Nonlinear Dynamics and Chaos: Discovering the Hidden Order and Unpredictability in Physics, Biology,

• [Option \bf B8b if taken as a half-unit. OSS paper code 2B47.]Overview This course aims to provide an introduction to the tools of dynamical systems theory, which are essential for the realistic modelling and study of many disciplines, including mathematical ecology and biology, fluid dynamics, granular media, mechanics, and more. The course will include the study of both nonlinear ordinary differential equations and maps. It will draw examples from appropriate model systems and various application areas. The problem sheets will require numerical computation (using programs such as Matlab).Learning Outcomes Students will have developed a sound knowledge and appreciation of some of the tools, concepts, and computations used in the study of nonlinear dynamical systems. They will also get some exposure to some modern research topics in the field. \textbfSynopsis Bifurcations and Nonlinear Oscillators [8 lectures](a) Bifurcation theory: standard codimension one examples (saddle-node, Hopf, etc.), normal forms and codimension two examples (briefly).(b) Non-conservative oscillators: Van der Pol's equation, limit cycles.(c) Conservative oscillators (introduction to Hamiltonian systems): Duffing's equation, forced pendulum.(d) Synchronization: synchronization in non-conservative oscillators, phase-only oscillators (e.g., Kuramoto model).

• Maps [2 lectures](a) Stability and periodic orbits, bifurcations of one-dimensional maps.(b) Poincar\'e sections and first-return maps (leads to part 3 topics)

• Chaos in Maps and Differential Equations [4 lectures](a) Maps: logistic map, Bernoulli shift map, symbolic dynamics, two-dimensional maps (examples could include Henon map, Chirikov-Taylor [``standard''] map, billiard systems)(b) Differential equations: Lyapunov exponents, chaos in conservative systems (e.g., forced pendulum, Henon-Heiles), chaos in non-conservative systems (e.g., Lorenz equations)

• Other topics [2 lectures or as time permits] Topics will vary from year to year and could include: dynamics on networks, solitary waves, spatio-temporal chaos, quantum chaos.

• Course Syllabus: Reading List: S. H. Strogatz, Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry and Engineering (Westview Press, 2000).

• E. Ott, Chaos in Dynamical Systems (Second edition, Cambridge University Press, Cambridge, 2002).

• P. Cvitanovic, et al, Chaos: Classical and Quantum (Niels Bohr Institute, Copenhagen 2008). [Available for free online at ]

• R. H. Rand, Lecture Notes on Nonlinear Vibrations. [Available for free online at ]

• J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields (Springer-Verlag, 1983).

• G. L. Baker and J. P. Gollub, \textit Chaotic Dynamics: An Introduction (Second edition, Cambridge University Press, Cambridge, 1996).

• P. G. Drazin, Nonlinear Systems (Cambridge University Press, Cambridge, 1992).

• S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Second edition, Springer, 2003).

• S. H. Strogatz, `From Kuramoto to Crawford: exploring the inset of synchronization in populations of coupled oscillators', Physica D 143 (2000) 1-20.

• Various additional books and review articles (especially for some of the `other topics').

Please note that e-book versions of many books in the reading lists can be found on SOLO and ORLO.

Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Stu

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MATH 4275/6275 Applied Dynamical Systems [Syllabus] An introduction to discrete and continuous dynamical systems. Study of nonlinear dynamical systems, leading to chaotic behavior. Topics include: Phase space. Linear and nonlinear systems. Poincare maps. Structual stability. Topological conjugacy. Types of equilibrium states and fixed points. Periodic orbits. Stability. Local bifurcations. Homoclinic orbits. Routes to chaos. Applications from physics, biology, population dynamics, economics.

Justification The idea that many simple nonlinear deterministic systems can behave in an apparently unpredictable and chaotic manner was first noticed by the great French mathematician Henri Poincare. Other early pioneering work in the field of chaotic dynamics were found in the mathematical literature by such luminaries as Birkhoff, Cartwright, Littlewood, Smale, Pontryagin, Andronov and his students, among others. In spite of this, the importance of chaos was not fully appreciated until the widespread availability of digital computers for numerical simulations and the demonstration of chaos in various real time systems. This realization has broad implications for many fields of science, and it is only within the past two decades that the field has undergone explosive growth. It is found that the ideas of chaos have been very fruitful in such diverse disciplines as biology, economics, chemistry, engineering, fluid mechanics, and physics. The field of nonlinear dynamics is very active. Furthermore, there are numerous applications in the physical and biological sciences, economics, etc. There are graduate programs in nonlinear dynamics, and there is an abundance of problems arising from recent subfields, such as the control of chaos. Courses in the nonlinear sciences would add to both our undergraduate and graduate programs due to both its timeliness and interesting applications. Many fascinating phenomena, such as our weather system for example, are intrinsically nonlinear. This course will provide an introduction to nonlinear behavior, principally through the study of differential and difference equations (maps). We will begin with a study of the stability of equilibrium states, fixed points and periodic orbits of one- and two-dimensional maps. This can be followed by the study of their bifurcations, and the analysis of saddle trajectories. This will lead us to emergence of strange attractors of fractal dimensions; the evolution of saddle orbits into chaos. It will lead to the idea that seemingly random behavior can emerge from perfectly deterministic systems. Many of the key ideas can be illustrated by fairly simple sets of equations that on a computer can be solved rapidly and accurately. Nonlinear dynamical systems are used as models in every field of science and engineering. Universal patters of behavior, including chaos, have been observed in large sets of examples. Mathematical theories describing geometrically the qualitative behavior of "generic" systems explain many of these patterns. This course will discuss dynamical system theory and its application. Several representative examples from different disciplines will be described at the beginning of the course and used throughout the semester to illustrate theoretical ideas. Emphasis will be placed upon bifurcations, the qualitative changes in solutions of differential and difference equations that occur as system parameters are varied. Computational methods for the analysis of dynamical systems will be also discussed. Examples come from population dynamics (Voltera-Lotka model, period doubling cascade), physics (van der Pol and Duffing equations) meteorology (Lorenz model), chemistry (Belousov reaction), finance (low-order stock models, chaos); and normal forms from mathematics

The impact of nonlinear science across the broad spectrum of natural, life,and medical sciences is due to the unity of its fundamental concepts. Investigatorsof nonlinear phenomena from diverse backgrounds are transforming importantproblems once considered intractable, or even ill-posed, into promising newfields. Methods developed in the study of nonlinear systems are now beingapplied at the Georgia Institute of Technology(GT) and Emory University (EU) to suchareas as pattern formation, quantum theory, biomotor control, microfluidics,dynamics of neuronal populations, biological fluids, and nonequilibrium transport.These problems cut across many fields, and the progress in solving them requiresbringing together researchers from widely diverse fields, such as mechanical,biomedical and electrical engineering, physics, applied and pure mathematics,chemistry, biological and health sciences.

The greatest challenge of modern nonlinear physics is to develop theoryand applications of dynamics in systems with many degrees of freedom. We envisiona multi-pronged attack on this grand problem through three cross-cuttingresearch efforts, with nonlinear science as the unifying theme. The CNScross-disciplinary program will be based on common concepts unifying a broadrange of problems of basic science, such as pattern formation, nonequilibriumdynamics, turbulence, classical and quantum chaos. Research and advancedtraining fostered by CNS will impact on a diverse range of applications ofnonlinear science to fields such as biophysics, neuroscience, engineeringproblems involving liquids, interface motion, novel materials, flows at microscales,and nonlinear control, through the three research efforts:

Chaotic dynamics of systems with effectively few degrees of freedom istoday well understood, and forms the basis for current applications of nonlinearmethods to a broad range of fields. However, when the dynamics involves manystrongly coupled degrees of freedom, available tools for describing chaoslargely fail. Experiments such as the high-resolution spectroscopy of highlyexcited (so-called ``Rydberg'') atoms in strong external fields pose botha formidable challenge to theorists and at the same time offer ideal physicallaboratories for investigating multi-dimensional dynamics. Developing thetheory for these experiments has so far been stymied by the fact that theclassical dynamics undergoes a radical change when the number of degreesof freedom exceeds two: beyond that threshold, a wealth of new physics becomespossible. This inability to handle high-dimensional chaos in a systematicway remains a fundamental barrier to the application of the methods of nonlinearscience in many disciplines. 2ff7e9595c