### MATH 4193.001/5103.001 - Introductory Mathematical Modeling / Mathematical Models - Spring 2020 TR 1:30--2:45 p.m., 122 PHSC

Instructor: Nikola Petrov, 1101 PHSC, npetrov AT math.ou.edu

Office Hours: Mon 2:30-3:30 p.m., Tue 10:30-11:30 a.m., or by appointment, in 1101 PHSC.

A brief (tentative) list of topics to be covered:

• One-dimensional 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, one-dimensional maps, fractals, strange attractors.

Homework:

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/dtV2/3 (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 first-order ODEs x'=f(x) for the unknown function x:[0,T]→Rn (T is a positive number), where f:RnRn is a given function;
• non-autonomous first-order systems: x'=f(x,t);
• converting a single higher-order ODE into a system of 1st-order ODEs;
• geometric representation of the solution of the IVP x'=f(x), x(0)=x(0) as a parameterized curve in Rn called flow, φt(x(0)), where for every t∈[0,T], φt:RnRn maps the initial condition x(0)Rn to the "present" position, x(t)=φt(x(0));
• the flow satisfies dφt(x(0))/dt=f(φt(x(0))), φ0(x(0))=x(0) (the latter can be written as φ0=IdRn (where IdRn is the identity function in Rn);
• semigroup property of the flow: φt(φs(x(0)))=φt+s(x(0)), i.e., φtφs=φt+s;
• corollaries of the semigroup property: φ0=IdRn, (φt)−1=φt.
Linear constant-coefficient systems of ODEs:
• (homogeneous) linear constant-coefficient 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 φt(x(0))=etAx(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−1DC, then etA=etC−1DC=C−1etDC.
• Lecture 3 (Tue, Jan 21): Fun fact:
• for a smooth function ƒ:RR, if Da(ƒ):=ƒ'(a) and hR, then exp(hDa)(ƒ)=ƒ(a+h).
From flows of ODEs to Poincaré maps:
• consider a system of (autonomous) ODEs x'=f(x) in Rn whose flow is φt; an integral line of the flow is a curve {φt(x(0)):tR} in Rn;
• a periodic solution with (minimal) period T: a solution such that φt+T(x(0))=φt(x(0)) for all tR;
• the integral line of a periodic solution is a closed curve in Rn;
• for a given flow φt, a Poincaré surface Σ is a (piece of) hypersurface (i.e., an (n−1)-dimensional surface in Rn) 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: Pk(x):=PP∘...∘P(x) (k times);
• usefulness of the Poincaré map: lowering the dimension;
• a digression on the importance of dimension:
• differences between 1-dimensional and 2-dimensional problems: a point moving on R from the negative half of R cannot go to the positive half without going through 0;
• differences between 2-dimensional and 3-dimensional problems: in 2 dimensions, a non-self-intersecting 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 ƒk(x) for very large k;
• iterating a one-dimensional map ƒ:RR graphically - a cobweb plot;
• fixed points on a cobweb plot - the x-coordinate 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 Rn).
• a picture of a flow in R2 and a Poincaré map (related to Problem 3 of Homework 1).
One-dimensional maps ƒ:RR:
• 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 ƒm;
• an example: the logistic map ƒμ:[0,1]→[0,1] given by ƒμ(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]):
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 graphical-iteration-of-1-dim-maps.nb taken from a code (link) written by Sander Huisman (University of Twente); the code plots 1000 iterations of the map ƒμ(x)=μx(1−x) with μ=3.1 and initial point x0=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):
One-dimensional maps ƒ:RR (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 ƒk;
• an exercise: take a calculator, make sure that it is set to working in radians, type any number x0, 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 long-time 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(ti+1)=x(tit)=x(ti)+x'(tit+(1/2!)x''(ti)(Δt)2+...≈x(ti)+x'(tit=x(ti)+ƒ(x(ti))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);
• semi-stable 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 x1*=1, ƒ(x)≈−256(x−1), so this FP is stable;
• deriving without derivatives that near the FP x2*=3, ƒ(x)≈−16(x−3)2, so this FP is semistable;
• exercise: do this for the FP x3*=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 x1*=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 x1*=1 is a straight line with slope −1;
• near x2*=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 x2*=3 is an upside-down parabola;
• near x3*=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 x3*=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 non-generic situation (when dramatic changes may occur due to arbitrarily small values of the parameters);
• an example: consider number of roots of an algebraic equation ƒμ(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 x2+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, x1=−|μ|1/2 and x2=|μ|1/2
• exercise: show that the equation x3x=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 non-generic 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 semi-stable FP at 3, and a repelling one at 5; the semi-stable 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 right-hand 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 right-hand side of the ODE) is also non-generic (this is not so obvious, we will study this soon).
Saddle-node (tangent, blue sky) bifurcation in a 1-parameter 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'=ƒμ(x)=μ+x2:
• 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 1-parameter family of ODEs x'=ƒμ(x)=μ−x−ex: writing ƒμ(x) as a difference of two functions: ƒμ(x)=φμ(x)−ψ(x), with φμ(x)=μ−x, ψ(x)=ex, and rewriting the condition for a fixed point, ƒμ(x)=0, as condition for equality φμ(x)=ψμ(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 Hirsch-Smale-Devaney (the most important thing for us are the pictures).
• Lecture 7 (Tue, Feb 4):
Saddle-node (tangent, blue sky) bifurcation in a 1-parameter family of ODEs on R (cont.):
• finishing the calculations from the complicated example from Lecture 6:
• computing the critical value μc=1 and the fixed point x*c=0 for μ=μc at which the graphs of φμ(x) and ψ(x) are tangent;
• Taylor series of a function of two variables,
• expansion of ƒμ(x) in a Taylor series near the point (μ,x)=(1,0): ƒμ(x)≈(μ−1)−(1/2)(x−0)2;
• computing the approximate values of the fixed points for μ slightly above μc, so that only the lowest-order 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 1-parameter family of ODEs on R:
• studying the bifurcation in the 1-parameter family x'=ƒμ(x)=−x3x: 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 Tc=1: for temperatures slightly below Tc, x*≈±[3(TTc)]1/2.
• Lecture 8 (Thu, Feb 6):
More complicated bifurcations:
• bifurcations in the 2-parameter family x'=ƒμ,μ(x)=−x3x+ν which exhibits a supercritical pitchfork bifurcation and two saddle-node bifurcations.
Logistic equation with predation:
• Lecture 9 (Tue, Feb 11):
Logistic equation with predation (cont.):
• empirical observations from looking at the intersections of the graphs of φr,k(x) and ψ(x) in order to locate the (non-zero) fixed points:
• the system can have 1, 2, or 3 (non-zero) FPs,
• if r is too large, then for any value of k the system has only one (non-zero) FP,
• if k is too small, then for any value of r the system has only one (non-zero) FP,
• if the straight line representing φr,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 (non-zero) FP;
• the system undergoes tangent (saddle-node, blue sky) bifurcations when the graphs of φr,k(x) and ψ(x) are tangent at some point;
• looking for tangent bifurcations: the conditions for tangency are
φr,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:
$k=\frac{2(x^*)^3}{(x^*)^2-1}, \ r=\frac{2(x^*)^3}{(1+(x^*)^2)^2};$
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 non-zero FP and where it has 3 non-zero FPs has a shape of a horn with a tip at (kc,rc)=(33/2,33/2/8)≈(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 higher-dimensional systems:
• general setup: t∈[0,∞) time, x:[0,∞)→Rn unknown function, f:Rn×[0,∞)→Rn given function;
ODE: x'=f(x,t), IC: x(t(0))=x(0), IVP=ODE+IC;
• classifications: autonomous vs. non-autonomous, 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, Fnet(x,v,t) total force acting on the particle (sum of all the forces acting on the particle), then x''=Fnet;
• 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 non-deformable 3-dimensional object: dof=6;
• a deformable 3-dimensional object: dof=∞;
• example: planar pendulum with air-resistance:
• coordinate: x=θ/(2π)∈S1, velocity: y=x'∈R, so that the phase space of the pendulum is the infinite cylinder S1×R;
• introduce air-resistance: Newton II is x''+γx'+(g/l)sin(x)=0, which can be written as a system of 1st order ODEs: set x1(t)=x(t), x2(t):=x1'(t), so that Newton II becomes x1'=x2, x2'=−(g/l)sin(x1)−γx2;
• FPs (steady-state solutions): (0,0) and (π,0);
• linearizing the system near the FP (0,0) (which should be stable by our physical intuition): for (x1,x2)≈(0,0), Newton II becomes x1'=x2, x2'=−(g/l)x1−γx2;
• converting the linearized system to a single 2nd order ODE: x''+2βx'+ω02x=0, where 2β:=γ>0, ω02:=g/l>0;
• characteristic equation: r2+2βr'+ω02=0, solutions: r1,2=−β±(β2−ω02)1/2.
• Lecture 11 (Tue, Feb 18):
Introduction to higher-dimensional 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 air-resistance 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 higher-dimensional 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 ui 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, Jij=∂ƒi/∂xi(x*).

• Lecture 13 (Tue, Feb 25):
Classification of linear systems:
• a detailed analysis of the linear system
$\mathbf{x}'=A\mathbf{x}, \ A=\left(\begin{array}{cc} a & 0 \\ 0 & -1 \end{array} \right):$
• a stable node (a<0, a≠−1),
• a star (a=−1),
• a line of fixed points (a=0),
• eigenvalues λj of a 2×2 matrix A - solutions of the quadratic equation λ2−τλ+Δ2=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 trace-determinant plane for a 2×2 matrix A:
• if Δ=τ2/4, then A has one double real eigenvalue;
• "inside" the parabola Δ=τ2/4 (i.e., if τ2/4<Δ), A has two complex conjugate non-real eigenvalues;
• "outside" the parabola Δ=τ2/4, 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:
• if the constant 2×2 matrix A in the ODE x'=Ax has non-real eigenvalues:
• λ1,2=α±iβ (α,β∈R, α≠0): stable spiral (α<0) or unstable spiral (α>0);
• λ1,2=±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.
• 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 higher-dimensional 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 λ1,...,λn are real and distinct, the general solution is a linear superposition of terms of the form eλjtvj (where vj is an eigenvector with eigenvalue λj);
• 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)tv=eαt[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:
• Hartman-Grobman 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+y4, y'=y - its linearization at the FP (1,0) is u1'=−u1, u2'=u2 (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 non-hyperbolic FP: the system x'=x2, y'=y has a non-hyperbolic fixed point at (0,0), the solution of the original system is x(t)=(x0−1t)−1, y(t)=y0et, the linearized system is u1'=0, u2'=−u2 with solution u1(t)=u10, u2(t)=u20et, so that the solutions of the original system and of its linearization at (0,0) behave very differently;
• example of a non-hyperbolic FP: the system x'=−yx(x2+y2)1/2, y'=xy(x2+y2)1/2 has a non-hyperbolic fixed point at (0,0), the integral curves of the linearized system, u1'=−u2, u2'=u1 are circles (i.e., the FP is a center); a polar change of variables transforms the original system into r'=μr2, 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 Fnet is conservative (potential), i.e., that it depends only on the coordinate and there exists a function V:RnR such that Fnet(x)=−∇V(x): in this case Newton's Second Law reads mx''=Fnet(x)=−∇V(x);
• multiply mx''=−∇V(x) by x' to show that the quantity (1/2)m|x'|2+V(x) is conserved, i.e., if x(t) is a solution of Newton's Second Law, then d/dt[(1/2)m|x'(t)|2+V(x(t))]=0;
• kinetic energy (1/2)m|x'|2 and potential energy V(x);
• recognizing that F:UR2, UR2, is conservative: if U is simply-connected and F(x)=P(x,y)i+Q(x,y)j, then F is conservative if and only if ∂P/∂y=∂Qx;
• recognizing that F:UR3, UR3, 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=∂Qx: 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 1-dimensional 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)m|x'|2+(1/2)kx2, and dE/dt=−γ|x'|2≤0, so that the power of the air resistance force is −γ|x'|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)[EV(x)]1/2;
• example: V(x)=−(1/2)x2+(1/4)x4, then F(x)=−V'(x)=xx3, 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 semi-attracting 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(x2−1)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 C1 function V:UR (with URn such that V(x)>0 for all xx*, 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'=−xy3, then (0,0) is a FP, and V(x,y)=x+4y2 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 R2, if ƒ(x) is of one sign, then Liénard equation has no periodic orbits;
• example: the system x'=−y+(x2+x2−1), y'=x+(x2+y2−1), has no periodic orbits inside the circle of radius 21/2 centered at (0,0);
• example: rabbits vs. sheep: the system x'=x(3−x−2y), y'=y(2−xy) 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:
• Poincare-Bendixson 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−r2)+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;
• saddle-node (blue sky, tangent) bifurcation, prototype: x'=μ+x2;
• supercritical pitchfork bifurcation, prototype: x'=μxx3;
• subcritical pitchfork bifurcation, prototype: x'=μx+x3;
• transcritical bifurcation, prototype: x'=x(x−μ) (see Problem 2 in Homework 2).
Bifurcations of flows in 2 dimensional systems:
• saddle-node bifurcation in a prototypical example: x'=μ+x2, 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+ey, y'=1−y;
• detecting saddle-node bifurcations by using nullclines - bifurcations occur when the x'=0 and the y'=0 nullclines are tangent;
• example: x'=−ax+y; y'=x2/(1+x2)−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'=x2/(1+x2)−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 R2:
• supercritical pitchfork bifurcation in the prototypical example x'=μxx3, 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'=xy: 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 1-17
• Lecture 23 (Tue, Apr 7):
Bifurcations of flows in 2 dimensional systems (cont.):
• supercritical Hopf bifurcation:
• a prototypical example: r'=μrr3, θ'=ω+br2;
• 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 μ1/2;
• converting the problem to Cartesian coordinates and linearizing it to get x'=μx−ωy, y'=ωxy; eigenvalues μ±iω
• general phenomena in the supercritical Hopf bifurcation:
• the pair of (complex conjugate) non-real 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 μc behaves as (μ−μc)1/2;
• subcritical Hopf bifurcation:
• a prototypical example: r'=μr+r3r5, θ'=ω+br2;
• for μ<−1/4, the origin is a stable fixed point, no limit cycles;
• at μ=−1/4, the r-equation 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 ƒext(t):
mx''+γx'+kxext(t), where k>0, γ>0 are constants;
• divide by m to get x''+2αx'+ω02x0cos(ωt), where where 2α:=γ/m>0, ω02:=k/m>0, and we took (1/mext(t)=ƒ0cos(ωt) where ω is some (external) frequncy;
• the general solution of the full equation, x''+2αx'+ω02x0cos(ωt), is a sum of the general solution xc(t) of the homogeneous equation x''+2αx'+ω02x=0 and a particular solution xp(t) of the full equation;
• general solution xc(t) of the homogeneous equation x''+2αx'+ω02x=0:
• roots of the characteristic equation: λ1,2=−α±(α2−ω02)1/2;
• α>ω0 (strong damping): both eigenvalues are real and negative, xc(t)=C1eλ1t+C2eλ2t, so the solution decays to 0 in a non-oscillatory fashion,
• α=ω0 (critical damping): one double negative (real) eigenvalue) λ=−α∈R: xc(t)=e−αt(C1+C2t), so the solution again decays to 0 in a non-oscillatory fashion,
• α<ω0 (weak damping): λ1,2=−α±iω1, where ω1:=(ω02−α2)1/2; the general solution is xc(t)=e−αt[C1sin(ω1t)+C2sin(ω2t)], so the solution decays to 0 in an oscillatory fashion;
• the general solution xc(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 xp(t) of the full equation;
• the particular solution xp(t) of the full equation is a superposition of cos(ωt) and sin(ωt), so we look for it in the form xp(t)=Acos(ωt)+Bsin(ωt);
• computing the values of A and B and rewriting the solution xp(t) in the form xp(t)=Ccos(ωt−φ0);
• plotting the amplitude C of xp(t) as a function of the external driving frequency ω;
• periodically forced nonlinear oscillator:
• Duffing's equation: x''+2αx'+ω02xx30cos(ωt);
• after the transient terms die off, the solution is of the form xp(t)=C(ω)cos(ωt−φ0), 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+x2y, y'=bayx2y, where x(t)>0 is the concentration of ADP (adenosine diphosphate), x(t)>0 is the concentration of F6P (fructose-6-phosphate), a>0 and b>0 are positive parameters;
• constructing the outer boundary of a trapping region in order to apply the Poincaré-Bendixson 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 saddle-node bifurcation of cycles, on the example of the system r'=μr+r3r3, θ'=ω+br2, studied in Lecture 23:
• the system undergoes a saddle-node 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;
• infinite-period bifurcation in the system r'=r(1−r2), θ'=μ−sin(θ), computing the period of the oscillatory motion for μ>μc=1, and studying its behavior as μ→μc+;
• homoclinic bifurcation: when a limit cycle "collides" with a saddle point; example: x'=y, y'=μy+xx2+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 2-dimensional torus T2 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 2-dimensional torus T2 with constant velocity: θ1'=ω1=const, θ2'=ω2=const - the trajectory is a straight line of slope ω21;
• question: what is the condition on the slope ω21 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:pZ,qN}, irrational numbers RQ;
• Theorem: both Q and RQ are dense in R;
• autonomous ODE on T2: θ'=ƒ(θ), i.e., θ1'=ƒ112), θ2'=ƒ212),
where ƒ=(ƒ12):T2R2 is given, and θ=(θ12):RT2 is the unknown function;
• flow with constant velocity, θ'=ω, where ω=(ω12) with ω21Q: the integral lines are closed orbits;
• a detailed analysis of the flow of θ'=ω with ω21=2/3:
• representing the orbit on the torus T2 (with T2 thought of as a square with identified opposide sides),
• representing the orbit on the torus T2 (with T2 thought of as a surface in R3),
• representing the orbit thought of as a curve in R3 - the trefoil know in R3!
• fact: if p≥2 and q≥2 have no common factors, the trajectories of θ'=ω where ω=(ω12) with ω21=p/q are always knotted closed curves in R3;
• flow with constant velocity, θ'=ω, where ω=(ω12) with ω21 irrational:
• the integral lines never close, proof by "unfolding" the torus;
• each trajectory is dense in T2;
• coupled oscillators - a model system: θ1'=ω1+K1sin(θ2−θ1), θ2'=ω2+K2sin(θ1−θ2):
• introducing the phase difference φ=θ1−θ2, ODE for φ: φ'=ω1−ω2−(K1+K2)sin(φ),
• plotting φ' as a funcion of φ: if |ω1−ω2|<(K1+K2) - no fixed points, if |ω1−ω2|=(K1+K2) - one FP, if |ω1−ω2|>(K1+K2) - two FPs of opposite stability;
• phase-locked solution with θ12*.
• Lecture 28 (Thu, Apr 23):
Coupled oscillators, motion on a torus (cont.):
• coupled oscillators - a model system: θ1'=ω1+K1sin(θ2−θ1), θ2'=ω2+K2sin(θ1−θ2):
• derivation of the expression for φ* if |ω1−ω2|>(K1+K2): φ* is defined implicitly by sin(φ*)=(ω1−ω2)/(K1+K2),
• compromise frequency: ω*=(K1ω2+K2ω1)/(K1+K2).
Flows on T2, Poincaré maps, circle maps:
• definition of the first return map of a flow on the torus T2 to the circle Σ=S1={θ2=0};
• the Poincaré map P:Σ→Σ of a flow θ'=ƒ(θ) on T2 is an example of a circle map P:S1S1;
• derivation of the Poincaré map for the flow on T2 of the ODE θ1'=1/10, θ1'=1;
• a FP of the Poincaré map P:S1S1 corresponds to a periodic orbit of the flow of θ'=ƒ(θ) on T2;
• 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=Pq;
• Arnol'd's map: Pα,β:S1S1 given by Pα,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 T2, Poincaré maps, circle maps (cont.):
• phase locking of Arnol'd's map for β∈[0,1/(2π)) - rational rotation number τ(Pα,β)=p/qQ - 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(Pα,β) as a function of α for a fixed value of β∈[0,1/(2π)): τ(Pα,β) is a continuous function of α that is locally constant if τ(Pα,β)∈Q and strictly increasing otherwise;
• self-similarity 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(xct) and g(x+ct);
• Fisher's equation (1937) describing logistic growth with diffusion: ut=Duxx+ru(1−u/K);
• D>0 diffusion coefficient, r>0 reproduction rate, K>0 carrying capacity;
• exercise: non-dimensionalizing to ut=uxx+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(xct) 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'=−UcV+U2;
• 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: cphys,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, "Multi-species waves and practical applications" (generalizations to several species)

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