## M454: Intro Dynamical Systems I

INSTRUCTOR

Mark Pernarowski

OFFICE HOURS

Schedule
TEXTBOOK:
Nonlinear Dynamics and Chaos, Steven Strogatz, 2nd ed.
The course % will be determined as follows:

 Percent Date Midterm M 25% Oct 21 in class Final F 25% 12/16 @8-9:50am Homework HW 50% see  HW1-HW5 100

Homework assignments will have different raw scores.

The HW % above is obtained from the sum of the raw scores.

Dates and details for the Midterm and Final exam will be

announced later in the course.

All grades will be recorded in D2L.

HOMEWORK:
Below the Homework and due dates will be posted.

 Due Date HW1 Thursday, Sept 16 HW2 Thursday, Sep 30 HW3 Thursday, Oct 14 HW4 Thursday, Nov 4 HW5 Thursday, Dec 2 FINAL Thursday Dec 16 (8-9:50am in class)

NOTES

Handwritten and/or typeset lecture notes here:

 Notes_0 Dynamical system examples, overview, systems, ode, Maps, PDE's and terminology Notes_1 First order ODE: stability and phase portraits Notes_2a First Order ODE: existence theorems, blowup, Picard iteration, Taylor Series Notes_2b Linear Stability for fixed points Notes_3 Comparison Theorem and Potential Functions Notes_4 Saddle and Transcritical Bifurcations Notes_5 Pitchfork bifurcations and Structural Stability Notes_6 Taylor Series, Implicit Function Theorem, Hyperbolicity, Normal Forms Notes_7 Perturbed TC and PF bifurcations and Budworm Population Model Notes_8 Dynamics on S1 (unit circle) brief overview Notes_9 Planar system notation, fixed point and stability definitions Notes_10 Brief review of linear algebra Notes_11 Linear planar systems: saddles,nodes,spirals,centers,line of equilibria Notes_12 Linear planar systems: classification and manifolds E^s, E^c and E^u (and here) Notes_13 Planar systems: nullclines, linearization,hyperbolicity,flow, H-G Theorem Notes_14 W^s,W^u, homo/heteroclinic orbits, Lotka Volterra, Conservative & Hamiltonians Notes_15 Nonlinear centers and Hamiltonian systems, Reversible systems Notes_16 Index Theory Notes_17 Polar coord, limit cycles, periods, gradient sys., dissipation, Liapunov fns. Notes_18 Dulac Criterion Notes_19 Poincare-Bendixson: positive invariance, trapping regions, omega-limit sets Notes_19a Poincare application to Glycolosis Oscillator model Notes_20 Relaxation Oscillations: Period estimates Notes_21 Bifurcations of Fixed points Notes_22 Hopf Bifurcations. Summary notes here Notes_23 Global Bifurcations of Periodic Orbits and Return Maps

MORE NOTES

Here are some supplementary notes I may refer to:    Book.pdf

EXAMS:
 Midterm 1:  (in class, date: Thursday Oct 21 - in class.) 1) Is open notes. You may bring any handwritten or typed notes but not the text. You may not use software on your computer or phone. The last page of the exam includes this List of bifurcation Theorems 2) You won't need scrap paper. You will answer questions on the exam iteself. 3) You'll need to know basic definitions and SN/TC/PF bifurcations for one T/F question. 4) The bulk of the exam will be taken from HW-HW3 5) Topics included and excluded: existence and uniqueness questions phase portraits of x'=f(x) - definitely Picard iteration - possibly Taylor series approximations of x'=f(x) - possibly Potential functions All definitions of Stability of fixed points - definitely Definition of hyperbolic points f=0, f_x=0 - definitely Saddle Node (SN), Transcritical (TC) and Pitchfork bifurcations - definitely Taylor series approximations for f(x,mu) near non-hyperbolic points- definitely Locus of nonhyperbolic in (mu,lambda)-plane - possibly (but nothing more than the locus, i.e., not the bifurcations of x, drawing 3-d surfaces. just whats on  homework 3)

 Final Exam  - (takehome, Due: TBA) 1) Topics: TBA 2) You may only use the text, your class notes or the online notes. 3) You may use software for drawing functions curves 4) You may not talk to any fellow students 5) You may ask me to clarify questions

TOPICS:
• Dynamics on R
• fixed points, stability, linear stability
• Existence, Uniqueness, Picard Iteration, Taylor series methods
• Blowup, Comparison arguments
• Elementary Bifurcations on R
• Saddle Node, Transcritical, Pitchfork bifurcations
• Normal Forms
• Stability Diagrams
• Dynamics on S^1
• Phase portraits, periodic orbits
• Bifurcations
• Asymptotic Period estimates and bottlenecks
• Compactification of dynamics on R onto S^1
• Linear Planar Systems
• Phase portraits, fixed points
• Fundamental matrix solutions
• Stability in from Tr(Df) and det(A)
• linear manifolds E^s, E^c and E^u
• Planar Systems Introduction
• Nullclines, Flow, Hyperbolic fixed points
• Homeomorphisms and Topological Equivalence
• Liapunov Stable, Attracting, isolated, asymptotically stable
• Hartman-Grobman Theorem
• Special Structures in Planar Systems
• Conservative Systems
• Hamiltonian Systems
• Reversible Systems
• Theorems for Nonlinear Centers
• Index Theory
• Definition and Calculation
• Integral formulation
• Key properties and Theorems
• Proving nonexistence of Periodic orbits
• Periodic Orbits in Planar Systems
• Conversion to Polar coordinates
• Limit Cycles and Stability Definitions
• Gradient Systems - no periodic orbits
• Dissipative systems - no periodic orbits
• Liapunov Functions - no periodic orbits
• Dulac Criterion
• Poincare-Bendixson Thms (trapping regions)
• Omega Limit sets, general Poincare-Bendixson.
• Poincare Return Maps
• 1-D Maps
• Euler's method, Newton's method, tent and logistic maps
• fixed points, stability, asymptotic stability, orbits, cobwebs
• Linear stability: taylor series and rigorous proofs
• Introductory examples including superstability
• Logistic Map: fixed points, period 2, period 3,...

COVID
POLICY

LEARNING
OUTCOMES:

Upon completion of this course, a student will be able to: Provide a qualitative bifurcation analysis of a simple one-dimensional, one parameter nonlinear differential equation; Understand and analyze basic types of linear and nonlinear oscillators; Linearize a two-dimensional non-linear system of differential equations at an equilibrium, and use this linearization to analyze the behavior of nearby solutions; Analyze dynamics of a two-dimensional nonlinear system of differential equations using a phase plane analysis.

Upon completion of this course, a student will be able to: Find fixed points and low period periodic points for simple one-dimensional maps both graphically and analyticaly; Analyze dynamics of one dimensional maps using symbolic dynamics; Understand and be able to reproduce construction of the Smale's horseshoe; Have an understanding of simple models of chaotic dynamics.

WAGA

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