Index: Numerical files Nonlinear Dynamics YFX1520 Index of .nb files The files (totalling 80) are used during the lectures to demonstrate the nonlinear phenomena dis- cussed. The .nb files can be opened with Wolfram Mathematica software. Use, modify and distribute under CC BY 4.0 licence. Lecture 1 Link File name Description nb#1 mag pendulum.nb Numerical integration of dynamical system describing magnetic pendulum's mo- tion. A pendulum in which a metal ball (or magnet) is attached to its end and which is oscillating over a plane where a set of attractive magnets are present (2{6 magnets). nb#2 numerics lecture1.nb Analytic solution to a nonlinear ODE. Numerical solution and phase portrait of 1-D flowx _ = sin x. nb#3 numerics lecture1b.nb Numerical solution and phase portrait of 1-D logistic equation. Lecture 2 Link File name Description nb#1 numerics lecture2.nb Bifurcation diagrams in 1-D systems: saddle node bifurcation, transcritical bi- furcation, pitchfork bifurcation (supercritical), pitchfork bifurcation (subcriti- cal). Lecture 3 Link File name Description nb#1 numerics lecture3.nb 1-D approximation corresponding to the bead on a hoop dynamics: 1-D phase portrait and a family of numerical solutions to initial value problem. nb#2 numerics lecture3b.nb Bead on a hoop dynamics: graphical determination of fixed points and the bifurcation diagram (first-order model). nb#3 numerics lecture3c.nb Numerical solution and phase portrait of the over-damped bead on a hoop problem (second-order model). Lecture 4 Link File name Description nb#1 numerics lecture4.nb Vector field plotting. Integrated solution and phase portrait of damped har- monic oscillator. nb#2 numerics lecture4b.nb Calculation of matrix determinant, trace, eigenvalues and eigenvectors using computer. Classification of fixed points in 2-D linear homogeneous systems. Lecture 5 Link File name Description nb#1 numerics lecture5.nb Effect of nonlinearity on a linear center (borderline case). Numerical solution and phase portrait of the same problem. nb#2 numerics lecture5b.nb Competitive cohabitation of sheep and rabbits (Lotka-Volterra predator{pray model). Linear analysis of fixed points and system manifolds. Numerical solu- tion and phase portrait. nb#3 numerics lecture5c.nb The Lotka-Volterra predator{pray model for sharks and fish (cyclic behaviour). Numerical solution and phase portrait. nb#4 numerics lecture5d.nb Interactive comparison of dynamics of the Li´enardequation and its linearisation. nb#5 numerics lecture5e.nb Particle in a double well potential, conservative system and homoclinic orbits. Linear analysis of fixed points, numerical solution and phase portrait. Lecture 6 Link File name Description nb#1 numerics lecture6.nb Mathematical pendulum and heteroclinic orbit. Integrated numerical solution. nb#2 numerics lecture6b.nb The Dulac's criterion and limit-cycles. A numerical example: integrated solu- tion and phase portrait. D. Kartofelev 1 K As of May 31, 2021 Index: Numerical files Nonlinear Dynamics YFX1520 nb#3 numerics lecture6c.nb The Dulac's criterion and limit-cycles. Another numerical example: integrated solution and phase portrait. nb#4 numerics lecture6d.nb The Poincar´e-Bendixsontheorem. Phase portrait of a nonlinear system in polar coordinates. nb#5 numerics lecture6e.nb Glycolysis phase portrait and null-clines. Numerical solution of glycolysis model. Lecture 7 Link File name Description nb#1 numerics lecture8.nb Examples of bifurcations in 2-D systems: saddle-node bifurcation, transcriti- cal bifurcation, supercritical and subcritical pitchfork bifurcations. Interactive code. nb#2 numerics lecture8b.nb Linear analysis of the system from the previous file. nb#3 numerics lecture8c.nb Examples of bifurcations in 2-D systems: supercritical Hopf bifurcation. Inte- grated solution. Bifurcation diagram. nb#4 numerics lecture8d.nb Examples of bifurcations in 2-D systems: subcritical Hopf bifurcation. Inte- grated solution. Bifurcation diagram. nb#5 numerics lecture8e.nb Examples of bifurcations in 2-D systems: saddle-node coalescence of limit- cycles. Bifurcation diagram. nb#6 numerics lecture8f.nb Examples of bifurcations in 2-D systems: SNIPER (saddle-node infinite period bifurcation). Integrated solution and phase portrait. nb#7 numerics lecture8g.nb Examples of bifurcations in 2-D systems: homoclinic bifurcation or saddle-loop bifurcation (solitons). Integrated solution and phase portrait. Lecture 8 Link File name Description nb#1 numerics 9.nb Trajectories on the surface of a torus, interactive 3-D plot. Periodic and quasi-periodic trajectories on the surface of a torus. Trefoil knot (for p = 3, q = 2). Cinquefoil knot (for p = 5 and q = 2). nb#2 numerics 9b.nb Quasi-periodic oscillators. Quasi-periodic decoupled system: periodic solution, quasi- periodic solution. The Fourier and power spectra of the solutions. nb#3 numerics 9c.nb Quasi-periodic coupled oscillations. The Fourier and power spectra of the solutions. nb#4 numerics 9d.nb Quasi-periodic coupled oscillations. Interactive code. nb#5 numerics 9e.nb Chaotic system example { the Lorenz mill. Power spectrum of the Lorenz mill solu- tions. Numerical solution plotting in 3-D. nb#6 numerics 9f.nb Chaotic system example { the Lorenz attractor. Numerical integration of the Lorenz attractor (3-D interactive, 2-D (x-z projection) interactive). nb#7 numerics 9g.nb Remark on 3-D phase portraits. 3-D phase portrait and flow visualisation of Lorenz attractor. nb#8 Analysis tool.nb Coursework analysis tools: Mathematica notebook for analysing 2-D and 3-D systems. Use of the tool does not require coding skills. Lecture 9 Link File name Description nb#1 numerics 10.nb Dissipative chaotic flow (the Lorenz attractor) visualisation via particle tracking. Cal- culation of divergence of 3-D and higher dimensional flows. nb#2 numerics 10b.nb The Lyapunov exponent of 3-D chaotic systems (visual demonstration). nb#3 numerics 10c.nb Demonstration of the effect of calculation accuracy and precision on the numerical solution of a chaotic system (predictability horizon). Transient and intermittent chaos. nb#4 numerics 10d.nb Numerical solutions of chaotic systems: Lorenz attractor, R¨osslerattractor, Chen attractor, modified Chen attractor, Lu attractor, Pan-Xu-Zhou attractor, Bouali at- tractor, and the algebraically simplest chaotic dissipative flow (J. C. Sprott). nb#5 numerics 10e.nb Numerical solutions of chaotic systems: double (mathematical) pendulum (one and two sets of initial conditions). Phase portrait of double pendulum. nb#6 numerics 10f.nb Numerical solutions of chaotic systems: gravitational 3 body problem (planar). Inter- active code with several dynamic visualisation styles. D. Kartofelev 2 K As of May 31, 2021 Index: Numerical files Nonlinear Dynamics YFX1520 nb#7 numerics 10g.nb Introduction to cobweb diagramming. Interactive cobweb diagrams of 1-D maps 2 xn+1 = rxn and xn+1 = r(xn − xn) where r is a constant. nb#8 numerics 10h.nb Interactive cobweb diagram and iterates of the Lorenz map. nb#9 numerics 10i.nb Quantitatively accurate Lorenz mapping. Lecture 10 Link File name Description nb#1 numerics 11.nb Cobweb diagram and iterates of the (generalised) Lorenz map. Lyapunov expo- nent λ(r) of the Logistic map. nb#2 numerics 11b.nb Cobweb diagram of the Logistic map. Orbit diagram of the Logistic map. Period doubling and orbit diagram. nb#3 numerics 11c.nb The Logistic map: cobweb diagram, orbit diagram and map iterates, Lyapunov exponent. nb#4 numerics 11d.nb Period-3 window and intermittency in the Logistic map (iterates xn). nb#5 numerics 11e.nb The sine map iterates, cobweb and orbit diagrams (xn 2 [0; 1]). nb#6 (not numerics 11f.nb The sine map iterates, cobweb and orbit diagrams (xn 2 [−1; 1]). required) Lecture 11 Link File name Description nb#1 numerics L11.nb Comparison of superstable and stable fixed points in the Logistic map. nb#2 numerics L11b.nb Calculation of universal limiting function g and value of the Feigenbaum constant α in the case of the Logistic map using power series expansion (one and four term approximations). nb#3 numerics L11c.nb Periodically forced damped Duffing oscillator. Numerical solution and phase por- trait. A 2π-periodic Poincar´esection of the Duffing oscillator (static). nb#4 numerics L11d.nb Periodically forced damped Duffing oscillator and a dynamic Poincar´esection ani- mation. nb#5 numerics L11e.nb Numerical solution of the R¨osslerattractor. The Poincar´esection of the R¨ossler attractor. nb#6 numerics L11f.nb Periodic orbits in the R¨osslerattractor for varied c value. nb#7 numerics L11g.nb Higher-dimensional attractor reconstruction from 1-D time-domain signals. Exam- ples: sine wave, quasi-periodic signal, y(t) of the Lorenz attractor. Lecture 12 Link File name Description nb#1 numerics 13.nb Dynamics of the R¨osslerattractor. Tracking particles in a 3-D flow (animation). nb#2 numerics 13b.nb The Cantor set fractal generator. nb#3 numerics 13c.nb The von Koch curve and Koch Snowflake fractal generator. nb#4 numerics 13d.nb The H´enonmap fixed points and iterates, chaotic and periodic (period-2 cycle) solutions. Zooming into local fractal microstructure of the H´enonmap. nb#5 numerics 13e.nb Mapping and iterating a rectangle with the H´enonmap (animation). nb#6 numerics 13f.nb Calculation of orbit diagram of the H´enonmap (x-a, x-b, y-a, y-b diagrams). nb#7 numerics 13g.nb Alternative ways of calculating the H´enonmap. nb#8 (not numerics 13h.nb Random examples of linear and nonlinear 2-D maps (nonlinear system matrix). required) Ways of calculating and plotting them. nb#9 numerics 13i.nb The Gingerbreadman map calculated for initial conditions x0 = −0:3, y0 = 0 showing 4 · 104 iterates. Lecture 13 Link File name Description 2 nb#1 numerics LM.nb Basin of attraction and 2-D map in polar coordinates frn+1 = rn; θn+1 = θn−sin θng (dynamic animation). nb#2 numerics LMb.nb Iterates of linear 2-D maps and their phase portraits.