**Links and References**

**Books**

Nonlinear Dynamics and Chaos: With Applications 2ed, Strogatz, 2014 (Main text)

Differential Equations, Blanchard, Devaney, Hall (more elementary treatment in the same modern spirit)

Nonlinear Differential Equations and Dynamical Systems, Verhulst (more rigorous treatment in the same modern spirit)

Stability, Instability, and Chaos, Glendinning (more rigorous treatment in the same modern spirit)

Chaos: An Introduction to Dynamical Systems, Alligood, Sauer, Yorke (more focus on discrete dynamical systems)

Online ODE texts

Dynamics and Bifurcations, Hale and Kocak (especially chapters on topological conjugacy and equivalence)

Differentiable Dynamical Systems, Meiss (especially chapters on topological conjugacy and equivalence)

Practical Bifurcation and Stability Analysis , Seydal (free GT download)

**Videos**

Strogatz Cornell lectures based on our main text

Strogatz lab demos

Simulation of bead on a rotating hoop

Interesting links

How Populations Grow: The exponential and logistic equations

Dimensional analysis and scaling

More About Nondimensionalization

Buckingham π theorem (Advanced Math)

Self-contained proof of the fundamental existence and uniqueness theorem for ODEs (Advanced Math)

ODE models of chemical reactions

The velocity dependence of aerodynamic drag

How terminal is terminal velocity?

Big O and little o

Nondimensionalizing a first order quadratic ODE

Thresholds and breakpoints in ecosystems with a multiplicity of stable states (May, Nature 1977)

Basic Lyapunov Theory (Boyd, EE363 Stanford)

SIS transmission model basics

Imperfect Vaccine Aggravates the Long-standing Dilemma of Voluntary Vaccination

The SIR model and the Foundations of Public Health

Kinetics, Microbial Growth (Panilov)

The Hill equation: a review of its capabilities in pharmacological modeling (Maire)

Overview of some theoretical approaches for derivation of the Monod equation (Liu)

Time-scale separation – Michaelis and Menten’s old idea, still bearing fruit (Gunawardena)

Buzz Holling and the Functional Response (Denny)

Some characteristics and simple types of predation and parasitism (Holling)

Analytic criterion for bifurcation for x'=f(x)

Single table summary of analytic criterion for bifurcation for x'=f(x)

Baby normal forms

Very elementary model of buckling column

Selected topics in bifurcation theory (advanced, but good discussion of beams)

The Effects of Temperature on the Stability of a Neuronal Oscillator (Winberg)

Synchronized fireflies (Great Smoky Mountains) parking pass lottery

The Synchronicity of Firefly Flashing (Stewart)

Control of flashing in fireflies (Buck)

Synchronous Rhythmic Flashing of Fireflies (Buck)

Synchronous Rhythmic Flashing of Fireflies II (Buck)

Toward a Functional Interpretation of Synchronous Flashing by Fireflies (Buck)

An adaptive model for synchrony in the firefly Pteroptyx malaccae (Ermentrout)

Synchronization of pulse-coupled biological oscillators (Strogatz)

Quick review of 2×2 matrices for ODEs (*)

Jordan Canonical Form for 2D matrices (*)

Solving 2x2 systems of ODEs (*)

Phase portraits of 2x2 systems of linear ODEs (summary) (*)

Phase portraits of 2x2 systems of linear ODEs (details) (*)

Invertible 2x2 matrices map unit circle to ellipse another proof another proof

Some dynamical models of love (Sprout)

Linear system model of sliding plates

Linear maps and ODEs (advanced)

Electrical circuits and linear ODEs

Model of sliding plates

Examples of phase portraits for nonlinear systems in the plane 1 2 3 4 5 6

Conjugate phase portraits of linear systems (McSwiggen and Meyer)

Introduction to homeomorphisms (see 1.6) (Math fans)

Meiss Chapter on topological equivalence conjugacy (Math fans)

An Undergraduate’s Guide to the Hartman-Grobman and Poincar'e-Bendixon Theorems

Topological Conjugacy and Equivalence and remarks of the Hartman-Grobman theorem

Stable and Unstable Manifolds for Planar Dynamical Systems (Smith)

Amazing Jordan curves (Ross)

Lyapunov functions and stability

Competition, Coexistence, and Evolution, Ayala

Conjugacies (Rauch)

Supercritical bifurcation of a spinning hoop (Physics fans)

A mechanical analog of first and second order phase transitions via bead on a rotating hoop (Physics fans)

A damped pendulum forced with a constant torque (experiment and theory)

Routh-Hurwitz criterion

Introduction to competition (slides from upper level ecology course)

Species Interactions and Competition

Dynamics of Predation

Population Dynamics of Mutualism

Quasispecies Made Simple (Bull)

What is a quasispecies? (Novak)

Proof of Morse Lemma

Slides on nonlinear control theory (Kwatny)

Invariant Manifolds of Dynamical Systems and an application to Space Exploration (Wirth)

A Numerical Study of Low-Frequency Discharge Oscillations in Hall Thrusters [equivalent to predator-prey model] (Fife)

Dynamics around non-hyperbolic fixed points

Mutualisms and Aquatic Community Structure: The Enemy of My Enemy Is My Friend

Some intuition about gradient systems

Dynamics of Predation

Awesome examples of simple closed curves (Ross)

The Jordan Curve Theorem, Formally and Informally (Hales)

Poincare-Bendixson theorem (Ciesielski)

The collapse of the Tacoma Narrows Bridge

Crowd synchrony on the Millennium Bridge

Large-Amplitude Periodic Oscillations in Suspension Bridges: Some New Connections with Nonlinear Analysis (advanced)

A piecewise linear suspension bridge model: nonlinear dynamics and orbit continuation

Large torsional oscillations in suspension bridges: fixing an old approximation

What to Say About the Tacoma Narrows Bridge to your Introductory Physics Class (an alternative linear model)

Oscillating chemical reactions

Alfred Lotka, Vito Volterra, and Population Cycles

Strict Lyapunov Functions for Control of Robot Manipulators (Kelley)

Strict Lyapunov function and Chetaev function for stability/instability of the pendulum (Santibanex)

Buckling and Oscillation of a Beam (via saddle node bifurcation)

Control of a Saddle Node Bifurcation in a Power System Using a PID Controller (Alvarez)

Criterion for Hopf bifurcation 1

Criterion for Hopf bifurcation 2

Briggs-Rauscher Reaction (I will present class demo)

Hopf bifurcation in fluid mechanics (Benard problem)

Introduction to bifurcation analysis (van Voorn)

Derivation of the Lorenz equations 1 2

Original paper by Lorenz

Properties (*)

Video of chaotic waterwheel modeled by Lorenz system

An introduction to the Lorenz system (Lanford)

An introduction to the Lorenz equations (Sparrow)

The Lorenz attractor exists (Tucker)

A rigorous ODE solver and Smale's 14th problem (Tucker)

1 page summary of properties of Lorenz system (Knill)

1 page summary of phase portraits and bifurcations of Lorenz system (Knill)

Preturbulence: A Regime Observed in a Fluid Flow Model of Lorenz (Kaplan and Yorke)

Visualizing the structure of chaos in the Lorenz system (Osinga and Krauskopf)

The fractal property of the Lorenz attractor (Viswanath)

Chaotic Dynamics of a Nonlinear Density Dependent Population Model (Ugarcovi and Weiss)

Chapter 2Chapter 2

How Populations Grow: The exponential and logistic equations

Dimensional analysis and scaling

More About Nondimensionalization

Buckingham π theorem (Advanced Math)

Self-contained proof of the fundamental existence and uniqueness theorem for ODEs (Advanced Math)

ODE models of chemical reactions

The velocity dependence of aerodynamic drag

How terminal is terminal velocity?

Big O and little o

Nondimensionalizing a first order quadratic ODE

Thresholds and breakpoints in ecosystems with a multiplicity of stable states (May, Nature 1977)

Basic Lyapunov Theory (Boyd, EE363 Stanford)

SIS transmission model basics

Imperfect Vaccine Aggravates the Long-standing Dilemma of Voluntary Vaccination

The SIR model and the Foundations of Public Health

Kinetics, Microbial Growth (Panilov)

The Hill equation: a review of its capabilities in pharmacological modeling (Maire)

Overview of some theoretical approaches for derivation of the Monod equation (Liu)

Time-scale separation – Michaelis and Menten’s old idea, still bearing fruit (Gunawardena)

Buzz Holling and the Functional Response (Denny)

Some characteristics and simple types of predation and parasitism (Holling)

**Chapter 3 (Bifurcations)**Analytic criterion for bifurcation for x'=f(x)

Single table summary of analytic criterion for bifurcation for x'=f(x)

Baby normal forms

Very elementary model of buckling column

Selected topics in bifurcation theory (advanced, but good discussion of beams)

The Effects of Temperature on the Stability of a Neuronal Oscillator (Winberg)

**Chapter 4 (Flows on****Circle)**Synchronized fireflies (Great Smoky Mountains) parking pass lottery

The Synchronicity of Firefly Flashing (Stewart)

Control of flashing in fireflies (Buck)

Synchronous Rhythmic Flashing of Fireflies (Buck)

Synchronous Rhythmic Flashing of Fireflies II (Buck)

Toward a Functional Interpretation of Synchronous Flashing by Fireflies (Buck)

An adaptive model for synchrony in the firefly Pteroptyx malaccae (Ermentrout)

Synchronization of pulse-coupled biological oscillators (Strogatz)

**Chapter 5 (Linear Systems of ODEs)**Quick review of 2×2 matrices for ODEs (*)

Jordan Canonical Form for 2D matrices (*)

Solving 2x2 systems of ODEs (*)

Phase portraits of 2x2 systems of linear ODEs (summary) (*)

Phase portraits of 2x2 systems of linear ODEs (details) (*)

Invertible 2x2 matrices map unit circle to ellipse another proof another proof

Some dynamical models of love (Sprout)

Linear system model of sliding plates

Linear maps and ODEs (advanced)

Electrical circuits and linear ODEs

Model of sliding plates

****

Chapter 6 (Phase Plane)Chapter 6 (Phase Plane)

Examples of phase portraits for nonlinear systems in the plane 1 2 3 4 5 6

Conjugate phase portraits of linear systems (McSwiggen and Meyer)

Introduction to homeomorphisms (see 1.6) (Math fans)

Meiss Chapter on topological equivalence conjugacy (Math fans)

An Undergraduate’s Guide to the Hartman-Grobman and Poincar'e-Bendixon Theorems

Topological Conjugacy and Equivalence and remarks of the Hartman-Grobman theorem

Stable and Unstable Manifolds for Planar Dynamical Systems (Smith)

Amazing Jordan curves (Ross)

Lyapunov functions and stability

Competition, Coexistence, and Evolution, Ayala

Conjugacies (Rauch)

Supercritical bifurcation of a spinning hoop (Physics fans)

A mechanical analog of first and second order phase transitions via bead on a rotating hoop (Physics fans)

A damped pendulum forced with a constant torque (experiment and theory)

Routh-Hurwitz criterion

Introduction to competition (slides from upper level ecology course)

Species Interactions and Competition

Dynamics of Predation

Population Dynamics of Mutualism

Quasispecies Made Simple (Bull)

What is a quasispecies? (Novak)

Proof of Morse Lemma

Slides on nonlinear control theory (Kwatny)

Invariant Manifolds of Dynamical Systems and an application to Space Exploration (Wirth)

A Numerical Study of Low-Frequency Discharge Oscillations in Hall Thrusters [equivalent to predator-prey model] (Fife)

Dynamics around non-hyperbolic fixed points

Mutualisms and Aquatic Community Structure: The Enemy of My Enemy Is My Friend

**(Hay)**

Construction of a genetic toggle switch in Escherichia coli (Collins)**Chapter 7 (Limit Cycles)**Some intuition about gradient systems

Dynamics of Predation

Awesome examples of simple closed curves (Ross)

The Jordan Curve Theorem, Formally and Informally (Hales)

Poincare-Bendixson theorem (Ciesielski)

The collapse of the Tacoma Narrows Bridge

Crowd synchrony on the Millennium Bridge

Large-Amplitude Periodic Oscillations in Suspension Bridges: Some New Connections with Nonlinear Analysis (advanced)

A piecewise linear suspension bridge model: nonlinear dynamics and orbit continuation

Large torsional oscillations in suspension bridges: fixing an old approximation

What to Say About the Tacoma Narrows Bridge to your Introductory Physics Class (an alternative linear model)

Oscillating chemical reactions

Alfred Lotka, Vito Volterra, and Population Cycles

Strict Lyapunov Functions for Control of Robot Manipulators (Kelley)

Strict Lyapunov function and Chetaev function for stability/instability of the pendulum (Santibanex)

**Chapter 8 (Bifurcations Revisited)**Buckling and Oscillation of a Beam (via saddle node bifurcation)

Control of a Saddle Node Bifurcation in a Power System Using a PID Controller (Alvarez)

Criterion for Hopf bifurcation 1

Criterion for Hopf bifurcation 2

Briggs-Rauscher Reaction (I will present class demo)

Hopf bifurcation in fluid mechanics (Benard problem)

Introduction to bifurcation analysis (van Voorn)

**Chapter 9 (Lorenz Equations)**Derivation of the Lorenz equations 1 2

Original paper by Lorenz

Properties (*)

Video of chaotic waterwheel modeled by Lorenz system

An introduction to the Lorenz system (Lanford)

An introduction to the Lorenz equations (Sparrow)

The Lorenz attractor exists (Tucker)

A rigorous ODE solver and Smale's 14th problem (Tucker)

1 page summary of properties of Lorenz system (Knill)

1 page summary of phase portraits and bifurcations of Lorenz system (Knill)

Preturbulence: A Regime Observed in a Fluid Flow Model of Lorenz (Kaplan and Yorke)

Visualizing the structure of chaos in the Lorenz system (Osinga and Krauskopf)

The fractal property of the Lorenz attractor (Viswanath)

Chaotic Dynamics of a Nonlinear Density Dependent Population Model (Ugarcovi and Weiss)

**Software**

pplane and dfield (Java versions) Users manual

Wolfram Alpha

2x2 matrices with Mathematica (1 page)

Brief Mathematica ODE Demo.nb.pdf (2 pages)

Mathematica cheat sheet

The Matlab ODE suite

Elementary Mechanics Using Matlab (good guide to solving problems in mechanics)

Solving ODEs in Matlab

Physical modeling in MATLAB (Downey) free textbook

LSODE

3 lines of Mathematical code to numerically solve x''+sin(x)=0, x(0)=1, x'(0)=-1 and plot solution

7 lines of Matlab code to numerically solve x''+sin(x)=0, x(0)=1, x'(0)=-1 and plot solution

Interactive Differential Equations (applets) These are very insightful but require you modify the JAVA security settings

Differential equations With Matlab (good book with intro to numerical issues solving ODEs)

Mathematica Notebook to study motion of particle in double well potential and undamped pendulum pdf version

Two disasters caused by computer arithmetic mistakes