed the Feynman Lectures III and Sakurai's Modern Quantum Mechanics.\\r\\n Continuum Mechanics, Spring 2013 Stress. Deformation. Strain. Elastic deformations. Stress modulus and stability of the material. The Navier equation. Elastic waves. Seismic applications. Waves in beams. Dislocations. Fluid kinematics. Conservation laws. Energy equation. Real, ideal and incompressible fluids. Helmholtz decomposition. Divergence, curl and circulation theorem. The vorticity equation. Bernoulli theorem. Sound waves. Jeans instability. Gravity wave. KH and RT instabilities. Quasi-linear waves. Shock. Velocity potential and stream function. Complex potential. Elementary solutions.Blasius formulae. Joukowsky transform. Kutta condition. Kutta-Joukowski theorem. General equations and boundary conditions. Planar, Couette and Poiseuille flows. Viscous non stationary interface. Viscous drag on a sphere. Pressure gradients. Turbolence. The equations of Magneto-Hydrodynamique. Generation of magnetic fields. Dynamics and conservation of magnetic flux. Magnetic stress. Alfven and magneto-sonic waves. Magnetic Reynolds and Pradtl number.Computational Science: Introduction to Scientific Computing I, Spring 2014Introduction to Linux and fundamental concepts in programming. Implementation of simple algorithms of numerical integration, solution of ordinary differential equations with applications in biology, chemistry and physics; Monte Carlo methods, simulation of stochastic processes (spread of disease); chaos\\/fractals (Mandelbrot set , Lorenz attractor); percolation (forest fires), partial differential equations (electron orbits in the electric field, diffusion); parallel computing.\\r\\n\\t\\n\\t\\t\\t\\n\\t\\t\\t\\t\\tPrevious\\n\\t\\t\\t\\/\\n\\t\\t\\tNext image\\n\\t\\t\\t\\t\\n\\t\\t\\t\\t()\\t\\t\\t\\t\\t\\n\\t\\t\\t\\tThumbnails\\n\\t\\t\\t\\n\\n\\n\\t\\t \\t\\t\\n \\t\\t\\n \\t\\t\\n\\t\\t\\t\\t \\n\\t\\t\\t\\t\\n\\t\\t\\n\\t\\t\\n\\t\\t\\t\\t\\t\\n\\t\\t\\n\\t\\t \\t\\n\",\"content_2x\":\"Advising:Ph.D. Thesis Committee, Omar Costa HamidoDr. Hamido completed a PhD in Music with emphasis in Integrated Composition, Improvisation, and Technology, at University of California, Irvine. I served on his Ph.D. committee. His Ph.D. explored emerging paradigms in Quantum Computing (QC) for new modes of artistic creation, and the collected works can be found on his thesis website Quantumland.art.\\r\\n\\r\\nBachelor's Thesis, Rafael K\\u00fcng, Fall 2010 Rafael worked with me to integrate some visualization and analysis capabilities specific to tracking halos in a cosmological simulation over time into AstroViz. He developed an additional module to AstroViz which can read in halo track information from an SQLite database of a standard format and visualize it in several interesting ways controlled by the user from the ParaView GUI. The final portion of the project integrated the ability to highlight tracks which experience collisions along their path as well as plotting capability for various physical quantities as a function of their collision history.Teaching: InstructorPastSummer App Space, Summer 2017.MIT MEET, Y2. Computer Science, Introduction to Java, Summer 2007.MIT MEET, Y3. Computer Science, Android Application Development, Summer 2013. MIT Global Startup Labs: University of the Philippines Diliman Program, Lead Technical Instructor, Summer 2013.Lecturer, UP Diliman Android Development Lecture Series, August, 2013.\\r\\nAssistantElectricity and Magnetism, ESG MIT, 2004.Mathematical Methods for Physicists I, Fall 2010 This course covered a broad swath of mathematical concepts: elementary calculus, linear algebra, vector calculus, Fourier series and transforms, ordinary differential equations, calculus of variations, tensor analysis, functions of a complex variable and more. We largely followed the excellent text by Prof. Mary Boas which I wholeheartedly recommend. Switching off with my co-assistant, the excellent Lucia Hosekova who has helpfully assisted the class previously, I prepared homework assignments, graded them, and gave a weekly review session. In addition, I gave a lecture on the gamma and beta functions and the Bromwich integral method of finding the inverse Laplace transform. Finally, I prepared the review sheet for the final written exam, helped give the review sessions, and prepared the final exam in consultation with both the professor, Professor George Lake and my co-assistant.\\r\\nMathematical Methods for Physicists II, Spring 2011This course took up where MMP I left off, including Sturm Liouville theory, a few more special functions, complex analysis, integral equations, group theory, nonlinear dynamics and chaos.Astrophysical Dynamics, Fall 2011Stars, planets, black holes, dark matter and more. This graduate course followed the canonical textbook in the field, Binney and Tremaine's Galactic Dynamics.Advanced Computational Science, Spring 2012Software Engineering.Quantum Mechanics I, Fall 2012 This course covered an introduction to non-relativistic single-particle quantum mechanics. In particular, the basic concepts of quantum mechanics, such as the quantisation of classical systems, the description of observables as operators on a Hilbert space, and the formulation of symmetries were discussed and illustrated with generic examples. We followed the Feynman Lectures III and Sakurai's Modern Quantum Mechanics.\\r\\n Continuum Mechanics, Spring 2013 Stress. Deformation. Strain. Elastic deformations. Stress modulus and stability of the material. The Navier equation. Elastic waves. Seismic applications. Waves in beams. Dislocations. Fluid kinematics. Conservation laws. Energy equation. Real, ideal and incompressible fluids. Helmholtz decomposition. Divergence, curl and circulation theorem. The vorticity equation. Bernoulli theorem. Sound waves. Jeans instability. Gravity wave. KH and RT instabilities. Quasi-linear waves. Shock. Velocity potential and stream function. Complex potential. Elementary solutions.Blasius formulae. Joukowsky transform. Kutta condition. Kutta-Joukowski theorem. General equations and boundary conditions. Planar, Couette and Poiseuille flows. Viscous non stationary interface. Viscous drag on a sphere. Pressure gradients. Turbolence. The equations of Magneto-Hydrodynamique. Generation of magnetic fields. Dynamics and conservation of magnetic flux. Magnetic stress. Alfven and magneto-sonic waves. Magnetic Reynolds and Pradtl number.Computational Science: Introduction to Scientific Computing I, Spring 2014Introduction to Linux and fundamental concepts in programming. Implementation of simple algorithms of numerical integration, solution of ordinary differential equations with applications in biology, chemistry and physics; Monte Carlo methods, simulation of stochastic processes (spread of disease); chaos\\/fractals (Mandelbrot set , Lorenz attractor); percolation (forest fires), partial differential equations (electron orbits in the electric field, diffusion); parallel computing.\\r\\n\\t\\n\\t\\t\\t\\n\\t\\t\\t\\t\\tPrevious\\n\\t\\t\\t\\/\\n\\t\\t\\tNext image\\n\\t\\t\\t\\t\\n\\t\\t\\t\\t()\\t\\t\\t\\t\\t\\n\\t\\t\\t\\tThumbnails\\n\\t\\t\\t\\n\\n\\n\\t\\t \\t\\t\\n \\t\\t\\n \\t\\t\\n\\t\\t\\t\\t \\n\\t\\t\\t\\t\\n\\t\\t\\n\\t\\t\\n\\t\\t\\t\\t\\t\\n\\t\\t\\n\\t\\t \\t\\n\",\"content_no_html\":\"Advising:Ph.D. Thesis Committee, Omar Costa HamidoDr. Hamido completed a PhD in Music with emphasis in Integrated Composition, Improvisation, and Technology, at University of California, Irvine. I served on his Ph.D. committee. His Ph.D. explored emerging paradigms in Quantum Computing (QC) for new modes of artistic creation, and the collected works can be found on his thesis website Quantumland.art.\\r\\n\\r\\nBachelor's Thesis, Rafael K\\u00fcng, Fall 2010 Rafael worked with me to integrate some visualization and analysis capabilities specific to tracking halos in a cosmological simulation over time into AstroViz. He developed an additional module to AstroViz which can read in halo track information from an SQLite database of a standard format and visualize it in several interesting ways controlled by the user from the ParaView GUI. The final portion of the project integrated the ability to highlight tracks which experience collisions along their path as well as plotting capability for various physical quantities as a function of their collision history.Teaching: InstructorPastSummer App Space, Summer 2017.MIT MEET, Y2. Computer Science, Introduction to Java, Summer 2007.MIT MEET, Y3. Computer Science, Android Application Development, Summer 2013. MIT Global Startup Labs: University of the Philippines Diliman Program, Lead Technical Instructor, Summer 2013.Lecturer, UP Diliman Android Development Lecture Series, August, 2013.\\r\\nAssistantElectricity and Magnetism, ESG MIT, 2004.Mathematical Methods for Physicists I, Fall 2010 This course covered a broad swath of mathematical concepts: elementary calculus, linear algebra, vector calculus, Fourier series and transforms, ordinary differential equations, calculus of variations, tensor analysis, functions of a complex variable and more. We largely followed the excellent text by Prof. Mary Boas which I wholeheartedly recommend. Switching off with my co-assistant, the excellent Lucia Hosekova who has helpfully assisted the class previously, I prepared homework assignments, graded them, and gave a weekly review session. In addition, I gave a lecture on the gamma and beta functions and the Bromwich integral method of finding the inverse Laplace transform. Finally, I prepared the review sheet for the final written exam, helped give the review sessions, and prepared the final exam in consultation with both the professor, Professor George Lake and my co-assistant.\\r\\nMathematical Methods for Physicists II, Spring 2011This course took up where MMP I left off, including Sturm Liouville theory, a few more special functions, complex analysis, integral equations, group theory, nonlinear dynamics and chaos.Astrophysical Dynamics, Fall 2011Stars, planets, black holes, dark matter and more. This graduate course followed the canonical textbook in the field, Binney and Tremaine's Galactic Dynamics.Advanced Computational Science, Spring 2012Software Engineering.Quantum Mechanics I, Fall 2012 This course covered an introduction to non-relativistic single-particle quantum mechanics. In particular, the basic concepts of quantum mechanics, such as the quantisation of classical systems, the description of observables as operators on a Hilbert space, and the formulation of symmetries were discussed and illustrated with generic examples. We followed the Feynman Lectures III and Sakurai's Modern Quantum Mechanics.\\r\\n Continuum Mechanics, Spring 2013 Stress. Deformation. Strain. Elastic deformations. Stress modulus and stability of the material. The Navier equation. Elastic waves. Seismic applications. Waves in beams. Dislocations. Fluid kinematics. Conservation laws. Energy equation. Real, ideal and incompressible fluids. Helmholtz decomposition. Divergence, curl and circulation theorem. The vorticity equation. Bernoulli theorem. Sound waves. Jeans instability. Gravity wave. KH and RT instabilities. Quasi-linear waves. Shock. Velocity potential and stream function. Complex potential. Elementary solutions.Blasius formulae. Joukowsky transform. Kutta condition. Kutta-Joukowski theorem. General equations and boundary conditions. Planar, Couette and Poiseuille flows. Viscous non stationary interface. Viscous drag on a sphere. Pressure gradients. Turbolence. The equations of Magneto-Hydrodynamique. Generation of magnetic fields. Dynamics and conservation of magnetic flux. Magnetic stress. Alfven and magneto-sonic waves. Magnetic Reynolds and Pradtl number.Computational Science: Introduction to Scientific Computing I, Spring 2014Introduction to Linux and fundamental concepts in programming. Implementation of simple algorithms of numerical integration, solution of ordinary differential equations with applications in biology, chemistry and physics; Monte Carlo methods, simulation of stochastic processes (spread of disease); chaos\\/fractals (Mandelbrot set , Lorenz attractor); percolation (forest fires), partial differential equations (electron orbits in the electric field, diffusion); parallel computing.\\r\\n{slideshow}{image 5}{image 4}{image 2}{image 1}{\\/slideshow}\",\"content_partial_html\":\"Advising:Ph.D. Thesis Committee, Omar Costa HamidoDr. Hamido completed a PhD in Music with emphasis in Integrated Composition, Improvisation, and Technology, at University of California, Irvine. I served on his Ph.D. committee. His Ph.D. explored emerging paradigms in Quantum Computing (QC) for new modes of artistic creation, and the collected works can be found on his thesis website Quantumland.art.\\r\\n\\r\\nBachelor's Thesis, Rafael K\\u00fcng, Fall 2010 Rafael worked with me to integrate some visualization and analysis capabilities specific to tracking halos in a cosmological simulation over time into AstroViz. He developed an additional module to AstroViz which can read in halo track information from an SQLite database of a standard format and visualize it in several interesting ways controlled by the user from the ParaView GUI. The final portion of the project integrated the ability to highlight tracks which experience collisions along their path as well as plotting capability for various physical quantities as a function of their collision history.Teaching: InstructorPastSummer App Space, Summer 2017.MIT MEET, Y2. Computer Science, Introduction to Java, Summer 2007.MIT MEET, Y3. Computer Science, Android Application Development, Summer 2013. MIT Global Startup Labs: University of the Philippines Diliman Program, Lead Technical Instructor, Summer 2013.Lecturer, UP Diliman Android Development Lecture Series, August, 2013.\\r\\nAssistantElectricity and Magnetism, ESG MIT, 2004.Mathematical Methods for Physicists I, Fall 2010 This course covered a broad swath of mathematical concepts: elementary calculus, linear algebra, vector calculus, Fourier series and transforms, ordinary differential equations, calculus of variations, tensor analysis, functions of a complex variable and more. We largely followed the excellent text by Prof. Mary Boas which I wholeheartedly recommend. Switching off with my co-assistant, the excellent Lucia Hosekova who has helpfully assisted the class previously, I prepared homework assignments, graded them, and gave a weekly review session. In addition, I gave a lecture on the gamma and beta functions and the Bromwich integral method of finding the inverse Laplace transform. Finally, I prepared the review sheet for the final written exam, helped give the review sessions, and prepared the final exam in consultation with both the professor, Professor George Lake and my co-assistant.\\r\\nMathematical Methods for Physicists II, Spring 2011This course took up where MMP I left off, including Sturm Liouville theory, a few more special functions, complex analysis, integral equations, group theory, nonlinear dynamics and chaos.Astrophysical Dynamics, Fall 2011Stars, planets, black holes, dark matter and more. This graduate course followed the canonical textbook in the field, Binney and Tremaine's Galactic Dynamics.Advanced Computational Science, Spring 2012Software Engineering.Quantum Mechanics I, Fall 2012 This course covered an introduction to non-relativistic single-particle quantum mechanics. In particular, the basic concepts of quantum mechanics, such as the quantisation of classical systems, the description of observables as operators on a Hilbert space, and the formulation of symmetries were discussed and illustrated with generic examples. We followed the Feynman Lectures III and Sakurai's Modern Quantum Mechanics.\\r\\n Continuum Mechanics, Spring 2013 Stress. Deformation. Strain. Elastic deformations. Stress modulus and stability of the material. The Navier equation. Elastic waves. Seismic applications. Waves in beams. Dislocations. Fluid kinematics. Conservation laws. Energy equation. Real, ideal and incompressible fluids. Helmholtz decomposition. Divergence, curl and circulation theorem. The vorticity equation. Bernoulli theorem. Sound waves. Jeans instability. Gravity wave. KH and RT instabilities. Quasi-linear waves. Shock. Velocity potential and stream function. Complex potential. Elementary solutions.Blasius formulae. Joukowsky transform. Kutta condition. Kutta-Joukowski theorem. General equations and boundary conditions. Planar, Couette and Poiseuille flows. Viscous non stationary interface. Viscous drag on a sphere. Pressure gradients. Turbolence. The equations of Magneto-Hydrodynamique. Generation of magnetic fields. Dynamics and conservation of magnetic flux. Magnetic stress. Alfven and magneto-sonic waves. Magnetic Reynolds and Pradtl number.Computational Science: Introduction to Scientific Computing I, Spring 2014Introduction to Linux and fundamental concepts in programming. Implementation of simple algorithms of numerical integration, solution of ordinary differential equations with applications in biology, chemistry and physics; Monte Carlo methods, simulation of stochastic processes (spread of disease); chaos\\/fractals (Mandelbrot set , Lorenz attractor); percolation (forest fires), partial differential equations (electron orbits in the electric field, diffusion); parallel computing.\\r\\n\\t\\n\\t\\t\\t\\n\\t\\t\\t\\t\\tPrevious\\n\\t\\t\\t\\/\\n\\t\\t\\tNext image\\n\\t\\t\\t\\t\\n\\t\\t\\t\\t()\\t\\t\\t\\t\\t\\n\\t\\t\\t\\tThumbnails\\n\\t\\t\\t\\n\\n\\n\\t\\t \\t\\t\\n \\t\\t\\n \\t\\t\\n\\t\\t\\t\\t \\n\\t\\t\\t\\t\\n\\t\\t\\n\\t\\t\\n\\t\\t\\t\\t\\t\\n\\t\\t\\n\\t\\t \\t\\n\",\"date\":\" 2010\",\"bgcolor\":\"\",\"comment_count\":0,\"thumb_url\":\"https:\\/\\/payload.cargocollective.com\\/1\\/5\\/178500\\/2444891\\/prt_1323698183.jpg\",\"thumb_url_2x\":\"https:\\/\\/payload.cargocollective.com\\/1\\/5\\/178500\\/2444891\\/prt_1323698183.jpg\",\"thumb_url_4x\":\"https:\\/\\/payload.cargocollective.com\\/1\\/5\\/178500\\/2444891\\/prt_1323698183.jpg\",\"thumb_width\":200,\"thumb_height\":134,\"default_thumb_url\":\"https:\\/\\/payload.cargocollective.com\\/1\\/5\\/178500\\/2444891\\/prt_1323698183.jpg\",\"default_thumb_url_2x\":\"https:\\/\\/payload.cargocollective.com\\/1\\/5\\/178500\\/2444891\\/prt_1323698183.jpg\",\"default_thumb_url_4x\":null,\"default_thumb_width\":200,\"default_thumb_height\":134,\"custom_thumb_url\":\"https:\\/\\/payload.cargocollective.com\\/1\\/5\\/178500\\/2444891\\/prt_1323698183.jpg\",\"custom_thumb_url_2x\":null,\"custom_thumb_url_4x\":null,\"custom_thumb_width\":200,\"custom_thumb_height\":134,\"sort\":22,\"set_id\":13902874,\"set_name\":\"Academic & Instruction\",\"set_key\":\"Academic-Instruction\",\"set_url\":\"https:\\/\\/www.christinecorbettmoran.com\\/Academic-Instruction\",\"set_count\":null,\"is_set\":false,\"views\":\"10,141\"} var _gaq = _gaq []; _gaq.push(['_setAccount', 'UA-27675639-1']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ' ' : ' ') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })();Filter: haloviz view allAcademic & InstructionTeaching and AdvisingAbout ContactCV/ResumeAdvising:Ph.D. Thesis Committee, Omar Costa HamidoDr. Hamido completed a PhD in Music with emphasis in Integrated Composition, Improvisation, and Technology, at University of California, Irvine. I served on his Ph.D. committee. His Ph.D. explored emerging paradigms in Quantum Computing (QC) for new modes of artistic creation, and the collected works can be found on his thesis website Quantumland.art.Bachelor's Thesis, Rafael Küng, Fall 2010 Rafael worked with me to integrate some visualization and analysis capabilities specific to tracking halos in a cosmological simulation over time into AstroViz. He developed an additional module to AstroViz which can read in halo track information from an SQLite database of a standard format and visualize it in several interesting ways controlled by the user from the ParaView GUI. The final portion of the project integrated the ability to highlight tracks which experience collisions along their path as well as plotting capability for various physical quantities as a function of their collision history. 153554b96e