**Lecture Schedule**

8/17 Course introduction, what is a dynamical system?, discrete and continuous dynamical systems, IVP

8/19 Geometric, physical, and analytic interpretations of x’=f(x), equilibrium points, logistic population model

8/21 Sketching solutions of x’=f(x), stability of an equilibrium point, linear stability analysis

8/24 Pathologies for IVP, fundamental existence and uniqueness theorem for IVP, applications

8/26 Monotonicity of solutions of x’=f(x) and applications, classification of solutions of IVP

8/28 Nondimensionalization of ODEs, logistic ODE with various forms of harvesting

8/31 Potential functions, stability via potential functions, singularly perturbed mechanical system, Monod growth law for bacteria

9/02 Intro to bifurcations, saddle-node bifurcation, bifurcation diagram, examples

9/04 Saddle-node bifurcations, normal form, analytic characterization

9/07 Labor Day - no class

9/09 Transcritical bifurcation, applications, pitchfork bifurcations

9/11 Pitchfork bifurcations, homework problem discussion, SIS transmission model (e.g., Gonorrhea)

9/14 SIS transmission model, RO, theoretical foundations of public health interventions, very quick into to SIR transmission model

9/16 Overdamped bead on a rotating hoop

9/18 Exam 1

9/21 Flows on the circle, nonlinear oscillator, square root scaling law

9/23 Overdamped pendulum with constant torque, quick review of linear systems of ODEs

9/25 Singular perturbation problem, Jordan Canonical form for 2x2 matrices, geometry of 2x2 matrices

9/28 Complete characterization of phase portraits for linear systems in the plane, stable and unstable manifolds, notions of stability of an equilibrium point

9/30 det(A)-trace(A) plane, exponential of a matrix

10/2 More on linear systems, hyperbolic equilibrium points, random linear systems, 1-parmmeter families of linear systems, structural stability of linear systems

10/5 Modeling love and relationships, introduction to nonlinear systems, pplane software

10/7 Introduction to nonlinear systems, equivalence and conjugacy of linear systems of ODEs

10/9 Equivalence of one and two dimensional linear ODEs

10/12 No class

10/14 Hartman-Grobman and Sternberg linearization theorems

10/16 Examples of linear stability analysis

10/19 More examples of linear stability analysis

10/21 Equilibrium points of Lorenz system, Routh-Hurwitz criterion

10/23 Competition between species, Lotka-Voltera competition model

10/26 Homework problem discussion (including power series approximation for stable and unstable manifolds)

10/28 Lotka-Voltera competition model

10/30 Exam II

11/02 Examples of conservative systems (including motion in double well potential), Introduction to Hamiltonian systems

11/04 More examples of conservative systems, properties of conservative systems, predator-prey system, undamped pendulum

11/06 Undamped pendulum continued, Morse lemma, more properties of conservative systems, Poincare-Lyapunov stability theorem (loose end)

11/09 Reversible systems

11/11 Reversible vs Hamiltonian systems, gradient systems

11/13 Gradient systems, limit cycles, global Lyapunov functions, Bendixson's criterion

11/16 Dulac's criterion, Lyapunov functions, quick intro to Poincare-Bendixson theorem

11/18 Poincare-Bendixson theorem

11/20 Saddle-node bifurcation in 2D, Hopf bifurcation

11/23 Hopf bifurcation (normal form, sufficient criterion)

11/25 No class

11/27 No class

11/30 Lorenz system

11/02 Lorenz system

12/04

12/11 Final exam

8/19 Geometric, physical, and analytic interpretations of x’=f(x), equilibrium points, logistic population model

8/21 Sketching solutions of x’=f(x), stability of an equilibrium point, linear stability analysis

8/24 Pathologies for IVP, fundamental existence and uniqueness theorem for IVP, applications

8/26 Monotonicity of solutions of x’=f(x) and applications, classification of solutions of IVP

8/28 Nondimensionalization of ODEs, logistic ODE with various forms of harvesting

8/31 Potential functions, stability via potential functions, singularly perturbed mechanical system, Monod growth law for bacteria

9/02 Intro to bifurcations, saddle-node bifurcation, bifurcation diagram, examples

9/04 Saddle-node bifurcations, normal form, analytic characterization

9/07 Labor Day - no class

9/09 Transcritical bifurcation, applications, pitchfork bifurcations

9/11 Pitchfork bifurcations, homework problem discussion, SIS transmission model (e.g., Gonorrhea)

9/14 SIS transmission model, RO, theoretical foundations of public health interventions, very quick into to SIR transmission model

9/16 Overdamped bead on a rotating hoop

9/18 Exam 1

9/21 Flows on the circle, nonlinear oscillator, square root scaling law

9/23 Overdamped pendulum with constant torque, quick review of linear systems of ODEs

9/25 Singular perturbation problem, Jordan Canonical form for 2x2 matrices, geometry of 2x2 matrices

9/28 Complete characterization of phase portraits for linear systems in the plane, stable and unstable manifolds, notions of stability of an equilibrium point

9/30 det(A)-trace(A) plane, exponential of a matrix

10/2 More on linear systems, hyperbolic equilibrium points, random linear systems, 1-parmmeter families of linear systems, structural stability of linear systems

10/5 Modeling love and relationships, introduction to nonlinear systems, pplane software

10/7 Introduction to nonlinear systems, equivalence and conjugacy of linear systems of ODEs

10/9 Equivalence of one and two dimensional linear ODEs

10/12 No class

10/14 Hartman-Grobman and Sternberg linearization theorems

10/16 Examples of linear stability analysis

10/19 More examples of linear stability analysis

10/21 Equilibrium points of Lorenz system, Routh-Hurwitz criterion

10/23 Competition between species, Lotka-Voltera competition model

10/26 Homework problem discussion (including power series approximation for stable and unstable manifolds)

10/28 Lotka-Voltera competition model

10/30 Exam II

11/02 Examples of conservative systems (including motion in double well potential), Introduction to Hamiltonian systems

11/04 More examples of conservative systems, properties of conservative systems, predator-prey system, undamped pendulum

11/06 Undamped pendulum continued, Morse lemma, more properties of conservative systems, Poincare-Lyapunov stability theorem (loose end)

11/09 Reversible systems

11/11 Reversible vs Hamiltonian systems, gradient systems

11/13 Gradient systems, limit cycles, global Lyapunov functions, Bendixson's criterion

11/16 Dulac's criterion, Lyapunov functions, quick intro to Poincare-Bendixson theorem

11/18 Poincare-Bendixson theorem

11/20 Saddle-node bifurcation in 2D, Hopf bifurcation

11/23 Hopf bifurcation (normal form, sufficient criterion)

11/25 No class

11/27 No class

11/30 Lorenz system

11/02 Lorenz system

12/04

12/11 Final exam