MATH 4193 - Introductory Mathematical Modeling, Section 001
MATH 5103 - Mathematical Models, Section 001
Fall 2005
MWF 10:30 - 11:20 a.m., 1105 PHSC
Instructor:
Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu,
http://www.math.ou.edu/~npetrov/.
Office Hours:
W 2-3 p.m. or by appointment.
Prerequisites for MATH 4073:
MATH 3113 (Intro to ODEs) or 3413 (Physical Math I);
MATH 3333 (Linear Algebra I);
or permission of the instructor.
Prerequisites for MATH 5103:
Admission to the M.S. program or permission of instructor.
Text:
M. W. Hirsch, S. Smale, R. L. Devaney.
Differential Equations, Dynamical Systems &
An Introduction to Chaos.
2nd ed, Elsevier/Academic Press, 2004,
ISBN 0-12-349703-5.
Homeworks
Content of the lectures:
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Lecture 1 (Mon, Aug 22):
Introduction: subject of theory of dynamical systems.
The simplest equation x'=ax:
general solution, initial value problem,
equilibrium solution,
behavior of solutions for different values of a,
solution graphs and phase lines
(Sec. 1.1).
-
Lecture 2 (Wed, Aug 24):
The logistic equation x'=x(1-x):
general solution, initial value problem,
slope field, solution graphs, phase line
(Sec. 1.2).
Constant harvesting equationx'=x(1-x)-h:
number of fixed points as a function of h,
bifurcation diagram (Sec. 1.3).
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Lecture 3 (Fri, Aug 26):
Periodic harvesting equation:
Poincare map,
periodic solutions and fixed points of the Poincare map,
iterating the Poincare map graphicallly (Sec. 1.4).
Read Sec. 1.5 by yourself (it is not a very easy reading).
-
Lecture 4 (Mon, Aug 29):
Linear systems of ODEs: autonomous and non-autonomous systems,
equilibrium points.
Second order ODEs:
harmonic oscillator.
Planar systems:
geometric picture, the harmonic oscillator as a planar system.
Preliminaries from linear algebra:
2-dimensional vectors, 2x2 matrices; determinant -
relationship with the number of solutions of a linear system,
geometric meaning;
linear dependence and independence;
basis, standard basis, Cartesian coordinates in R2
(Sec. 2.1-2.3.)
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Lecture 5 (Wed, Aug 31):
Planar linear systems of ODEs:
number equilibrium points and relation with det(A),
eigenvalues and eigenvectors, characteristic equation,
solving planar linear systems, linearity principle
(Sec. 2.4-2.7).
Phase portraits for planar systems:
real distinct eigenvalues (saddle and sink cases)
(Sec. 3.1).
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Lecture 6 (Fri, Sep 2):
Phase portraits for planar systems (cont.):
real distinct eigenvalues (source case),
complex eigenvalues (center, spiral sink, spiral source),
repeated eigenvalues (for A diagonal)
(Sec. 3.2, 3.3.).
-
Lecture 7 (Wed, Sep 7):
Phase portraits for planar systems (cont.):
repeated eigenvalues with only one linearly independent
eigenvector.
The trace-determinant plane:
trace and determinant and their relationship
with the types of eigenvalues of the characteristic equation
(Sec. 3.3, 4.1).
-
Lecture 8 (Fri, Sep 9):
The trace-determinant plane (cont.):
a complete classification of the types of phase portraits
of 2-dimensional linear autonomous systems of ODEs
(Sec. 4.1).
-
Lecture 9 (Mon, Sep 12):
Changing coordinates in the plane:
matrices as linear transformations of R2,
invertibility and nonzero determinants, inverse matrix,
canonical forms of 2-dimensional systems of ODEs,
algorithm for converting a 2-dimensional system of ODEs
to its canonical form
(Sec. 3.4).
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Lecture 10 (Wed, Sep 14):
Dynamical classification of planar linear systems of ODEs:
flow of a system of ODEs, time t map of a flow,
topological conjugacy,
hyperbolic matrices and systems of ODEs,
classification of planar hyperbolic systems
(Sec. 4.2).
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Lecture 11 (Fri, Sep 16):
Dynamical classification of planar linear systems of ODEs (cont.):
classification of planar hyperbolic systems, an example
(Sec. 4.2).
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Lecture 12 (Mon, Sep 19):
The exponential of a matrix:
definition and properties of exp(tA),
using exp(tA) to write the solution
of a system of linear constant-coefficient ODEs
(Sec. 6.4).
-
Lecture 13 (Wed, Sep 21):
Nonautonomous linear systems:
Nonautonomous linear systems,
the particular case of X' = AX+G(t)
with constant matrix A,
example: forced damped harmonic oscillator;
variation of parameters, example:
existence of a periodic solution of the
forced damped harmonic oscillator
with period equal to the period of the driving
(Sec. 6.5).
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Lecture 14 (Fri, Sep 23):
Nonautonomous linear systems (cont.):
more on periodically forced damped or undamped
harmonic oscillator, resonance.
Dynamical systems:
description of physical/biological systems,
degrees of freedom, flow,
discrete and continuous dynamical systems.
The existence and uniqueness theorem:
examples of non-existence or non-uniqueness
of solutions of differential equations
(Sec. 6.5, 7.1, 7.2).
-
Lecture 15 (Mon, Sep 26):
The existence and uniqueness theorem (cont.):
statement and idea of proof (Picard iteration).
Continuous dependence of solutions:
statement, exponential divergence of solutions
starting at nearby initial points
(Sec. 7.2, 7.3).
-
Lecture 16 (Wed, Sep 28):
The variational equation:
derivation and examples
(Sec. 7.4).
-
Lecture 17 (Fri, Sep 30):
Equilibria in nonlinear systems: illustrative examples:
linearization around an equilibrium point,
an example: a global linearizing change of variables
(Sec. 8.1).
-
Lecture 18 (Mon, Oct 3):
Equilibria in nonlinear systems: illustrative examples (cont.):
an example: non-existence of a global linearizing change of variables;
an example: non-existence of a local linearizing change of variables
around a non-hyperbolic equilibrium point
(Sec. 8.1).
-
Lecture 19 (Wed, Oct 5):
Nonlinear sinks and sources:
transforming a nonlinear sink into a "canonical" form,
x'=-ax
+h1(x,y),
y'=-bx
+h2(x,y)
(where a>0, b>0);
the linearization theorem
(Sec. 8.2).
-
Lecture 20 (Mon, Oct 10):
Saddles:
linearization around a saddle point,
stable and unstable curves/surfaces/manifolds,
an example
(Sec. 8.3).
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Lecture 21 (Wed, Oct 12):
Stability:
definitions of stability, asymtptotic stability
and instability of equlibrium points.
Bifurcations:
generalities,
change of a hyperbolic equlibrium point
due to change of the parameter
(Sec. 8.4, 8.5).
-
Lecture 22 (Fri, Oct 14):
Bifurcations (cont.):
theorem about saddle-node bifurcations
(Sec. 8.5).
-
Lecture 23 (Mon, Oct 17):
Bifurcations (cont.):
pitchfork bifurcation, examples;
Hopf bifurcation - definition
(Sec. 8.5).
-
Lecture 24 (Wed, Oct 19):
Bifurcations (cont.):
Hopf bifurcation - examples,
physical systems exhibiting Hopf bifurcation
- rotating pendulum, turbulent flow
(Sec. 8.5).
-
Lecture 25 (Fri, Oct 21):
Nullclines:
definition, examples,
heteroclinic connection and heteroclinic bifurcation
(Sec. 9.1).
-
Lecture 26 (Mon, Oct 24):
Stability of equilibria:
Lyapunov stability theorem;
nonlinear pendulum with damping
- derivation of the equation from Newton's 2nd law,
non-dimensionalizing the equation,
equilibrium solutions
(Sec. 9.2).
-
Lecture 27 (Wed, Oct 26):
Stability of equilibria (cont.):
nonlinear pendulum without damping
- phase space, phase portrait, homoclinic connections,
linearization around the equilibrium solutions
(Sec. 9.2).
-
Lecture 28 (Fri, Oct 28):
Stability of equilibria (cont.):
nonlinear pendulum with damping
- linear analysis of the equilibrium solutions,
energy, Lyapunov function,
nonlinear stability of the equilibrium solutions,
phase portrait
(Sec. 9.2).
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Lecture 29 (Mon, Oct 31):
Stability of equilibria (cont.):
invariant sets, positively and negatively
invariant sets, alpha- and omega-limit sets.
Gradient systems:
definition, conditions for being a gradient
system in one- and two-dimensional cases,
interpretation of gradient systems
(Sec. 9.2, 9.3).
-
Lecture 30 (Wed, Nov 2):
Gradient systems (cont.):
Lyapunov functions for gradient systems,
types of critical points of V,
isolated minima of V and stable equilibria,
level surfaces, local representation
of V-1(c)
as a graph of a function in a neighborhood
of a regular point of V,
orthogonality of the solutions of a gradient system
and the level curves of V
(Sec. 9.3).
-
Lecture 31 (Fri, Nov 4):
Hamiltonian systems:
generalized coordinates and generalized velocities,
Lagrangian, Euler-Lagrange equations,
generalized momenta, Hamiltonian,
Hamilton's equations, energy conservation,
examples
(Sec. 9.4).
-
Lecture 32 (Mon, Nov 7):
Hamiltonian systems (cont.):
area-preservation property
of the flow in the phase plahe
of a 1-degree-of-freedom Hamiltonian system.
Limit sets:
alpha- and omega-limit sets, properties.
Local sections and flow boxes:
transversal line to the flow of a system of 2 ODEs
through a point in the phase plane,
local section
(Sec. 9.4, 10.1, 10.2).
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Lecture 33 (Wed, Nov 9):
Local sections and flow boxes (cont.):
construction of a local conjugacy
between the flow of the constant vector field
(1,0) in the (s,u) plane
and the vector field F(X).
flow box, time of first arrival.
The Poincare map:
studying the stability of a periodic orbit
of an autonomous planar system of ODEs
(Sec. 10.2, 10.3).
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Lecture 34 (Fri, Nov 11):
Monotone sequences in planar dynamical systems:
monotone sequences along a solution of a planar system
of ODEs, monotone sequence along a line segment,
proposition on monotonicity along a local section
of a monotone sequence along a solution of the sytem
(Sec. 10.4).
-
Lecture 35 (Wed, Nov 16):
An example from physics:
analysis of the motion of a bead on a rotating hoop.
-
Lecture 36 (Fri, Nov 18):
Monotone sequences in planar dynamical systems (cont.):
restrictions on number of crossings of an integral line
through an omega-limit point and any local section.
Poincare-Bendixson theorem:
statement of the theorem,
an example of a complicated set
(triadic Cantor set in [0,1])
(Sec. 10.4, 10.5).
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Lecture 37 (Mon, Nov 21):
Ruling out closed orbits:
nonexistence of closed orbits
in gradient systems and in systems
admitting a strict Lyapunov function,
Dulac's criterion, examples.
-
Lecture 38 (Mon, Nov 28):
Ruling out closed orbits (cont.):
Bendixson's criterion, examples -
nonlinear oscillator with coordinate-dependent
spring constant and resistance coefficient,
van der Pol equation;
Liouville's formula for rate of change of phase volume,
example - phase volume in the Lorenz system.
-
Lecture 39 (Wed, Nov 30):
Applications of Poincare-Bendixson theorem:
practical issues - constructing a trapping region;
an example - existence of a limit cycle in an annulus;
remarks on constructing systems of ODEs in chemical kinetics,
fast and slow manifolds.
-
Lecture 40 (Fri, Dec 2):
Applications of Poincare-Bendixson theorem (cont.):
oscillatory glycolysis - isoclines, equilibrium solutions,
preparation for applying Poincare-Bendixson theorem.
-
Lecture 41 (Mon, Dec 5):
Applications of Poincare-Bendixson theorem (cont.):
oscillatory glycolysis - constructing the trapping region
without equilibrium solutions,
condition on the parameters for existance of periodic solution,
comparison with results of computer simulations.
-
Lecture 42 (Wed, Dec 7):
Bifurcations of cycles in plane systems:
Hopf bifurcation, saddle-node bifurcations of cycles,
infinite-period bifurcations, homoclinic bifurcations;
behavior of the amplitude and the period of the cicle
around the critical value of the parameter.
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Lecture 43 (Fri, Dec 9):
Conclusion:
a brief history of nonlinear dynamics.
A brief discussion of the problems from the take-home final.
Just for fun:
Course Objectives:
This course is intended to be an introduction
to some aspects of the general theory of dynamical systems,
which is the theory that aims to describe the long-term behavior
of systems whose law of evolution is known,
but whose exact solution is not available.
The point of view adopted is that the way of understanding
the behavior of the system is to find characteristics
that organize the long-term behavior of the system.
We will study classification of linear systems of differential equations,
will develop techniques for nonlinear systems and will
disscuss applications of these techniques.
The last third of the class will be devoted to
developing the basics of the theory of discrete dynamical systems
(i.e., maps).
Course Content:
The course content below is only a rough outline;
it can be modified depending on the interest of the students.
-
First-order equations - bifurcations, periodic solutions, Poincare map.
-
Planar linear systems - eigenvalues and eigenvectors, phase portraits,
classification.
-
Higher-dimensional linear systems, the concept of genericity.
-
Nonlinear systems - existence and uniqueness theorem, continuous dependence,
variational equations.
-
Equilibria in nonlinear systems - sinks, sources and saddles, stability and bifurcations.
-
Global nonlinear techniques - stabilit of equilibria, gradient and Hamiltonian systems,
limit sets, Poincare map, Poincare-Bendison theorem.
-
Discrete dynamical systems - examples, bifurcation diagrams, symbolic dynamics,
the shift map, the Cantor set.
Grading:
Homework 40%, midterm 25%, final exam 35%.
Attendance:
You are required to attend class on those days when an
examination is being given; attendance during other class periods is also
strongly encouraged. Be advised that you are fully responsible for the
material covered in each class, whether or not you attend.
Make-ups for missed exams will be given only if
there is a compelling reason for the absence ,
which I know about beforehand
and can document independently of your testimony
(for example, via a note or phone call from a
doctor, parent, or clergyman).
Some Important Dates :
- Final Day to Register or Add a Class: Friday, August 26
- Last day to drop a class with a refund: Friday, September 2.
- Last day to drop a class without recorded grade: Friday, September 2.
- Last day to withdraw with an automatic W:
Friday, September 30.
- Last day to withdraw with a W/F without petition:
Sunday, October 30.
- Midterm exam: TBA.
- Final exam: TBA.
Academic Misconduct: All cases of suspected academic misconduct will
be referred to the Dean of the College of Arts and Sciences for prosecution
under the University's Academic Misconduct Code. The penalties can be quite
severe. Don't do it!
For more details on the University's
policies concerning academic misconduct
click here.
Students With Disabilities:
The University of Oklahoma is committed to providing reasonable accommodation
for all students with disabilities. Students with disabilities who require
accommodations in this course are requested to speak with the instructor
as early in the semester as possible. Students with disabilities must be
registered with the Office of Disability Services prior to receiving
accommodations in this course. The Office of Disability Services is located
in Goddard Health Center, Suite 166: phone 405-325-3852 or TDD (only)
405-325-4173.