MATH 4193.001/5103.001  Introductory Mathematical Modeling / Mathematical Models  Spring 2020
TR 1:302:45 p.m., 122 PHSC
Instructor:
Nikola Petrov, 1101 PHSC, npetrov AT math.ou.edu
Office Hours:
Mon 2:303:30 p.m., Tue 10:3011:30 a.m., or by appointment, in 1101 PHSC.
First day handout
A brief (tentative) list of topics to be covered:

Onedimensional systems:
flows on the line,
bifurcations,
flows on the circle.

Two (and higher)dimensional flows:
linear systems,
the phase plane,
limit cycles,
bifurcations.

Chaos:
the Lorenz equations,
onedimensional maps,
fractals,
strange attractors.
Homework:

Homework 1, due on Tuesday, February 4.

Homework 2, due on Tuesday, February 18.

Homework 3, due on Tuesday, March 3.

Homework 4, due on
Gradescope
by 11:59 p.m. on Thursday, March 26.

Homework 5, due on
Gradescope
by 11:59 p.m. on Saturday, April 18.

Homework 6, due on
Gradescope
by 11:59 p.m. on Saturday, May 2.
A project for MATH 5103:
Content of the lectures:

Lecture 1 (Tue, Jan 14):
Introduction:

ODEs vs. PDEs;

ODE + initial condition (IC) = initial value problem (IVP);

examples from population dynamics:

simplest model P'(t)=kP(t),
problems with this model (unbounded exponential growth),

correction accounting for the limited amount of resources
 logistic equation,
P'=kP(1−P/M),
where M is the carrying capacity of the system;

making the model more realistic: seasonal variation of the living conditions, harvesting,
existence of predators, random events or conditions,...

an example of an IVP without solution:
x'=ƒ(x),
where ƒ(x)=−1 for x≥0 and
ƒ(x)=1 for x<0;

an example of an IVP with infinitely many solutions:
dV/dt=βV^{2/3} (with β=const>0),
V(0)=0, describing the volume of a water droplet
in oversaturated vapor right at the moment of its formation;

linear vs. nonlinear equations;

nonlinear equations are so hard to solve because the principle
of superposition fails (hence no representation of the solution
as a sum of independent solutions, no Fourier transform,
no Laplace transform...).

Lecture 2 (Thu, Jan 16):
Systems of ODEs and their flows:

systems of autonomous firstorder ODEs
x'=f(x) for the unknown function
x:[0,T]→R^{n}
(T is a positive number), where
f:R^{n}→R^{n} is a given function;

nonautonomous firstorder systems: x'=f(x,t);

converting a single higherorder ODE into a system of 1storder ODEs;

geometric representation of the solution of the IVP
x'=f(x), x(0)=x^{(0)}
as a parameterized curve in R^{n} called flow,
φ_{t}(x^{(0)}),
where for every t∈[0,T],
φ_{t}:R^{n}→R^{n}
maps the initial condition x^{(0)}∈R^{n}
to the "present" position, x(t)=φ_{t}(x^{(0)});

the flow satisfies
dφ_{t}(x^{(0)})/dt=f(φ_{t}(x^{(0)})),
φ_{0}(x^{(0)})=x^{(0)}
(the latter can be written as
φ_{0}=Id_{Rn}
(where Id_{Rn} is the identity function in
R^{n});

semigroup property of the flow:
φ_{t}(φ_{s}(x^{(0)}))=φ_{t+s}(x^{(0)}),
i.e.,
φ_{t}∘φ_{s}=φ_{t+s};

corollaries of the semigroup property:
φ_{0}=Id_{Rn},
(φ_{t})^{−1}=φ_{−t}.
Linear constantcoefficient systems of ODEs:

(homogeneous) linear constantcoefficient systems of ODEs:
x'=Ax, where A is a constant n×n matrix;

definition of an exponent of a square matrix;

proof that the solution of the IVP x'=Ax,
x(0)=x^{(0)} is given by
φ_{t}(x^{(0)})=e^{tA}x^{(0)};

example: exponentiating a diagonal matrix with distinct diagonal elements;

if the n×n matrix A has real and distinct eigenvalues,
then it can be diagonalized, i.e., there exists an invertible matrix C such that
D=CAC^{−1} is diagonal;

if A=C^{−1}DC,
then
e^{tA}=e^{tC−1DC}=C^{−1}e^{tD}C.

Lecture 3 (Tue, Jan 21):
Fun fact:

for a smooth function ƒ:R→R, if D_{a}(ƒ):=ƒ'(a) and h∈R,
then exp(hD_{a})(ƒ)=ƒ(a+h).
From flows of ODEs to Poincaré maps:

consider a system of (autonomous) ODEs x'=f(x) in R^{n}
whose flow is φ_{t};
an integral line of the flow is a curve
{φ_{t}(x^{(0)}):t∈R} in R^{n};

a periodic solution with (minimal) period T: a solution such that
φ_{t+T}(x^{(0)})=φ_{t}(x^{(0)})
for all t∈R;

the integral line of a periodic solution is a closed curve in R^{n};

for a given flow φ_{t},
a Poincaré surface Σ is a (piece of) hypersurface
(i.e., an (n−1)dimensional surface in R^{n})
that is "transversal" to the integral lines of the ODE
(i.e., at a point of intersection of an integral line
and Σ, the tangent line to the integral line does not belong
to the tangent plane to Σ at this point);

a digression: span of a set of vectors, transversality;

remark: the integral lines do not intersect (unless f becomes zero);

definition of the Poincaré map P:Σ→Σ for the ODE x'=f(x)
(and a given choice of Σ);

consecutive intersections of a given integral line with Σ can be interpreted as iterations of the
Poincaré map: P^{k}(x):=P∘P∘...∘P(x)
(k times);

usefulness of the Poincaré map: lowering the dimension;

a digression on the importance of dimension:

differences between 1dimensional and 2dimensional problems:
a point moving on R from the negative half of R cannot
go to the positive half without going through 0;

differences between 2dimensional and 3dimensional problems:
in 2 dimensions, a nonselfintersecting line that "spirals in"
must "spiral out" in order to leave the domain.
Maps  basic concepts:

a map  a function that is meant to be iterated;

goal: to understand the behavior of the high iterates
ƒ^{k}(x) for very large k;

iterating a onedimensional map ƒ:R→R graphically  a cobweb plot;

fixed points on a cobweb plot  the xcoordinate
of the intersection of the graph of ƒ with the diagonal y=x;

stable and unstable fixed points.

Lecture 4 (Thu, Jan 23):
From flows of ODEs to Poincaré maps (cont.):

a periodic orbit of the Poincaré map corresponds to a periodic orbit
of the ODE (i.e., a closed curve in R^{n}).

a picture of a flow in R^{2} and a Poincaré map
(related to Problem 3 of Homework 1).
Onedimensional maps ƒ:R→R:

stability of a fixed point of a map;
deriving a criterion for stability of a FP x^{*}:
if ƒ(x^{*})<1, then the FP x^{*} is stable,
if ƒ(x^{*})>1, then the FP x^{*} is unstable;

observation: a periodic point of ƒ of period m
is a fixed point of the iterated map ƒ^{m};

an example: the logistic map ƒ_{μ}:[0,1]→[0,1] given by
ƒ_{μ}(x)=μx(1−x) (where μ>0 is a parameter):

computing the fixed points of the map:
x^{*}=0 and x^{*}=(μ−1)/μ;

computing the stability of the fixed point x^{*}=(μ−1)/μ
for μ>1 (if μ<1, the fixed point is not in the interval [0,1]):
ƒ_{μ}'(x^{*})=2−μ, so
x^{*}=(μ−1)/μ is stable for μ∈(1,3)
and unstable for μ>3;

numerical illustrations of the phenomena by using Mathematica;
to experiment with this yourself, download and run the Mathematica notebooks
graphicaliterationof1dimmaps.nb
taken from a code
(link) written by
Sander Huisman (University of Twente);
the code plots 1000 iterations of the map ƒ_{μ}(x)=μx(1−x)
with μ=3.1 and initial point x_{0}=0.1;
note that, to execute a Mathematica command, you have to put the cursor on the command,
press and hold down the SHIFT key and, while holding it down, press RETURN;

observing in Mathematica how, when the parameter μ increases,
the map undergoes a period doubling, i.e., the fixed point
x^{*}=(μ−1)/μ loses stability as μ passes through 3,
and a stable periodic orbit of period 2 is born; for the values at which consecutive period doublings occur,
see the second table in the Wikipedia page on
Feigenbaum constants.

Lecture 5 (Tue, Jan 28):
Onedimensional maps ƒ:R→R (cont.):

Intermediate Value Theorem (IVT): if ƒ:[a,b]→R is continuous
and K is a value between ƒ(a) and ƒ(b),
then there exists c∈[a,b] such that ƒ(c)=K;

Condition for uniqueness of the point c from the IVT:
assume additionally that ƒ is differentiable
and ƒ'(x) is either >0 for any x∈[a,b]
or <0 for any x∈[a,b], then c is unique;

using the IVT to prove the existence of a fixed point x^{*}
of a continuous map ƒ:[a,b]→R
if ƒ([a,b])⊆[a,b]
 a graphical proof and a formal proof (apply the IVT
to the function g(x)=ƒ(x)−x
to conclude that there exists a point
x^{*}∈[a,b] such that
g(x^{*})=ƒ(x^{*})−x^{*}=0,
i.e., ƒ(x^{*})=x^{*});

on the uniqueness of the FP x^{*} of ƒ:[a,b]→R:
if ƒ is differentiable and ƒ'(x)<1 for any x∈[a,b],
then the FP x^{*} is unique;

this recipe can be used to prove existence of periodic points:
recall that a periodic point of ƒ of period k is a fixed point of the iterated map
ƒ^{k};

an exercise: take a calculator, make sure that it is set to working in radians,
type any number x_{0}, and start pressing COS many times
(i.e., you are iterating the function cos(x))  the iterates will converge
to x^{*}=0.73908513...; it is obvious that this fixed point
of the map ƒ(x)=cos(x) is attracting (why?).
Flows on the line  a geometric picture:

goal: to understand the longtime behavior of all solutions of the ODE x'=ƒ(x);

tool: looking at the graph of ƒ in the (x,x')plane;

a digression:
Euler method for approximate solution of an ODE:
x(t_{i+1})=x(t_{i}+Δt)=x(t_{i})+x'(t_{i})Δt+(1/2!)x''(t_{i})(Δt)^{2}+...≈x(t_{i})+x'(t_{i})Δt=x(t_{i})+ƒ(x(t_{i}))Δt;

if x^{*}∈R is a value such that ƒ(x^{*})=0,
then the (unique under appropriate condition on ƒ) solution of the IVP
x'=ƒ(x), x(0)=x^{*}
is x(t)=x^{*} for all t≥0;

studying the stability of a fixed point x^{*}
directly from the graph of ƒ in the (x,x')plane
(often called the phase plane) of the system:
look for the values of x for which ƒ(x) is zero;

an example:
x'=(x−1)(x−3)^{2}(x−5)^{3};

stable FPs (atracting FPs, attractors, sinks),
unstable FPs (repelling FPs, repellers, sources);

semistable fixed points, like the fixed point x^{*} in the ODE
x'=(x−1)(x−3)^{2}(x−5)^{3}.

Lecture 6 (Thu, Jan 30):
Flows on the line  a geometric picture (cont.):

continuation of the qualititative analysis of the behavior of the solutions of the ODE
x'=(x−1)(x−3)^{2}(x−5)^{3}:

deriving without derivatives that near the FP x_{1}^{*}=1,
ƒ(x)≈−256(x−1), so this FP is stable;

deriving without derivatives that near the FP x_{2}^{*}=3,
ƒ(x)≈−16(x−3)^{2}, so this FP is semistable;

exercise: do this for the FP x_{3}^{*}=5 to find the stability of this FP;

Taylor expansions of
ƒ(x)=(x−1)(x−3)^{2}(x−5)^{3}
near the fixed points:

near x_{1}^{*}=1:
ƒ(x)=ƒ(1)+ƒ'(1)(x−1)+(1/2!)ƒ''(1)(x−1)^{2}+...≈ƒ'(1)(x−1)=−256(x−1),
so the graph near x_{1}^{*}=1 is a straight line with slope −1;

near x_{2}^{*}=3:
ƒ(x)=ƒ(3)+ƒ'(3)(x−3)+(1/2!)ƒ''(3)(x−3)^{2}+...≈(1/2)ƒ''(3)(x−1)^{2}=−16(x−3)^{2},
so the graph near x_{2}^{*}=3 is an upsidedown parabola;

near x_{3}^{*}=5:
ƒ(x)=ƒ(5)+ƒ'(5)(x−5)+(1/2!)ƒ''(5)(x−5)^{2}+(1/3!)ƒ'''(5)(x−5)^{3}+...≈(1/3!)ƒ'''(5)(x−5)^{2}=16(x−5)^{3},
so the graph near x_{3}^{*}=5 is a cubic parabola;

idea of a generic situation (when the number and/or type of fixed points
does not change dramatically under arbitrarily small changes of the values of the parameters),
and of a nongeneric situation (when dramatic changes may occur due to arbitrarily small values of the parameters);

an example:
consider number of roots of an algebraic equation ƒ_{μ}(x)=0 depending on a parameter μ:

consider the linear equation ax+b=0 with a≠0:
if the values of a and b depend on μ, then for very small changes of μ
the equation still has one root;

consider the quadratic equation x^{2}+μ=0:

if μ>0, the equation does not have any roots;

if μ=0, the equation has one (double) root, x=0;

if μ<0, the equation has two distinct roots, x_{1}=−μ^{1/2} and
x_{2}=μ^{1/2}

exercise: show that the equation x^{3}+μx=0 has

one simple root if μ>0 (a root x^{*} of the equation ƒ(x)=0
is said to be simple if ƒ(x^{*})=0 and
ƒ'(x^{*})≠0);

one triple root if μ=0 (a root x^{*} of the equation ƒ(x)=0
is said to be triple if ƒ(x^{*})=0,
ƒ'(x^{*})=0, ƒ''(x^{*})=0,
but ƒ'(x^{*})≠0);

three simple roots if μ<0 (you can easily find the values of these roots);

a nongeneric situation in an ODE : the ODE
x'=(x−1)(x−3)^{2}(x−5)^{3}
has three fixed points: an attracting FP at 1, a semistable FP at 3, and a repelling one at 5;
the semistable fixed point x_{*}=3 in the ODE
x'=(x−1)(x−3)^{2}(x−5)^{3}
can be destroyed easily by adding or subtracting an arbitrarily small number
to the righthand side: for arbitrarily small ε>0, the ODE
x'=(x−1)(x−3)^{2}(x−5)^{3}−ε
has only two FPs (a stable FP slightly smaller than 1, and an unstable FP slightly bigger than 5);
on the other hand, for arbitrarily small ε>0, the ODE
x'=(x−1)(x−3)^{2}(x−5)^{3}+ε
has four fixed points (a stable FP slightly bigger than 1, an unstable FP slightly smaller than 3,
a stable FP slightly bigger than 3, and an unstable FP slightly smaller than 5)
 all this is obvious from the graph of ƒ in the phase plane;
in fact, the FP at 5 of the original equation (which comes from the factor
(x−5)sup>3 in the righthand side of the ODE)
is also nongeneric (this is not so obvious, we will study this soon).
Saddlenode (tangent, blue sky) bifurcation in a 1parameter family of ODEs on R:

a bifurcation  an abrupt change of the behavior of the solutions
for an arbitrarily small change of the parameters;

example: x'=ƒ_{μ}(x)=μ+x^{2}:

for μ>0, there are no fixed points,the solution x(t) increases unboundedly;

for μ=0, 0 is a semistable fixed point;

for μ<0, there are two fixed points: an attracting one, −μ^{1/2},
and a repelling one, μ^{1/2};

graphs of solutions x(t) for different initial conditions;

a bifurcation diagram  a plot of the position of the fixed poitns as functions of the parameter μ;

a more complicated example  the 1parameter family of ODEs
x'=ƒ_{μ}(x)=μ−x−e^{−x}:
writing ƒ_{μ}(x) as a difference of two functions:
ƒ_{μ}(x)=φ_{μ}(x)−ψ(x),
with φ_{μ}(x)=μ−x,
ψ(x)=e^{−x},
and rewriting the condition for a fixed point,
ƒ_{μ}(x)=0,
as condition for equality φ_{μ}(x)=ψ_{μ}(x),
in order to perform a graphical analysis.
Suggested reading (if the discussion in class was not enough):
Sec. 1.7 of Layek (try to understand the pictures),
Sec. 1.2 and 1.3 of HirschSmaleDevaney (the most important thing for us are the pictures).

Lecture 7 (Tue, Feb 4):
Saddlenode (tangent, blue sky) bifurcation in a 1parameter family of ODEs on R (cont.):

finishing the calculations from the complicated example from Lecture 6:

computing the critical value μ_{c}=1 and the fixed point
x^{*}_{c}=0 for μ=μ_{c}
at which the graphs of φ_{μ}(x) and ψ(x) are tangent;

Taylor series of a function of two variables,

expansion of ƒ_{μ}(x) in a Taylor series near the point (μ,x)=(1,0):
ƒ_{μ}(x)≈(μ−1)−(1/2)(x−0)^{2};

computing the approximate values of the fixed points for μ
slightly above μ_{c}, so that only the lowestorder terms
in the Taylor expansion can be retained while the rest of the terms are ignored:
x^{*}≈±[2(μ−1)]^{1/2};

bifurcation diagram  positions of the fixed points as functions of the parameter μ.
Supercritical pitchfork bifurcation in a 1parameter family of ODEs on R:

studying the bifurcation in the 1parameter family
x'=ƒ_{μ}(x)=−x^{3}+μx:
when μ<0, there is only one stable FP, namely 0,
while for μ>0, the FP 0 becomes unstable while two stable FPs are born (one positive, one negative);

bifurcation diagram of the supercritical pitchfork bifurcation;

a physical example of a system that undergoes a supercritical pitchfork bifurcaition:
a ferromagnet, whose magnetization M is described by the equation M=tanh(M/T)
(where T is related to the temperature)
 introducing a new variable x:=M/T,
graphical analysis of the equation Tx=tanh(x),
analysis near the critical value T_{c}=1:
for temperatures slightly below T_{c},
x^{*}≈±[3(T−T_{c})]^{1/2}.

Lecture 8 (Thu, Feb 6):
More complicated bifurcations:

bifurcations in the 2parameter family
x'=ƒ_{μ,μ}(x)=−x^{3}+μx+ν
which exhibits a supercritical pitchfork bifurcation and two saddlenode bifurcations.
Logistic equation with predation:

derivation of a system modeling the dynamics of the budworm population accounting
for the limited resources and the predation:
where R>0 is the reproductive rate, K>0 is the carrying capacity,
and A>0 and B>0 are positive constants characterizing the predation;

remark: there are two very different timescales in the system (the life expectancy of the worms
and the life expectancy of the birds), so the population of birds is considered constant;

reducing the number of parameters from 4 to 2 by introducing nondimensional
variables x:=X/A and t:=BT/A
and nondimensional parameters k:=K/A>0 and
r:=RA/B>0, so that the equation becomes

x_{*}=0 is always an unstable fixed point;

the other fixed points are solutions of the equation
φ_{r,k}(x)=ψ(x),
where
φ_{r,k}(x)=r(1−x/k),
ψ(x)=x/(1+x^{2});

elementary facts about ψ(x)=x/(1+x^{2}):
it has a maximum at 1 with ψ(1)=1/2 and an inflection point at 3^{1/2}
with ψ(3^{1/2})=3^{1/2}/4 and ψ'(3^{1/2})=−1/8.

Lecture 9 (Tue, Feb 11):
Logistic equation with predation (cont.):

empirical observations from looking at the intersections of the graphs
of φ_{r,k}(x)
and ψ(x) in order to locate the (nonzero) fixed points:

the system can have 1, 2, or 3 (nonzero) FPs,

if r is too large, then for any value of k the system has only one
(nonzero) FP,

if k is too small, then for any value of r the system has only one
(nonzero) FP,

if the straight line representing φ_{r,k}(x)
is steeper than −1/8 (which is the slope of the tangent line
to the graph of ψ(x) at the inflection point),
then there cannot be more than one (nonzero) FP;

the system undergoes tangent (saddlenode, blue sky) bifurcations
when the graphs of φ_{r,k}(x)
and ψ(x) are tangent at some point;

looking for tangent bifurcations:
the conditions for tangency are
φ_{r,k}(x^{*})= ψ(x^{*})
(common point),
φ'_{r,k}(x^{*})= ψ'(x^{*})
(common slope);

expressing the pairs of values (k,r) at which tangent bifurcations
occur as a parameterized curve in the (k,r)plane:
since k>0 (and x>0), from the expression for k we see that x^{*} must be
greater than 1;

the bifurcation curve separating the domains in the (k,r)plane
where the system has 1 nonzero FP and where it has 3 nonzero FPs
has a shape of a horn with a tip at
(k_{c},r_{c})=(3^{3/2},3^{3/2}/8)≈(5.196,0.6495);

interpretation of the mathematical results:
"refuge" vs. "outbreak" (in the case when there are a total of 4 FPs)
 bistability;

hysteresis in the system  the state of the system may depend on the "history"
(i.e., how the sytem came to this state).

Lecture 10 (Thu, Feb 13):
Introduction to higherdimensional systems:

general setup: t∈[0,∞) time,
x:[0,∞)→R^{n} unknown function,
f:R^{n}×[0,∞)→R^{n} given function;
ODE: x'=f(x,t), IC: x(t^{(0)})=x^{(0)},
IVP=ODE+IC;

classifications: autonomous vs. nonautonomous, linear vs. nonlinear;

example: Newton's second law for a point particle of mass m:
if x(t) position at time t,
v(t)=x'(t) velocity at time t,
a(t)=v'(t)=x''(t) acceleration at time t,
F_{net}(x,v,t) total force acting on the particle
(sum of all the forces acting on the particle),
then x''=F_{net};

writing Newton II for a point particle as a system of 6 ODEs, each of order 1;

number of degrees of freedom (dof) of a system of particles; examples:

one point particle: dof=3,

two point particles: dof=6,

two point particles at the ends of a hard rod: dof=5,

three point particles attached to the vertices of a hard triangle: dof=6;

a nondeformable 3dimensional object: dof=6;

a deformable 3dimensional object: dof=∞;

example: planar pendulum with airresistance:

coordinate: x=θ/(2π)∈S^{1},
velocity: y=x'∈R,
so that the phase space of the pendulum is the infinite cylinder
S^{1}×R;

introduce airresistance: Newton II is
x''+γx'+(g/l)sin(x)=0,
which can be written as a system of 1st order ODEs: set
x_{1}(t)=x(t),
x_{2}(t):=x_{1}'(t),
so that Newton II becomes
x_{1}'=x_{2},
x_{2}'=−(g/l)sin(x_{1})−γx_{2};

FPs (steadystate solutions): (0,0) and (π,0);

linearizing the system near the FP (0,0) (which should be stable by our physical intuition):
for (x_{1},x_{2})≈(0,0),
Newton II becomes
x_{1}'=x_{2},
x_{2}'=−(g/l)x_{1}−γx_{2};

converting the linearized system to a single 2nd order ODE:
x''+2βx'+ω_{0}^{2}x=0,
where 2β:=γ>0, ω_{0}^{2}:=g/l>0;

characteristic equation:
r^{2}+2βr'+ω_{0}^{2}=0,
solutions:
r_{1,2}=−β±(β^{2}−ω_{0}^{2})^{1/2}.

Lecture 11 (Tue, Feb 18):
Introduction to higherdimensional systems (cont.):

derivation of energy conservation from Newton's 2nd law;

derivation of the rate of change of the total mechanical energy
of the planar pendulum expressed in terms of the power of the airresistance force;

conservative (potential) forces in 1 spatial dimension: for which
F(x)=−U'(x)
(where U(x) is the potential energy);

for potential forces: plotting the total energy and the potential energy on the same plot
(as functions of the coordinate), determining the classically allowed region;

generalization: conservative forces in 3 spatial dimensions:
F(x)=−∇U(x);

symmetries imply conservation laws:

homogeneity of time implies conservation of energy,

homogeneity of space implies conservation of momentum,

isotropy of space implies conservation of angular momentum;

back to the planar pendulum in the absence of air resistance:
the energy conservation can be used to plot the velocity
θ' as a function of the position θ
in the (θ,θ')plane;
the integral lines of the Newton's 2nd law
(or, equivalently, the system of two 1st order ODEs)
coincide with the level curves of the energy in the (θ,θ')plane (why?).

Lecture 12 (Thu, Feb 20):
Introduction to higherdimensional systems (cont.):

conserved quantities (first integrals);

proof that the energy is a first integral for the planar pendulum
in absence of air resistance;

global phase space diagram of the planar pendulum:
stable and unstable equilibria, separatrices, librations and rotations;

linearization of the system of the 1st order equations of the planar pendulum
(in absence of air resistance) near the fixed point (θ,θ')=(0,0);

solution of the linearized system (ellipse in the phase plane).
Linearization of nonlinear systems:

consider a general autonomous system
dx=f(x) with a fixed point x^{*}
determined by f(x^{*})=0;

Taylor expansion of f(x) near the fixed point x^{*}:
f(x^{*}+u)=f(x^{*})+Df(x^{*})u+h.o.t.=Df(x^{*})u+h.o.t.
(h.o.t. stands for "higher order terms", i.e., terms that are of order higher than linear
with respect to the components u_{i} of the small displacement u);

set x(t)=x^{*}+u(t) in the original ODE
and linearize about the FP x^{*} to derive the linearized equation
u'=Ju, where J=Df(x^{*})
is the Jacobian (derivative) matrix with constant entries,
J_{ij}=∂ƒ_{i}/∂x_{i}(x^{*}).

Lecture 13 (Tue, Feb 25):
Classification of linear systems:

a detailed analysis of the linear system

a stable node (a<0, a≠−1),

a star (a=−1),

a line of fixed points (a=0),

a saddle point (a>0);

eigenvalues λ_{j} of a 2×2 matrix A
 solutions of the quadratic equation
λ^{2}−τλ+Δ^{2}=0,
where τ=tr(A), δ=det(A);

proof that the roots of a polynomial equation with real coefficients
are either real or they come as pairs of complex conjugate numbers;

corollary: the eigenvalues of an n×n matrix with real entries
are either real or come in pairs of complex conjugate numbers;
in particular, if n is odd, then there is at least one real eigenvalue;

the tracedeterminant plane for a 2×2 matrix A:

if Δ=τ^{2}/4, then A has one double real eigenvalue;

"inside" the parabola Δ=τ^{2}/4 (i.e., if τ^{2}/4<Δ),
A has two complex conjugate nonreal eigenvalues;

"outside" the parabola Δ=τ^{2}/4,
A has two distinct real eigenvalues.

Lecture 14 (Thu, Feb 27):
Classification of linear systems (cont.):

Mathematica command to plot integral lines of the system
x'=3x+y,y'=2y
for x and y in the range [−2,2]:
StreamPlot[{3*x, 2*y}, {x, 2, 2}, {y, 2, 2}]

if the constant 2×2 matrix A in the ODE x'=Ax has real eigenvalues:

λ_{1}<λ_{2}<0: stable node;

λ_{1}<0<λ_{2}: saddle point;

0<λ_{1}<λ_{2}: unstable node;

λ_{1}=0, λ_{2}≠0: a line of attracting (if λ_{2}<0)
or repelling (if λ_{2}>0) fixed points;

λ_{1}=λ_{2}=0: all points are fixed;

λ_{1}=λ_{2}:

A diagonalizable, i.e.,
:
in this case A has two distinct eigenvectors: the phase portrait is
a stable star (λ<0) or an unstable star (λ>0);

A nondiagonalizable, in which case
:
in this case A has only one eigenvector: the phase portrait is
a stable degenerate node (λ<0) or an unstable degenerate node (λ>0);

if the constant 2×2 matrix A in the ODE x'=Ax has nonreal eigenvalues:

λ_{1,2}=α±iβ (α,β∈R, α≠0):
stable spiral (α<0) or unstable spiral (α>0);

λ_{1,2}=±iβ (β∈R): center;

proof of the invariance of the trace and determinant with respect of similarity transformations:
tr(CAC^{−1})=tr(A),
det(CAC^{−1})=det(A).
Assembling the global picture from local information:

rabbits versus sheep:

linear analysis of the fixed points;

basins of attraction of the stable fixed points,

stable and unstable manifolds of the saddle point.
Suggested reading (mandatory):
Suggested reading (optional):

in Chapters 2 and 3 of the book by Hirsch, Smale, and Devaney
you can find a detailed analysis of 2×2 matrices and
the corresponding linear systems of ODEs;

a readable introduction to higherdimensional Linear Algebra with many examples
can be found in Chapter 5 of the book by Hirsch, Smale, and Devaney.

Lecture 15 (Tue, Mar 3):
Assembling the global picture from local information (cont.):

more old and new concepts:

fixed point,

periodic orbit,

stable and unstable manifolds,

heteroclinic connection,

homoclinic connection
(an example: the separatrix curve in the phase portrait of a pendulum);

a digression: solving a linear system of ODEs
u'=Ju (with J a constant matrix) by using Linear Algebra:

simple case 1: if all eigenvalues
λ_{1},...,λ_{n} are real and distinct,
the general solution is a linear superposition of terms of the form
e^{λjt}v_{j}
(where v_{j} is an eigenvector with eigenvalue
λ_{j});

simple case 2: if α±iβ are simple eigenvalues,
find a complex eigenvector v corresponding to α+iβ,
then the real and the imaginary parts of
e^{(α+iβ)t}v=e^{αt}[cos(βt)+i sin(βt)]v
are linearly independent solutions of the ODE.

Lecture 16 (Thu, Mar 5):
On the importance of the hyperbolicity of a fixed point:

HartmanGrobman Theorem: If x^{*}
is a hyperbolic fixed point of x'=f(x),
then there exists a change of variables in a neighborhood
of x^{*} that transforms the original system into
its linearization,
u'=Df(x^{*})⋅u, at x^{*};

example of a hyperbolic fixed point (Problem 1 from Homework 3):
x'=1−x+y^{4}, y'=y
 its linearization at the FP (1,0) is
u_{1}'=−u_{1},
u_{2}'=u_{2}
(hence the FP is hyperbolic),
the change of variables
X(t)=x(t)−(1/5)y(t)^{4}−1,
Y(t)=y(t)
transforms the original system into its linearization at the FP (0,0),
visually the zoom near the FP (0,0) into the integral lines of the original system
looks like the linearized system;

example of a nonhyperbolic FP: the system x'=x^{2}, y'=y
has a nonhyperbolic fixed point at (0,0),
the solution of the original system is
x(t)=(x_{0}^{−1}−t)^{−1},
y(t)=y_{0}e^{−t},
the linearized system is
u_{1}'=0, u_{2}'=−u_{2}
with solution
u_{1}(t)=u_{10}, u_{2}(t)=u_{20}e^{−t},
so that the solutions of the original system and of its linearization at (0,0)
behave very differently;

example of a nonhyperbolic FP: the system
x'=−y+μx(x^{2}+y^{2})^{1/2},
y'=x+μy(x^{2}+y^{2})^{1/2}
has a nonhyperbolic fixed point at (0,0),
the integral curves of the linearized system,
u_{1}'=−u_{2}, u_{2}'=u_{1}
are circles (i.e., the FP is a center);
a polar change of variables transforms the original system into
r'=μr^{2}, θ'=1,
whose solutions spiral in for μ<0 and spiral out for μ>0.

Lecture 17 (Tue, Mar 10):
Energy conservation:

assume that the net force F_{net} is conservative (potential),
i.e., that it depends only on the coordinate
and there exists a function V:R^{n}→R
such that F_{net}(x)=−∇V(x):
in this case Newton's Second Law reads
mx''=F_{net}(x)=−∇V(x);

multiply mx''=−∇V(x) by x'
to show that the quantity (1/2)mx'^{2}+V(x)
is conserved, i.e., if x(t) is a solution of Newton's Second Law,
then d/dt[(1/2)mx'(t)^{2}+V(x(t))]=0;

kinetic energy (1/2)mx'^{2}
and potential energy V(x);

recognizing that F:U→R^{2}, U⊆R^{2},
is conservative: if U is simplyconnected and
F(x)=P(x,y)i+Q(x,y)j,
then
F is conservative if and only if ∂P/∂y=∂Q∂x;

recognizing that F:U→R^{3}, U⊆R^{3},
is conservative: if U is a ball, then
F is conservative if and only if curl F=0;

example of computing the potential energy V(x) from the force
F(x)=P(x,y)i+Q(x,y)j
in two dimensions if ∂P/∂y=∂Q∂x:
if F(x)=−cos(y)i+(xsin(y)−5)j,
then V(x)=xcos(y)+5y+C
(where C is a constant);

example of a 1dimensional dissipative system:
oscillator with air resistance:
mx''+γx'+kx=0,
with γ>0, k>0,
where the term γx' corresponds to the air resistance:
the energy is E=(1/2)mx'^{2}+(1/2)kx^{2},
and dE/dt=−γx'^{2}≤0,
so that the power of the air resistance force is −γx'^{2}≤0.

Lecture 18 (Thu, Mar 12):
Energy conservation (cont.):

in a system with one degree of freedom, if the energy E is conserved,
then the problem can be solved exactly by integrating
dx/dt=±[(2/m)[E−V(x)]^{1/2};

example: V(x)=−(1/2)x^{2}+(1/4)x^{4},
then F(x)=−V'(x)=x−x^{3},
writing Newton's Second Law as a system for
x(t) and y(t)=x'(t),
finding the fixed points and linearizing the system at each fixed point,
homoclinic connections, finding the frequencies of the small oscillations
near the centers (elliptic fixed points);
analyzing the problem by looking at the graph of V(x).
Limit cycles:

limit cycle  an isolated closed trajectory;

attracting, repelling, and semiattracting limit cycles;

remark: in linear systems there are no limit cycles
(periodic trajectories may exist, but they are not isolated);

van der Pol equation (1926):
x''+μ(x^{2}−1)x'+x=0
(μ>0 parameter)
 exhibits a limit cycle.
Lyapunov functions:

a globally asymptotically stable FP x^{*} of the ODE x'=f(x):
a FP such that for every initial condition in an some open neigborhood of
x^{*}, the solution x(t) tends to x^{*}
as t→∞;

a Lyapunov function for the equation x'=f(x) with a FP x^{*}:
a C^{1} function V:U→R (with U⊆R^{n}
such that V(x)>0 for all x≠x^{*},
V(x^{*})=0
and such that if x(t) is a solution of the ODE
starting in some open neighborhood of x^{*} with x(t)≠x^{*},
then (d/dt)V(x(t))<0;

Theorem: If a Lyapunov function exists near the FP x^{*},
then x^{*} is globally asymptotically stable;

example: x'=−x+4y, y'=−x−y^{3},
then (0,0) is a FP, and
V(x,y)=x+4y^{2} is a Lyapunov function,
therefore (0,0) is a globally asymptotically stable FP.

Lecture 19 (Tue, Mar 24):
Ruling out closed orbits:

Bendixson's Negative Criterion,
proof (using Greene's Theorem);

example: Liénard equation:

x''+ƒ(x)x'+g(x)=0;

physical interpretation (a point mass subjected to a restoring force and nonlinear damping);

the van der Pol equation is a particular case of a Liénard equation;

application of the Bendixson's Negative Criterion:
in a simply connected region in R^{2},
if ƒ(x) is of one sign, then Liénard equation has no periodic orbits;

example: the system
x'=−y+(x^{2}+x^{2}−1),
y'=x+(x^{2}+y^{2}−1),
has no periodic orbits inside the circle of radius 2^{1/2} centered at (0,0);

example: rabbits vs. sheep: the system
x'=x(3−x−2y),
y'=y(2−x−y)
has no periodic orbits in 1st quadrant that intersect the straight line
5−3x−4y=0;

Dulac's Criterion for ruling out periodic orbits,
proof (using Greene's Theorem);

using Dulac's Criterion for ruling out periodic orbits in 1st quadrant
in the rabbits vs. sheep problem.
Establishiing the existence of periodic orbits:

PoincareBendixson Theorem;

construction of a trapping region
(guaranteeing the existence of a trajectory that is "confined"
in the domain R);

example: constructing a trapping region
and proving the existence of a periodic orbit of the system
r'=r(1−r^{2})+μrcos(θ),
θ'=1, for small positive values of μ.
Optional reading: Sections 5.4.3, 5.4.4, 5.5.1 of Layek's book

Lecture 20 (Thu, Mar 26):
Review of bifurcations of flows in 1 dimensional systems:

"definition" of bifurcation;

fixed points of a system of ODEs;

stability of the fixed points  definition,
visual and analytical determination of the stability;

saddlenode (blue sky, tangent) bifurcation,
prototype: x'=μ+x^{2};

supercritical pitchfork bifurcation,
prototype: x'=μx−x^{3};

subcritical pitchfork bifurcation,
prototype: x'=μx+x^{3};

transcritical bifurcation,
prototype: x'=x(x−μ)
(see Problem 2 in Homework 2).
Bifurcations of flows in 2 dimensional systems:

saddlenode bifurcation in a prototypical example:
x'=μ+x^{2},
y'=−y;

"ghosts"  regions of very slow motion near the locations
of the FPs that "have just disappeared";

nullclines; FPs are the intersections of the x'=0
and the y'=0 nullclines;

example of finding the nullclines:
x'=x+e^{−y},
y'=1−y;

detecting saddlenode bifurcations by using nullclines
 bifurcations occur when the x'=0
and the y'=0 nullclines are tangent;

example:
x'=−ax+y;
y'=x^{2}/(1+x^{2})−by.
Optional reading: Section 6.5.1 of Layek's book.

Lecture 21 (Tue, Mar 31):
Bifurcations of flows in 2 dimensional systems (cont.):

example:
x'=−ax+y;
y'=x^{2}/(1+x^{2})−by.

equations of the nullclines;

computing the FPs;

deriving a condition for tangent bifurcation
by finding the relation between the parameters a and b
when two of the fixed points merge
(alternatively, one can write the condition for tangency
between the two nullclines);

linearization of the ODE at the fixed points;

locating the linearizations in the (Δ,τ) plane;

linearization analysis of all fixed points
and a sketch of the global phase portrait;

supercritical pitchfork bifurcation in R^{2}:

supercritical pitchfork bifurcation in the prototypical example
x'=μx−x^{3},
y'=−y;

locating the roots of the linearization of the prototypical system
at the fixed point (0,0) in the complex plane;

supercritical pitchfork bifurcation in the complicated example
x'=μx+y+sin(x),
y'=x−y:
a conjectured pitchfork bifurcation at μ=−2,
study of the system near the origin for μ near −2.

Lecture 22 (Thu, Apr 2):
Midterm Exam, on the material
covered in Lectures 117

Lecture 23 (Tue, Apr 7):
Bifurcations of flows in 2 dimensional systems (cont.):

supercritical Hopf bifurcation:

a prototypical example:
r'=μr−r^{3},
θ'=ω+br^{2};

for μ≤0 the origin is a stable fixed point
(with exponential decay of r with time when μ<0,
and algebraic decay of r with time when μ=0);

for μ>0 the origin is an unstable fixed point,
and a stable limit cycle appears, which is a circle
of radius μ^{1/2};

converting the problem to Cartesian coordinates and linearizing it to get
x'=μx−ωy,
y'=ωx+μy;
eigenvalues μ±iω

general phenomena in the supercritical Hopf bifurcation:

the pair of (complex conjugate) nonreal eigenvalues cross over from the left half
to the right half of the complex plane,

the frequency of the limit cycle "right after birth" is
ω=θ'≈Im(λ)_{μ=μc},
and the period is T=2π/ω,

the radius of the limit cycle grows continuously from 0 and
for μ larger than and close to μ_{c} behaves as
(μ−μ_{c})^{1/2};

subcritical Hopf bifurcation:

a prototypical example:
r'=μr+r^{3}−r^{5},
θ'=ω+br^{2};

for μ<−1/4, the origin is a stable fixed point,
no limit cycles;

at μ=−1/4, the requation unergoes a tangent bifurcation,
which results in the creation of a limit cycle at a finite distance from the origin
(i.e., this limit cicle's radius does not grow from zero);

for μ∈(−1/4,0), the origin is a stable FP,
there is a smaller unstable and a larger stable limit cycles;

at μ=0, the system undergoes a subcritical Hopf bifurcation:
the radius of the unstable limit cycle becomes zero
and the origin becomes an unstable FP;

after the disappearance of the "safe zone" around the origin
(i.e., the region inside the unstable limit cycle for μ∈(−1/4,0)),
the trajectories go to a distant fixed point, or another limit cycle,
or even to infinity  a dramatic and potentially dangerous transition!

A movie of the
Tacoma Narrows bridge collapse (likely caused by a subcritical Hopf bifurcation);

a different kind of disaster:
a movie
of the London Millenium Bridge oscillations.

Lecture 24 (Thu, Apr 9):
Different kinds of oscillations:

linear oscillator with a periodic external driving:

Newton's second law in the presence of a restoring force
−kx ("Hooke's law"),
a damping force −γx' (i.e., air resistance),
and a periodic external driving force ƒ_{ext}(t):
mx''+γx'+kx=ƒ_{ext}(t),
where k>0, γ>0 are constants;

divide by m to get
x''+2αx'+ω_{0}^{2}x=ƒ_{0}cos(ωt),
where
where 2α:=γ/m>0,
ω_{0}^{2}:=k/m>0,
and we took
(1/m)ƒ_{ext}(t)=ƒ_{0}cos(ωt)
where ω is some (external) frequncy;

the general solution of the full equation,
x''+2αx'+ω_{0}^{2}x=ƒ_{0}cos(ωt),
is a sum of the general solution x_{c}(t)
of the homogeneous equation
x''+2αx'+ω_{0}^{2}x=0
and a particular solution x_{p}(t) of the full equation;

general solution x_{c}(t) of the homogeneous equation
x''+2αx'+ω_{0}^{2}x=0:

roots of the characteristic equation:
λ_{1,2}=−α±(α^{2}−ω_{0}^{2})^{1/2};

α>ω_{0} (strong damping):
both eigenvalues are real and negative,
x_{c}(t)=C_{1}e^{λ1t}+C_{2}e^{λ2t},
so the solution decays to 0 in a nonoscillatory fashion,

α=ω_{0} (critical damping):
one double negative (real) eigenvalue) λ=−α∈R:
x_{c}(t)=e^{−αt}(C_{1}+C_{2}t),
so the solution again decays to 0 in a nonoscillatory fashion,

α<ω_{0} (weak damping):
λ_{1,2}=−α±iω_{1},
where ω_{1}:=(ω_{0}^{2}−α^{2})^{1/2}; the general solution is
x_{c}(t)=e^{−αt}[C_{1}sin(ω_{1}t)+C_{2}sin(ω_{2}t)],
so the solution decays to 0 in an oscillatory fashion;

the general solution x_{c}(t)
of the homogeneous equation always decays to 0
(so it is called the "transient" part of the solution),
so asymptotically (i.e., as t→∞)
the solution of the full equation is only the particular solution
x_{p}(t) of the full equation;

the particular solution x_{p}(t) of the full equation
is a superposition of cos(ωt) and sin(ωt),
so we look for it in the form
x_{p}(t)=Acos(ωt)+Bsin(ωt);

computing the values of A and B
and rewriting the solution x_{p}(t) in the form
x_{p}(t)=Ccos(ωt−φ_{0});

plotting the amplitude C of x_{p}(t)
as a function of the external driving frequency ω;

application of resonance: radio;

periodically forced nonlinear oscillator:

Duffing's equation:
x''+2αx'+ω_{0}^{2}x+βx^{3}=ƒ_{0}cos(ωt);

after the transient terms die off, the solution is of the form
x_{p}(t)=C(ω)cos(ωt−φ_{0}), where C(ω) may not be a function of ω
(in the sense that it may not satisfy the vertical line test);

hysteresis phenomenon  the amplitude of the oscillations may depend on how we got there;

parameteric resonance:

there is no extermal driving, but the parameters of the system change periodically;

example: a pendulum equation with a variable length;

Mathieu's equation:
x''+[α+βcos(t)]x=0;

for some choices of α and β in Mathieu's equation,
there exist unstable solutions;
toungues of instability in the (α,β)plane;

the frequency of the change of parameters may be different
from the frequency of oscillation of the pendulum;

compare the above oscillations with the oscillations occurring
after a supercritical Hopf bifurcation, when there is no periodic forcing.

Lecture 25 (Tue, Apr 14):
An example: Sel'kov model of glycolysis:

glycolysis  process of the breakdown of glucose, releasing energy
(see the
Wikipedia article);

Sel'kov (1968) equations in dimensionless form:
x'=−x+ay+x^{2}y,
y'=b−ay−x^{2}y,
where x(t)>0 is the concentration of ADP (adenosine diphosphate),
x(t)>0 is the concentration of F6P (fructose6phosphate),
a>0 and b>0 are positive parameters;

constructing the outer boundary of a trapping region in order to apply
the PoincaréBendixson Theorem to prove the existence of limit cycles;

"cutting out" the fixed point and studying when the fixed point is repelling
(by linearization) to finish the construction of a trapping region;

plot of the region in the (a,b)plane where a stable limit cycle exists.

Lecture 26 (Thu, Apr 16):
Global bifurcations:

a global bifurcation  a bifurcation that involves a large region
of the phase plane (not only a fixed point);

an example of a global bifurcation:
a saddlenode bifurcation of cycles, on the example of the system
r'=μr+r^{3}−r^{3},
θ'=ω+br^{2},
studied in Lecture 23:

the system undergoes a saddlenode bifurcation of cycles (global) at μ=−1/4,
which results in the creation of a limit cycle at a finite distance from the origin
(i.e., this limit cicle's radius does not grow from zero);

the system exhibits a subcritical Hopf bifurcation at μ=0:
the radius of the unstable limit cycle becomes zero
and the origin becomes an unstable FP;

infiniteperiod bifurcation in the system
r'=r(1−r^{2}),
θ'=μ−sin(θ),
computing the period of the oscillatory motion for μ>μ_{c}=1,
and studying its behavior as μ→μ_{c}^{+};

homoclinic bifurcation: when a limit cycle "collides" with a saddle point;
example:
x'=y,
y'=μy+x−x^{2}+xy.
Coupled oscillators, motion on a torus:

constructing a circle by identifying the ends of the interval [0,2π];

constructing an infinite cylinder by identifying the ends of the interval
[0,2π] in [0,2π]×R;

constructing a
Möbius strip
by identifying the opposite sides of a rectangle with a twist;

constructing a 2dimensional torus T^{2}
by identifying the opposite sides of the square [0,2π]×[0,2π];

constructing a
Klein bottle
by identifying the opposite sides of the square [0,2π]×[0,2π]
with a twist in one identification;

motion on the 2dimensional torus T^{2}
with constant velocity:
θ_{1}'=ω_{1}=const,
θ_{2}'=ω_{2}=const
 the trajectory is a straight line of slope ω_{2}/ω_{1};

question: what is the condition on the slope ω_{2}/ω_{1}
so that the point returns to its original position (without restriction on how long it takes)?

Lecture 27 (Tue, Apr 21):
Coupled oscillators, motion on a torus (cont.):

natural numbers N, integers Z,
rational numbers Q={p/q:p∈Z,q∈N},
irrational numbers R∖Q;

Theorem: both Q and R∖Q are dense in R;

autonomous ODE on T^{2}:
θ'=ƒ(θ), i.e.,
θ_{1}'=ƒ_{1}(θ_{1},θ_{2}),
θ_{2}'=ƒ_{2}(θ_{1},θ_{2}),
where
ƒ=(ƒ_{1},ƒ_{2}):T^{2}→R^{2}
is given, and
θ=(θ_{1},θ_{2}):R→T^{2}
is the unknown function;

flow with constant velocity, θ'=ω,
where ω=(ω_{1},ω_{2}) with
ω_{2}/ω_{1}∈Q:
the integral lines are closed orbits;

a detailed analysis of the flow of θ'=ω
with ω_{2}/ω_{1}=2/3:

representing the orbit on the torus T^{2}
(with T^{2} thought of as a square with identified opposide sides),

representing the orbit on the torus T^{2}
(with T^{2} thought of as a surface in R^{3}),

representing the orbit thought of as a curve in R^{3}
 the trefoil know in R^{3}!

fact: if p≥2 and q≥2
have no common factors, the trajectories of θ'=ω
where
ω=(ω_{1},ω_{2})
with ω_{2}/ω_{1}=p/q
are always knotted closed curves in R^{3};

flow with constant velocity, θ'=ω,
where ω=(ω_{1},ω_{2}) with
ω_{2}/ω_{1} irrational:

the integral lines never close, proof by "unfolding" the torus;

each trajectory is dense in T^{2};

coupled oscillators 
a model system:
θ_{1}'=ω_{1}+K_{1}sin(θ_{2}−θ_{1}),
θ_{2}'=ω_{2}+K_{2}sin(θ_{1}−θ_{2}):

introducing the phase difference
φ=θ_{1}−θ_{2},
ODE for φ:
φ'=ω_{1}−ω_{2}−(K_{1}+K_{2})sin(φ),

plotting φ' as a funcion of φ:
if ω_{1}−ω_{2}<(K_{1}+K_{2})
 no fixed points,
if ω_{1}−ω_{2}=(K_{1}+K_{2})
 one FP,
if ω_{1}−ω_{2}>(K_{1}+K_{2})
 two FPs of opposite stability;

phaselocked solution with θ_{1}=θ_{2}+φ^{*}.

Lecture 28 (Thu, Apr 23):
Coupled oscillators, motion on a torus (cont.):

coupled oscillators 
a model system:
θ_{1}'=ω_{1}+K_{1}sin(θ_{2}−θ_{1}),
θ_{2}'=ω_{2}+K_{2}sin(θ_{1}−θ_{2}):

derivation of the expression for φ^{*}
if ω_{1}−ω_{2}>(K_{1}+K_{2}):
φ^{*} is defined implicitly by
sin(φ^{*})=(ω_{1}−ω_{2})/(K_{1}+K_{2}),

compromise frequency:
ω^{*}=(K_{1}ω_{2}+K_{2}ω_{1})/(K_{1}+K_{2}).
Flows on T^{2}, Poincaré maps, circle maps:

definition of the first return map of a flow on the torus T^{2}
to the circle Σ=S^{1}={θ_{2}=0};

the Poincaré map P:Σ→Σ
of a flow θ'=ƒ(θ) on T^{2}
is an example of a circle map
P:S^{1}→S^{1};

derivation of the Poincaré map for the flow on T^{2}
of the ODE θ_{1}'=1/10, θ_{1}'=1;

a FP of the Poincaré map P:S^{1}→S^{1}
corresponds to a periodic orbit of the flow of
θ'=ƒ(θ) on T^{2};

finding the FPs of the Poincaré map graphically;

tangent bifurcation of a circle map;

rotation number of a circle map;

example: rigid rotation by 2/5;

a periodic point of period q of a map P
is a FP of the iterated map F=P^{q};

Arnol'd's map:
P_{α,β}:S^{1}→S^{1}
given by
P_{α,β}(θ)=[θ+α+βsin(2πθ)]mod 1
(x mod 1 stands for the fractional part of the number x).

Lecture 29 (Tue, Apr 28):
Flows on T^{2}, Poincaré maps, circle maps (cont.):

phase locking of Arnol'd's map for β∈[0,1/(2π))  rational rotation number
τ(P_{α,β})=p/q∈Q
 generically, there is an attracting and a repelling periodic orbit,
both of period q;

Arnol'd tongues  regions in the (α,β)plane where the rotation number
is rational;

behavior of the rotation number τ(P_{α,β})
as a function of α for a fixed value of β∈[0,1/(2π)):
τ(P_{α,β}) is a continuous function of α
that is locally constant if
τ(P_{α,β})∈Q
and strictly increasing otherwise;

selfsimilarity of functions in dynamical systems.
Fisher's equation:

PDEs vs. ODEs;

heat/diffusion equation, physical interpretation,
solution in 1D for initial condition δ(x);

wave equation, physical interpretation,
solution in 1D  d'Alembert's formula,
physical meaning of the traveling wave solutions
ƒ(x−ct) and
g(x+ct);

Fisher's equation (1937) describing logistic growth with diffusion:
u_{t}=Du_{xx}+ru(1−u/K);

D>0 diffusion coefficient,
r>0 reproduction rate,
K>0 carrying capacity;

exercise: nondimensionalizing to
u_{t}=u_{xx}+u(1−u);

obvious solutions:
u(x,t)=0 or
u(x,t)=1 for all
x and t.

Lecture 30 (Thu, Apr 30)
Fisher's equation (cont.):

looking for a solution u(x,t) of the (nondimensionalized)
Fisher's equation that, for every fixed t,
tends to 1 for x→−∞
and to 0 for x→∞;

looking for a solution u(x,t) of the (nondimensionalized)
Fisher's equation in the form of a traveling wave:
u(x,t)=U(x−ct)
for some c>0;

derivation of an ODE for the function U(z):
U+cU+U(1−U)=0
with 0≤U(z)≤1,
U(z)→1 for z→−∞,
and U(z)→0 for z→∞;

rewriting the 2nd order ODE for U(z) as a 1st order system:
U'=V,
V'=−U−cV+U^{2};

linearization of the 1st order system for U(z) and
V(z) at the fixed points (0,0) and (1,0);

from the linearization at (0,0) conclude that the solution 0≤U(z)
is violated for c<2, so that the only physically interesting case is c≥2;

construction of a trapping region;

proof of the existence of a trajectory in the (U,V)plane
that tends to (1,0) as z→−∞
and tends to (0,0) ad z→∞;

minimal speed of propagation in physical units:
c_{phys,min}=2(rD)^{1/2}.
Bibliography on Fisher's equation:

C. Chicone, Ordinary Differential Equations with Applications,
2nd ed, Springer, 2006,
Sec. 3.6.3, "Traveling waves"

J. D. Murray, Mathematical Biology, Vol. I: An Introduction,
3rd ed, Springer, 2002,
Sec. 13.2,
"Fisher–Kolmogoroff equation and propagating wave solutions"

J. D. Murray, Mathematical Biology, Vol. II:
Spatial Models and Biomedical Applications,
3rd ed, Springer, 2002,
Ch. 1,
"Multispecies waves and practical applications"
(generalizations to several species)
Good to know: