PH4273 / PH6423 :  Nonlinear Dynamics
(January 2024 semester)

This is a first course on nonlinear dynamics meant for students from all disciplines with a background in calculus and linear algebra.

Course contents : ∙ Download as PDF file
Books :
(Reference numbers mentioned later refer to these books.)

1. Nonlinear dynamics and chaos by Steven Strogatz
(Classic introductory text for students from any science background.)

2. Chaos : Introduction to dynamical systems by K. T. Alligood, T. D. Sauer and J. A. Yorke
(This textbook has mathemtical flavour to it.)

3. Chaos in dynamical systems  by Edward Ott
(The chaos textbook for physicists. A bit too condensed and fast-paced.)

4. Differential equations, dynamical systems and an introduction to chaos  by M. W. Hirsch, Stephen Smale and Robert Devaney
(Good for the chaos part)

5. Chaos and nonlinear dynamics : An introduction for scientists and engineers by Robert Hilborn
(A good nonlinear dynamics textbook for physicists. Good for those seeking physical intuition and explanations.)

Evaulation :
Quiz (2 to 3) : 30%
Midsem : 30%
Endsem : 40%

Announcements :

18.4.2024 :
● Test 2 question paper

12.4.2024 :
● End-sem exam on 29.4.2024. This exam will cover everything taught in class, though almost 60% of the questions will be from the part taught after midsem exam (Lectures 14 to 25).

5.4.2024 :
● Test 1 question paper
Midsem question paper

27.3.2024 :
● Second test will be held on 6th April, 2024 (Saturday) at 11:00AM. This test will cover everything taught in class after midsem exam and until 21.3.2024 (Lectures 14 to 21).

25.3.2024 :
● Next test will be in the first week of April. Mostly on 6th April, 2024 (exact date/time will be announced soon). This test will cover everything taught in class after midsem exam and until 21.3.2024 (Lectures 14 to 21).

14.2.2024 :
● Mid-semester examination will take place on 23.2.2024 from 3.00 AM to 5.00 AM. This test will everything taught in class until 15.2.2024 (Lectures 1 to 13).

20.1.2024 :
● First test will be held on 6.2.2024 from 9.00 AM to 9.45 AM during the usual class hours. This test will cover all portions taught until 25.1.2024.

║ 18.4.2024 ║ Lecture 26 ║
Beyond the course and wading into research questions : Introductory ideas (not meant for exams) about integrable systems, KdV equations and solitons, nonlinear Schrodinger equation, classical and quantum chaos in Hamiltonian systems, and complex networks.
Ref : Solitons, Quantum chaos, Complex networks
║ 4.4.2024 ║ Lecture 25 ║
Strange attractor, self-similarity dimension, Box-counting dimension, Cantor set, Koch curve.
Ref : Section 11.0 to 11.4 of Ref [1]
║ 2.4.2024 ║ Lecture 24 ║
Lorentz model, dissipative system, chaos in Lorentz model, strange attractor.
Ref : Section 9.2 and initial parts of section 9.3 of Ref [1]
║ 28.3.2024 ║ Lecture 23 ║
Feigenbaum scaling in single-humped maps. Feigenbaum constant.
Brief notes for Feigenbaum scaling
Ref : Section 10.7 of Ref [1]
║ 26.3.2024 ║ Lecture 22 ║
Hierarchy of randomness -- ergodicity, mixing and chaotic.
Intro to Feigenbaum scaling in single-humped maps. Super-stable periodic points.
Ref : Section 10.7 of Ref [1] for scaling in maps.
║ 21.3.2024 ║ Lecture 21 ║
Shift map, what is a shift and its relation to chaos, Counting periodic orbits of shift map
Ref : Sec 15.6 of Ref [4] given in the list of books above.
Brief notes for Shift map
Brief notes for Counting periodic orbits
║ 19.3.2024 ║ Lecture 20 ║
Defining chaos, doubling map, proof that doubling map and tent map are chaotic
Ref : Sec 15.4 of Ref [4] given in the list of books above.
Brief notes for chaos definition
Some additional reading : Various definitions of chaos
║ 14.3.2024 ║ Lecture 19 ║
Iteneraries, sensitivity to initial conditions, Lyapunov exponent, application to tent and logistic maps
Ref : Sec 1.8 of Ref [2] given in the list of books above.
║ 12.3.2024 ║ Lecture 18 ║
Conjugacy relation, conjugacy between tent and logistic maps, invariant density of logistic map using conjugacy
Ref : Sec 15.4 of Ref [4] given in the list of books above.
║ 7.3.2024 ║ Lecture 17 ║
Tent map, invariant density and Perron-Frobenius equation, invariant density of tent map.
Ref : Sec 2.3.3 of Ref [3] given in the list of books above.
║ 5.3.2024 ║ Lecture 16 ║
Logistic map, period-k fixed points, bifurcation diagram for logistic map, cobweb diagram.
Try logistic map simulation online
║ 29.2.2024 ║ Lecture 15 ║
Introduction to one-dimensional maps, linear stability, fixed points.
║ 27.2.2024 ║ Lecture 14 ║
Bifurcations in 2D, Hopf bifurcations
║ 15.2.2024 ║ Lecture 13 ║
Limit cycle oscillations, Poincare-Bendixon theorem and how to apply it.


Belousov-Zhabotinsky oscillating chemical reaction.
║ 13.2.2024 ║ Lecture 12 ║
Nonlinear systems in two dimensions (continued). Structural stability of saddles, problems with linear stability analysis, conservative systems, robustness of centres in conservative systems.
║ 8.2.2024 ║ Lecture 11 ║
Nonlinear systems in two dimensions. Methods of analysis, examples, stability picture in the trace - determinant plane.
║ 1.2.2024 ║ Lecture 10 ║
Linear systems in two dimensions. Some interesting special cases including the case of degenerate eigenvalues, stability picture in the trace - determinant plane.
║ 30.1.2024 ║ Lecture 9 ║
Linear systems in two dimensions. Types of stability and fixed points.
║ 25.1.2024 ║ Lecture 8 ║
Flows on a circle. Saddle-node bifurcation and passage through bottleneck region, square-root scaling law in the vicinity of saddle-node bifurcation.
║ 23.1.2024 ║ Lecture 7 ║
Bead in a rotating hoop. Example of pitchfork bifurcation, non-dimensionalising the equation of motion.
║ 18.1.2024 ║ Lecture 6 ║
Bifurcations. Transcritical and pitchfork bifurcation. Stabilising unstable systems. Hysterisis. What all this means in practice ?
▶ Additional readings if you are interested (not mandatory for the course) :
Anticipating critical transitions
Pitchfork bifurcation and critical temperature transition
║ 16.1.2024 ║ Lecture 5 ║
Properties of flows in 1D. Bifurcations. Saddle-node bifurcation.
║ 11.1.2024 ║ Lecture 4 ║
Flows in one dimension. Figuring out solution without actually solving them. Existence and uniqueness of solutions for ODEs.
║ 9.1.2024 ║ Lecture 3 ║
Nonlinear oscillators, separatrix, time-period, calculating time-period of oscillatory motion.
║ 4.1.2024 ║ Lecture 2 ║
Nonlinear systems. Exact solution of linear and nonlinear pendulum in terms of elliptic functions. Time-period of oscillation.
▶ Nonlinear pendulum : Comprehensive solution, and one more resource for analytical solution.
║ 2.1.2024 ║ Lecture 1 ║
Introduction to nonlinearity. Effects of nonlinearity (e.g, Millenium bridge collapse), synchronisation, basics.


Additional reading (not mandatory for the course) : A research articleon this bridge collapse


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