18353spring 2001

2050J12006J18353J Nonlinear Dynamics I Chaos Fall 2012

MATLAB primer 2

The objective of this document is to get you to a position where you can complete the MATLAB portion on the PSet 6

In the previous assignments we were able to plot a phase portrait using the function streamslice This plotting tool does a good job of representing the qualitative features of the phase portrait However the user is at the mercy of the code and doesnrsquot necessarily have the ability to control where trajectories start Additionally this function does not give any time dependent information What will be explained in this MATLAB primer is a function called ode45 This function numerically solves the system for a given initial condition and lets the users manipulate the results however they please

1 ode45 Set Up

When trying to solve a system with ode45 two codes will actually have to be developed These two codes are the ldquosystemrdquo and ldquosolverrdquo codes Both these codes will have to be in the same folder that you are working out of Within the ldquosystemrdquo code you will create a set of state equations which will be used by the solver These state equations must be of the form

dx1 = f1(x1 x2 )

dt dx2

= f2(x1 x2 )dt =

dxn = fn(x1 x2 )

dt

Within the ldquosolverrdquo code you will select the initial conditions and have the solver command line included

2 The solver command line

A sample solver command line is as follows gt gt [T Y] = ode45(sys[0110][1 0])

The solver ode45 has three inputs The first input lsquosysrsquo is a function handle It tells ode45 which function to use as the system (this example will call the m-file sysm) The next input rsquo[0 1 10]rsquo is the vector of time-steps The system will be integrated from t0 = 0 to tfinal = 10 with a time-step 01 Finally the last input is rsquo[1 0]rsquo is the vector of initial conditions This example is a second order system which has the initial conditions x(0) = 1 and x prime(0) = 0

There are two outputs for this command line The first is rsquoT rsquo which is the time output by the command and the other output rsquoY rsquo is the system values as a function of time In this example there are 101 time steps and the dimensions of T are 101x1 and the dimensions of Y are 101x2 In this case the value Y (3 1) is the value of the first system variable (ie x) at the third time step and the value of Y (10 2) is the value of the second system variable (ie x) at the tenth time step

1

Y

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5 minus6 minus4 minus2 0 2 4 6

X

Figure 1 Phase plane created using ode45 starting a variety of initial conditions

Figure 2 Phase plane created using streamslice starting a variety of initial conditions

2

minus6 minus4 minus2 0 2 4 6minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

Y

X

MIT OpenCourseWarehttpocwmitedu

18353J 2050J 12006J Nonlinear Dynamics I ChaosFall 2012

For information about citing these materials or our Terms of Use visit httpocwmiteduterms


Matlab primer

The objective of this document is to get you to a position where you can complete the MATLAB portion on the second problem set It is by no means a comprehensive introduction to MATLAB

Figure 1 Example of a MATLAB window GUI

1 Download MATLAB

If you do not already have MATLAB on your computer you will need to download it on the MIT website httpwebmitedustudent-matlab This is a free package that only requires you have an MIT certificate on your computer Downloading it might take a while If you donrsquot have a certificate you can

use Matlab at Athena clusters

2 Download MATLAB Codes from Course Website

Once MATLAB is up and running on your computer you will need to download the codes from the course website Make sure you put this file in the Matlab working directory (to be discussed in the next section)

Athena is MITs UNIX-based computing environment OCW does not provide access to it

1

3 Working in the MATLAB environment

After the download is completed open the application You will be brought to the MATLAB window (see an example in Figure 1) At the top center of the GUI is the command directory location bar You can put in whatever directory you would like to work out of Put the M-files you download from the course website in the working directory To the left of the GUI is the list of files in your Command Directory make sure that the M-files are there Finally the command line is where you will start to run different commands ( try a couple things in this window for example 1+1 sqrt(9) a = [1 2 3]rsquo figure(1) plot(a) ) To make sure everything is working fine type in help name-of-the-filem in the command line If the files are in place a brief message should come up in the Command Window

If you need help and examples with any matlab function for example plot type in help plot A description of the function with several examples will appear

4 Working with the M-files

Within MATLAB there is the ability to write programs which can complete certain tasks These files are called M-files and you just downloaded the example ones for this Pset You will start to write your own M-files as the course goes along In order to open the M-files find the file you would like to open in the Current Directory window on the left side of the GUI Alternatively you can use the Open selection in the File drop down in the top left of the GUI You will have to modify the file you got for this Pset a little bit The examples within the code that you can use as a template

5 Comments

We will continue to use MATLAB on a regular basis and you will be expected to have developed a certain skill set after each Pset In this case we are expecting that you are capable of opening MATLAB working with M-files inputing data and working with plots There is a great set of support files built into MATLAB Simply type in helpdesk into the command line

Also within the help files there is a section on matrices and arrays that you should be familiar with This section is located in MATLAB gtgt Getting Started gtgt Matrices and Arrays The opening page of this section is shown in Figure 2

6 Other Matlab resources at MIT

Software at MIT httpwebmitedumatlabwww

2

Figure 2 Example of Matrices and Arrays in helpdesk

3




Before we used ode45 only for autonomous systems

Here once you rewrite equation as a second-order system there is

time dependence on the RHS (ie its not autonomous)

To use ode45 the same way we did before you can treat it as a 3-dim system

where time is a new independent variable z

x = y

y = g(xyz)

z = 1

(here x = theta y = theta z = time)

What changes in the code

- you have to give initial conditions z = 0 at t = 0

- in the program that computes the RHS the vector of the RHS is 3-dimensional and the third

component = 1





Due at noon on Friday Sept 14th in the box provided (to the right of the entrance) No late psets are accepted If you collaborated with another student list their names on the title sheet The work that you submit must be your own

Main concepts phase space (phase line phase plane) fixed points their stability review of linear algebra (matrices eigenvectors eigenvalues) and its application to linear systems of ODEs

Reading Strogatz Ch2 1803 notes (LS1-6) (see below)

Helpful review material from 1803

bull Review of 2nd order ODEs with constant coefficients ndash video lectures ndash see Lecture 9 httpocwmiteducoursesmathematics18-03-differential-equations-spring-2010video-lectures

bull 1803 notes are on httpmathmitedusuppnotessuppnotes03indexhtml

Problem 1 1D dynamical systems Population growth The Allee effect

Problem 2 1D dynamical systems Potentials Intro to bifurcations

The dynamics of first order systems x = f(x) can be pictured as a particle sliding down the walls of a dV (x)potential well where the potential V (x) is defined by f(x) = minus (Reference Strogatz ch 27) dx

1 Given the system x = sin(x) + α where α is a real number find the potential V (x) sketch it show graphically how a particle would move for different values of α (α = 0 09 1 11)

2 For the same values α find fixed points their stability sketch a phase portrait (phase line) with arrows representing the flow between the fixed points label the fixed points as stable unstable semistable

3 Sketch a graph of the x-coordinate of the fixed points as a function of α for 0 le α le 12 Describe what happens when α passes through 1 You just witnessed the phenomenon of a rdquobifurcationrdquo

1


Strogatz 234

How would you define this concept based on the one example of a bifurcation that you just witnessed (Do not look this concept up in the book it has to be your definition that results from what you observe)

4 In another system x = (x + 1)(x minus α) with a parameter α sketch the graphs of the x-coordinate of the fixed points mark on the graph where the fixed point is stable or unstable

5 The phenomenon of a rdquobifurcationrdquo occurs again at α = minus1 Describe what happens and modify your definition of a rdquobifurcationrdquo that you had above to include this case in the definition Sketch the phase line for the values of α slightly below at and slightly above the bifurcation value α = 1

6 For the system x = (x+1)(xminusα) sketch the phase line for the two values of α before the bifurcation and after the bifurcation

Problem 3 Linear oscillators (review of 1803)

Given the equations for linear oscillators

bull find the general solution

bull classify the oscillators as underdamped overdamped critically damped or undamped

bull plot several trajectories as a function of time with different initial conditions

uml a) θ + 10 θ + 41 θ = 0 uml b) θ + θ = 0 uml c) θ + 11 θ + 30 θ = 0 uml d) θ + 6 θ + 9 θ = 0

Problem 4 Linear systems of ODEs

For each of the equations in Problem 3

1 change variables to (x y) where x = θ and y = θ

2 rewrite the second order ODE as a system of first order ODEs for x and y and a matrix A

3 find eigenvalues and eigenvectors of the matrix A

4 find a general solution to the ODE

5 (matlab) in the (x y) plane plot some solutions (with different initial conditions) for each of the cases plot eigenvectors (if they are real) and label the systems as underdamped overdamped critically damped or undamped What can you say about the stability of the origin (x y) = (0 0)

2




in the box provided (until 5pm)No late psets are accepted I f you collaborated with other students in the class list their names on the title sheet

T he work that you submit must b e your own

Strogatz C h 5 61-64

there is a sample M AT L A B code with examples of how to plot phase portraits and

1

Due by the end of day on Thursday

Reading

MATLAB help

Problem 1 Stability of fixed points of linear systems of ODEs


Problem Set 2

vector fields of a dynamical system on a plane

Main concepts phase portraits for linear systems fixed points of nonlinear systems their stabilityphase portraits in the vicinity of a fixed point

Strogatz 5110

Problem 2 Romeo and Juliet Linear systems of ODEs

If x is a measure of how attracted Juliet is to Romeo and y is a measure of how attracted Romeo is to Juliet and we model the evolution of their relationship as a linear system x = ax + by y = cx + dy then

1 do opposites attract (ie analyze x = ax + by y = minusbx minus ay)

2 is there a happy ending in the story of romantic clones x = ax + by y = bx + ay

Problem 3 Rabbits versus Sheep Phase portrait for nonlinear systems of ODEs

The evolution of the populations of rabbits (x ge 0) and sheep (y ge 0) is described by x = x(3 minus 2x minus y) y = y(2 minus x minus y)

bull Find fixed points linearize the system near the fixed points find eigenvalues eigenvectors determine the stability of the fixed point and sketch the local phase portrait

bull Attempt to complete the sketch of the phase portrait for the full system

bull (MATLAB) plot the phase portrait Compare the plot to your sketch note any differences

bull Describe different scenarios that the populations of rabbits and sheep would undergo depending on the populations at time zero

Problem 4 Nonlinear systems which depend on a parameter (preview of bifurcations)

bull Find fixed points classify their stability (it will depend on α)

bull Identify the values αlowast of the parameter α such that when α passes through αlowast the phase portrait qualitatively changes (fixed points appear disappear change stability etc) We say that at these values of α the rdquobifurcationrdquo phenomenon occurs

bull Sketch (or plot with MATLAB) local phase portraits when α is just below at and just above αlowast

bull As α passes through αlowast does the vector field far away from the fixed points change a lot

2a) x = αx minus x y = minusy 3b) x = αx + x y = minusy

Problem 5 Centers are not structurally stable

For the following nonlinear system x = minusy + αx3 y = x + αy3 one of the fixed points is at the origin Which phase portrait should the point have It turns out that for systems with Re(λi) = 0 at the fixed point the linear analysis is inconclusive We demonstrate it below

a) linearize the system at the fixed point at the origin and confirm that linear analysis suggests that the fixed point is a center

b) change variables from (x y) to polar coordinates (r θ) using x = r cos θ y = r sin θ and the fact that 2 2x + y = r2 and θ = tanminus1(yx)

c) for this new system r = f(r θ) θ = g(r θ) analyze the first equation (which you will probably have in the form r = f(r x y)) and the fixed point r = 0 Show that for some values of α the origin (x y) = (0 0) (which corresponds to r = 0) is stable while for other values it is unstable Comment on why this disagrees with the conclusion of the linear theory

2





Problem Set 3

Due at 1201 pm on Friday September 28th in the box provided No late psets are accepted If you collaborated with other students in the class list their names on the title sheet The work that you submit must be your own

Main concepts fixed points in nonlinear systems their stability phase portraits conservative systems and nonlinear centers non-dimensionalization

Reading Strogatz Ch 6

Problem 1 NL systems linearization phase portraits

Problem 2 Conservative systems Glider

1

Strogatz 6519

Strogatz 6514

Problem 3 Conservative systems General relativity and planetary orbits

Problem 4 Conservative systems

For the following system x = (2 cos x minus 1) sin x

1 Find the potential and the conserved quantity

2 Rewrite the equation as a system in the (x y) plane using y = x find the fixed points their stability

3 Sketch the phase portrait plot it in MATLAB

2

Strogatz 657





Problem Set 4

Due at 1201 pm on Friday October 5th in the box provided No late psets are accepted If you collaborated with other students in the class list their names on the title sheet The work that you submit must be your own

Main concepts Three bifurcations in 1D dynamical systems (saddle-node transcritical and pitchfork)


Problem 1 Bifurcations of 1D dynamical systems Normal forms

For each of the following problems

bull sketch all the qualitatively different vector fields that occur as r is varied

bull Find at which x and for which critical values of the parameter r the bifurcation occurs

bull Which bifurcation is it (saddle-node transcritical supercritical pitchfork or subcritical pitchfork)

lowastbull Finally sketch or plot in MATLAB the bifurcation diagram of fixed points x versus r and mark the stable branches as solid lines while the unstable branches as dashed lines (Note for some of the cases it is simpler to plot the bifurcation in MATLAB)

(a) x = r minus cosh x

x(b) x = xr minus xe

(c) x = rx minus sinh x

(d) x = 5 minus reminusx2

3x(e) x = rx + 1+x2

(f) x = r minus cos x

Problem 2 Hierarchy of models of fishery

(based on Strogatz 373 and 374)

To understand the underlying dynamics of a real-world problem one frequently considers a hierarchy of models ndash from the simplest one to a more complicated one In this problem we consider three models of fisheries of increasing complexity Your task is to analyze them using the material from the class and to see if one model is better than the other (for example the dynamics of which one is more physical)

The simplest model for fishery would be that in the absence of fishing the population of fish is assumed ˙to grow logistically N = rN(1 minus NK) Then to model the fishing rate the simplest choices are

˙(a) N = rN(1 minus NK) minus H where the fish is harvested at a constant rate H ˙(b) N = rN(1 minus NK) minus HN ˙ N(c) N = rN(1 minus NK) minus H A+N

1

1 Interpret physically the form of the harvesting rate in (b) and (c) What does the parameter A stand for physically Which model of a harvesting rate do you think is more physical

2 Non-dimensionalize the models to obtain the forms a) x = x(1 minus x) minus h b) x = x(1 minus x) minus hx

xc) x = x(1 minus x) minus h a+x

3 For models (a) and (b) find the fixed points and determine their stability for different values of h Are any of them unphysical

4 (MATLAB) plot the bifurcation diagrams for the cases (a) and (b) Label stable unstable branches and bifurcation points

5 Do the normal form analysis in the cases (a) and (b) and obtain analytically the results you got using MATLAB in the item above

For model (c)

6 Show that the system can have one two or three fixed points depending on the values of a and h Classify the stability of the fixed points in each case

7 Analyze the dynamics near x = 0 and show that the bifurcation occurs when h = a What type of bifurcation is it

18 Show that another bifurcation occurs when h = (a + 1)2 for a lt ac where ac is to be determined 4 Classify the bifurcation

9 Plot the stability diagram of the system in (a h) parameter space (The stability diagrams in 2shydimensional parameter space will be shown in class on Tuesday) Can hysteresis occur in any of the stability regions

And finally your conclusions

10 Now that you know the dynamics of the three models and its dependence on parameters conclude which model is more realistic and which important (or not) physical effects it takes into account Which if any important effects does it miss

Matlab help

There is an easy way to plot parametrized curves in MATLAB which once you see it is obvious It is very useful for plotting bifurcation curves

Usually one is plotting a curve for example x = 00110 plot(xsin(x))

However say you need to plot a bifurcation diagram for x = r + sin(x) where r is a parameter on the (r x) plane Then you can reverse the arguments in plot x = -2022 plot(-sin(x)x)

In general when given a parameterized curve x(θ) = cos(θ) y(θ) = sin(θ) to plot it in MATLAB do theta = [0012pi] x = cos(theta) y = sin(theta) plot(xy) axis equal

2





Problem Set 5


Main concepts Bifurcations in 2D Index theory

Reading Strogatz Ch 81 68

Problem 1 Bifurcations in higher dimensions

1 Find and classify all bifurcations for the system x = y minus ax y = minusby + x(1 + x) Plot or draw the stability diagram

2 (Preview of limit cycles) In polar form the system is given by r = r(r2 minus micror + 1) θ = minus1 where micro is a parameter Discuss the bifurcations of the r equation treating it as a 1D system In 2D (ie in the plane (x y)) sketch all the different phase portraits What happens in 2D when micro goes through a bifurcation value for the r-equation

Problem 2 Index theory

1 Use index theory to show which of the systems cannot have a closed trajectory a) x = x(4 minus y minus x2) y = y(x minus 1) b) x = x(4 minus y minus x2) y = minusy(x minus 1) (Hint paying attention to where the nullclines are is helpful in this problem)

2 A local bifurcation of a smooth vector field happens where two fixed points are born rdquoout of the blue skyrdquo What kinds of points can these be (nodes saddles spirals etc) Explain

3 A local bifurcation of a smooth vector field happens where one saddle point becomes three other fixed points What are the admissible types of these three points Explain

4 A smooth vector field on the phase plane is known to have a closed trajectory C inside of which there are two other closed trajectories C1 and C2 neither of them are inside of each other Sketch the arrangement of the orbits and conclude where and which kind of fixed points must be present Note the solution is not unique so you can comment on what is the minimal number of fixed points should be present

1





Problem Set 6


Main concepts Limit cycles Van der Pol oscillator


Problem 1 Limit cycles

1 Sketch the phase portrait in the (x y)-plane for the system given in polar coordinates θ = r2 minus 2 r = r(1 minus r2)(r + 3)

2 Sketch the phase portrait in the (x y)-plane for the system θ = 1 r = micror + r3 for micro = minus1 minus036 minus001 0 03 Describe what happens when micro passes through 0

Problem 2 Numerical study of the Van der Pol oscillator

Consider an electric circuit shown in Figure 1 with the input-output relation for the tunnel diode given by

i = φt(v) = φ(v minus E0) + I0

with φ(v) = γv3 minus αv Here i is current and v is voltage The equation for the circuit can be written as

1V = (minusφ(V ) minus W )

C 1

W = V (1)L

Figure 1 An electrical circuit with a tunnel diode for the van der Pol oscillator

1

E0I0V C

W

L

i

I0

i=φt(v)

E0 v

Image by MIT OpenCourseWare

1 Show that the two above equations can be rewritten as

uml V minus 1(α minus 3γV 2)V +

1 V = 0

C LC

Non-dimensionalize it and bring it to the standard form of the Van der Pol equation

x+ ε(x 2 minus 1) x + x = 0

2 Show that for small ε ltlt 1 the amplitude of the limit cycle is about 2 What is the period of the limit cycle Does the period depend on the initial conditions Contrast this behavior to the limit case ε = 0 Document your answers with graphs of both the phase space and the displacement

3 For large ε gtgt 1 the oscillations become relaxation oscillations Show (numerically) that in this case the amplitude of the limit cycle is still equal to 2 To keep the values of the velocity bounded choose ε less than or about equal to 10 Again document your answer with graphs of both the phase space and the displacement Does the period depend on the initial conditions For very large ε gtgt 1 one can find an approximate period of the Van der Pol oscillator (see Strogatz p 212-215) Estimate the period of the limit cycle from your computed graphs How does this period compare with the period derived for large ε gtgt 1 in Strogatz

Problem 3 Ruling out the limit cycles

Show that there are no limit cycles in the system using Index theory or (Tue lecture) Dulacrsquos criterion Liapunov function or the fact that the system is a gradient system

1 x = x + y2 y = sin x + 4y

2 x = x(1 minus y) y = y(1 minus x)

3 53 x = y minus x y = minusx minus y2(Hint try a Liapunov function of the form V (x y) = ax + by2 with suitable a and b)

4 x = 2xy + cos(y) y = x2 minus x sin(y)

2





Problem Set 7

Due at 1201 pm on Friday November 2nd in the box provided No late psets are accepted If you collaborated with other students in the class list their names on the title sheet The work that you submit must be your own

Main concepts Poincare-Bendixon theorem trapping region Hopf bifurcation

Reading Strogatz Ch 73 82-3

Problem 1 Poincare-Bendixon theorem constructing a trapping region

Consider the system 3 3 x = minusmicrox minus y + x y = x minus microy + y

(a) Show that there exists a periodic orbit for micro = 1 by constructing a trapping region (b) (MATLAB) Plot the phase portrait for micro = 1 and micro = minus1 (c) Find the value of micro for which the Hopf bifurcation happens

Problem 2 A circular limit cycle

Consider a nonlinear oscillator

x+ ax(x 2 + x 2 minus 1) + x = 0

(a) Find and classify all fixed points (in the phase plane (x x)) for a gt 0 a lt 0 and a = 0 (b) Plot the phase portrait for a = 1 and a = minus1 (After plotting them in MATLAB write the command axis equal It scales the x and the y axis the same so that your circular orbits look circular and not ellipsoidal) The phase portraits look strikingly similar what is the transformation to change one of them into the other one (c) Show analytically that there is a limit cycle for a = 0 What is its stability for a gt 0 and a lt 0 (d) Determine the radius of the limit cycle analytically (e) Determine the period of the limit cycle analytically check the answer with your numerics (f) Check that the condition for the Hopf bifurcation for the eigenvalues near a = 0 is satisfied is the bifurcation supercritical subcritical or degenerate

Problem 3 Oscillating chemical reaction

1

Strogatz 832

Problem 4 Numerics versus a Liapunov function

For this system a Hopf bifurcation occurs at micro = 0

x = microx + y 2 y = minusx + microy minus x y

(a) MATLAB Plot the phase portraits for some micro lt 0 and micro gt 0 Deduce if the bifurcation is supercritical or subcritical (Note use ode45 since streamslice plots will be inconclusive) (b) For micro = 0 check analytically if the origin is stable or not does it give you the same prediction as numerics (Hint for micro = 0 the system can be easily rewritten as a second-order equation for x which models a nonlinear oscillator this will help you to construct a Liapunov function) (c) Sketch a bifurcation diagram

2





Problem Set 8

Due at 1201 pm on Friday November 9th in the box provided No late psets are accepted If you collaborated with other students in the class list their names on the title sheet The work that you submit must be your own

Main concepts Bifurcations of limit cycles Poicare map

Reading Strogatz 82-3 87

Problem 1 Global bifurcations of cycles


Discuss the bifurcations of the system r = r(micro +5 minus r2 +6r) θ = 1+ r Plot all the possible topologically different phase portraits in (x y)

Note If there are 3 of them for values of micro = 7 8 9 for example to save time and paper you can plot them in subplots mu = [789] figure(123) for i = 13 subplot(22i) streamslice(xyxdotydot) axis equal grid on xlabel(rsquoxrsquo) ylabel(rsquoyrsquo) title([rsquomu = rsquo num2str(mu(i))]) end

Problem 3 Poincare maps

Given a vector-field J 2x = x minus y minus x x2 + y J 2y = x + y minus y x2 + y

1 Plot the phase portrait in MATLAB Do you see any periodic solutions

2 Rewrite the system in polar coordinates Is there a stable limit cycle

3 You should obtain in (2) that the cycle makes one revolution in Δθ = 2π Given that integrate the r-equation between some value r1 for which θ = 0 to a value r2 for which θ = 2π then write r2 as a function of r1 You just obtained the analytical form of the Poincare map rn+1 = f(rn)

1

Strogatz 841

4 Use MATLAB or any other computational software to compute the first 10 iterations of the Poincare map (ie x1 = f(x0) x2 = f(x1) etc) starting from the initial conditions r = 02 θ = 0 Submit those numbers Do they approach any particular value

5 In MATLAB plot the function that you just obtained and on top of it plot rn+1 = rn Use the commands grid on axis equal Print this out and iterate this map by hand (cobwebs) from the initial conditions r = 02 θ = 0 Is it the same as in the previous question

6 Repeat the previous two items for a trajectory starting at r = 05 θ = 0 Does the trajectory converge to the same number You just found a stable fixed point of a 1-dimensional map

7 Compute the multipliers (Lecture on Tue)

Problem 4 Poincare maps ndash iterating Poincare maps

Given two Poincare maps below

1 Print their graphs out as in Problem 35 (with the grid on and the plotted diagonal rn = rn+1) Iterate them by hand using graphical technique of cobwebs and find the approximate values of the stable and unstable fixed points (which correspond to stable and unstable limit cycles)

2 Then iterate them using the computer from the same initial conditions as you did by hand and this will give you numerical values of the fixed points Report those

3 What is the difference in the way that the iterations approach a stable fixed point in (a) versus in (b) What condition on the derivative of the map at the fixed point would give you one type of approach versus the other one

(a) rn+1 = rn + 05 sin(rn) 0 le rn le 8 (b) rn+1 = rn minus ln(rn) 0 lt rn le 3

2





Problem Set 9


Main concepts Matthieursquos equation Floquet theory

Reading Lecture on Matthieursquos equation is posted online

In class analyzed a vertically driven pendulum described by the Matthieursquos equation Physically this is a frictionless pendulum in oscillating gravitational field

θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)

For the pset you have to study the stability of the periodic solution with friction ie equation

uml θ + γθ + ω02(1 + h cos(2ωt))θ = 0 γ gt 0 (2)

1 Describe what the terms mean physically consider h = 0 and γ = 0 cases

2 (Numerical) Pick some values of the constants (eg h = 02 ω0 = 1) and look numerically for the typical examples of the behavior of the system (both as a time-evolution θ(t) and on the phase plane (θ θ)) For the same initial conditions plot in MATLAB both the time-evolution and the trajectory in phase plane for different values of γ (for examples γ = 0 01 and 03) Try ω close to ω0 Describe which distinct physical regimes you have depending on the damping parameter amplitude and frequency of the forcing

Note this system has time-dependence in the coefficients

3 (Analytical) Since after adding damping this is still a linear system with periodic coefficients you can apply the Floquet theory It states that the solution can be written in the form of an exponential function multiplying a periodic function which we represent as a sum of cosines

infin0 θ(t) = emicrot an cos(nωt +Φn)

n=1

Plug this expression in (2) collect the terms in front of a1 and set them equal to zero Explain the reason behind this

Drop the high frequency terms (as in lecture when we dropped cos(3ωt + Φ)) It can be shown that that term is negligible

4 In the expression that you obtained collect the terms in front of sin(ωt) and cos(ωt) Set both of these groups equal to zero Explain the reason behind the step

5 The two equations that you obtained can be analyzed as a system of two linear equations for cos(φ) and sin(φ) When does this system have a solution Deduce the criterion for existence of a non-trivial (ie not identically zero) solution This will be your fourth order equation for micro the coefficients will depend on ω ω0 and h

1

6 Find the boundary of instability ie find where micro = 0 as a function of h and ωω0 To which side of that boundary will the solutions be unstable Plot it on the (h ωω0) plane for small h

7 The criterion for instability in the undamped case derived in class was

ω h gt 2

1 minus

(3)ω0

Plot it in Matlab denote the regions of stability instability For infinitesimally small h what is the frequency at which the pendulum is unstable How does this frequency relate to the frequency of unperturbed pendulum

Note It is not the same Itrsquos a different instability threshold than that of the undamped forced pendulum x + ω0

2x = cos(2ωt) for which the resonance happened when the forcing frequency 2ω was equal to the natural frequency ω0 Because of the relation that yoursquoll find the instability is called subharmonic

8 On the same Matlab plot plot the boundary of instability for the damped case for two values of γω0 = 01 and γω0 = 05 Interpret the change of the instability domain physically

9 Now that you analyzed the system analytically check your numerical experiments in question 2 and explain the behavior that you saw given that now you have a new perspective

2





MIDTERM (At-home portion)

Problem 1 Bifurcations ndash a biochemical switch

A gene G usually inactive is activated by a biochemical substance S to produce a pigment or other geneproduct when the concentration S exceeds a certain threshold

Let g(t) denote the concentration of the gene product and assume that the concentration s0 of S is fixedThe model is

k 2

g = k1s0 minus 3gk2g + (1)

k24 + g2

where krsquos are positive constants The production of g is stimulated by s0 at a rate k1 and by autocatalytic(positive) feedback process modeled by the nonlinear term There is also a linear degradation of g at arate k2

1 Nondimensionalize the equations and bring them to the form

dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)

2 Show that if s = 0 there are two positive fixed points x if r lt rlowast c where rc is to be determined

3 Find parametric equations for the bifurcation curves in (r s) space

4 (MATLAB) plot quantitatively accurate plot of the stability diagram in (r s) space

5 Classify the bifurcations that occur

6 Assume that initially there is no gene product ie g(0) = 0 and suppose that s is slowly increasedfrom zero (the activating signal is turned on) What happened to g(t) What happens if s thengoes back to zero Does the gene turn off again

Problem 2 Nonlinear oscillator

Given an oscillator x+ bxminus kx+ x3 = 0 b k can be positive negative or zero

1 Interpret the terms physically for different values of b and k

2 Find the bifurcation curves in (b k) plane state which bifurcation happens and what kind of fixedpoints one has to each side of the bifurcation curve

1

Problem 3 Numerical study of the displaced Van der Pol oscillator

The equations for a ldquodisplacedrdquo Van der Pol oscillator are given by

x = y minus a y = minusx+ δ(1 minus x2)y a gt 0 δ gt 0

Consider a small

1 Show that the system has two equilibrium points one of which is a saddle Find approximateformula for small a of this fixed point Study this system numerically with ode45

2 Submit the plots of phase plane with trajectories starting at different points (to illustrate thedynamics) for δ = 2 a = 01 02 04 and observe that the saddle point approaches the limit cycleof the Van der Pol equation

3 Find numerically the value of the parameter a to two decimal points when the saddle point collideswith the limit cycle What happens to the limit cycle after this collision

2





Midterm Practice Problems

The midterm exam will have a classroom written portion and a take-home computational portion

The best preparation for the exam is problem sets (the solutions are posted online) The solution to the PSet 6 will be online on Friday just after it is due

Topics

You now know how to analyze models of physical phenomena that are modeled as one and two-dimensional systems of ODEs that can depend on parameter The steps in the study are

1 Non-dimensionalize the system thus reducing the number of parameters in the problem

2 Find fixed points study the local phase portrait near these fixed points

(a) if the fixed points are hyperbolic then the linear theory gives the correct phase portrait

(b) if the linear system predicts centers then one has to do extra work Nonlinear centers appear in the cases of conservative systems reversible systems Otherwise more work is needed to determine if the phase portrait is a spiral or a center or something else (Strogatz Ch 7)

(c) in case of zero eigenvalues one has to use different techniques (for example graphical techniques) to deduce the stability of a fixed point For example x = x2 (x = 0 is semi-stable) x = x3

(x = 0 is unstable) etc

3 Deduce some of the characteristics of the global phase portrait (look at nullclines rdquofences and funnelsrdquo techniques from 1803 perhaps you can find a trapping region)

4 Check if there could be closed orbits in the system or you can rule out their existence

(a) in case of Lienard systems there exists a unique stable limit cycle

(b) check the index theory perhaps you can show that there are no closed orbits

(c) check if the system is conservative (then in 2D you can only have saddles and centers)

(d) check if you can rule out closed orbits completely (Dulacrsquos criterion existence of a Liapunov function gradient systems)

5 Study dependence of the system on parameters ndash bifurcation theory

(a) Bifurcations we studied in both 1D and 2D are saddle-node transcritical and pitchfork bifurshycations (but in PSets you had examples of others)

(b) Find where they occur (in 1D for x = f(x r) r-parameter can look at the system f(x r) = 0 partf(x r)partx = 0)

(c) Expand f(x r) near a critical point x = xlowast (fixed point) and r = rc (where the bifurcation happens) find the normal form deduce the type of the bifurcation

1

(d) Sometimes it is easier to see graphically when a bifurcation occurs For example if minusxx = f(x) = eminusx minus (x + a) then the bifurcation occurs when the curve y = e touches the

straight line y = x + a The same graphical technique is applicable to the second order systems when you can check when the nullclines start to intersect as you vary a parameter

6 Last but probably the most important item interpret your answer physically For example in the laser system you find a transcritical bifurcation and that a stable fixed point at zero loses its stability at the bifurcation point and a new stable equilibrium exists for the larger value of parameter However you should reinterpret your answer physically that below the critical value of the parameter the system acts like a lamp while above the rdquolaser thresholdrdquo it acts like a laser

Sample problems

1 Non-dimensionalization Pset 2 3 6 Extra sample problems 375 644-6 647 8110 (these are all long problems they all have good examples of non-dimensionalization but if you try to fully complete all the questions in them it will take a very long time)

2 Stability of fixed points linear phase portraits all psets Special cases Pset 2 problem 1 Extra sample problems x = x3 y = y

2x = minusx y = minusy Strogatz problems 523 - 5210

3 Nonlinear centers Pst 2 problem 5 Pset 3 (conservative systems) reversible systems (Strogatz problems in 66) Extra sample problems 6513

4 Using nullclines to deduce something about the system in case of bifurcations (Strogatz example 811) in case of a trapping region (Strogatz Glycosis example 732) Extra sample problems 816

5 Closed orbits (1 prove that there are some) NL centers (pset 3) 653 665 611 (2 rule out closed orbits) Pset 6 687 726 728 7212 7214 (3 limit cycles) Pset 6 742 717 (4 non-isolated periodic orbits) everything about linear and nonlinear centers

6 Closed orbits limit cycle versus centers Pet 6 problem 1 (case ε = 0 versus ε gt 0) Extra sample problems

7 Bifurcations in 1D 1 parameter Pset 4 Extra sample problems 311-4 321-4 341-4 345-10 3113 Plot bifurcation diagrams 354

8 Bifurcations in 1D 2 parametrs Pset 4 Extra sample problems 375

2

9 Bifurcations in 2D PSet 5 problem 1 Extra sample problems 811-7 8110 8113

10 Index theory Pset 5 Extra sample problems 682-5 (unusual examples) 689 686 Can there be two fixed points inside a limit cycle If so give an example (graphically do not make up equations for it)

11 Oscillators linear oscillators (1803) underdamped undamped critically damped oscillators (Pset 1) nonlinear oscillators x+ bx+ sin x = 0 Van der Pol (Pset 6 first lectures)

Matlab

For the take-home section you should be able to plot things in Matlab and be able to use it to make some research conclusions Sample problems are of the type

1 Use streamslice to plot the phase portrait for any of the 2D systems listed above

2 Use plot to plot bifurcation diagrams and trajectories x(t) y(t) as a function of time

3 For many problems streamslice does not give good results at all (examples in the last Matlab primer) Use ode45 in those cases and choose where you start the trajectories yourself

You should be able to plot phase portraits for any one of the problems listed above and in the psets Make sure that your plots are not inconclusive For example

1 if you want to show that a trajectory converges to a fixed point and you compute the trajectory via ode45 choose a long enough time-interval to show that it does

2 if you want to show that trajectories are attracted to a fixed point from all directions pick the initial conditions all around the fixed point and not in one small area to one side of the fixed point

3 label your plots accordingly It is useful to put the parameter value in the title when you vary the parameter and have many plots In the following example we vary micro between 0 and 1 in steps of 01 then plotting something with the title reading mu = 03

4 We are dealing with solutions to ODEs They are all differentiable Therefore in your plots the solutions should never have any corners (which happen if you plot a function with low resolution for example x = [0110] plot(xsin(x)) So always make sure that you have enough resolution when you plot the trajectories (in phase space or time-evolution plots)

mu = [0011] x = [0013] for i = 1length(mu) figure(1) plot(xsin(x)mu(i)) title([rsquomu = rsquonum2str(mu(i))]) end

3




Y

5

4

3

2

1

0

minus1

minus2

minus3

minus4

minus5 minus6 minus4 minus2 0 2 4 6

X

Figure 1 Phase plane created using ode45 starting a variety of initial conditions

Figure 2 Phase plane created using streamslice starting a variety of initial conditions

2

minus6 minus4 minus2 0 2 4 6minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

Y

X





Matlab primer



1 Download MATLAB






1






5 Comments





2


3









x = y

y = g(xyz)

z = 1





component = 1

















1


Strogatz 234


















2








1


Reading

MATLAB help



Problem Set 2



Strogatz 5110






















2





Problem Set 3






1

Strogatz 6519

Strogatz 6514







2

Strogatz 657





Problem Set 4














3x(e) x = rx + 1+x2







1







For model (c)







Matlab help





2





Problem Set 5












1





Problem Set 6












C 1

W = V (1)L


1

E0I0V C

W

L

i

I0

i=φt(v)

E0 v




1 V = 0

C LC


x+ ε(x 2 minus 1) x + x = 0





1 x = x + y2 y = sin x + 4y




2





Problem Set 7









x+ ax(x 2 + x 2 minus 1) + x = 0



1

Strogatz 832





2





Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3








Matlab primer



1 Download MATLAB






1






5 Comments





2


3









x = y

y = g(xyz)

z = 1





component = 1

















1


Strogatz 234


















2








1


Reading

MATLAB help



Problem Set 2



Strogatz 5110






















2





Problem Set 3






1

Strogatz 6519

Strogatz 6514







2

Strogatz 657





Problem Set 4














3x(e) x = rx + 1+x2







1







For model (c)







Matlab help





2





Problem Set 5












1





Problem Set 6












C 1

W = V (1)L


1

E0I0V C

W

L

i

I0

i=φt(v)

E0 v




1 V = 0

C LC


x+ ε(x 2 minus 1) x + x = 0





1 x = x + y2 y = sin x + 4y




2





Problem Set 7









x+ ax(x 2 + x 2 minus 1) + x = 0



1

Strogatz 832





2





Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3









5 Comments





2


3









x = y

y = g(xyz)

z = 1





component = 1

















1


Strogatz 234


















2








1


Reading

MATLAB help



Problem Set 2



Strogatz 5110






















2





Problem Set 3






1

Strogatz 6519

Strogatz 6514







2

Strogatz 657





Problem Set 4














3x(e) x = rx + 1+x2







1







For model (c)







Matlab help





2





Problem Set 5












1





Problem Set 6












C 1

W = V (1)L


1

E0I0V C

W

L

i

I0

i=φt(v)

E0 v




1 V = 0

C LC


x+ ε(x 2 minus 1) x + x = 0





1 x = x + y2 y = sin x + 4y




2





Problem Set 7









x+ ax(x 2 + x 2 minus 1) + x = 0



1

Strogatz 832





2





Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3





3









x = y

y = g(xyz)

z = 1





component = 1

















1


Strogatz 234


















2








1


Reading

MATLAB help



Problem Set 2



Strogatz 5110






















2





Problem Set 3






1

Strogatz 6519

Strogatz 6514







2

Strogatz 657





Problem Set 4














3x(e) x = rx + 1+x2







1







For model (c)







Matlab help





2





Problem Set 5












1





Problem Set 6












C 1

W = V (1)L


1

E0I0V C

W

L

i

I0

i=φt(v)

E0 v




1 V = 0

C LC


x+ ε(x 2 minus 1) x + x = 0





1 x = x + y2 y = sin x + 4y




2





Problem Set 7









x+ ax(x 2 + x 2 minus 1) + x = 0



1

Strogatz 832





2





Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3












x = y

y = g(xyz)

z = 1





component = 1

















1


Strogatz 234


















2








1


Reading

MATLAB help



Problem Set 2



Strogatz 5110






















2





Problem Set 3






1

Strogatz 6519

Strogatz 6514







2

Strogatz 657





Problem Set 4














3x(e) x = rx + 1+x2







1







For model (c)







Matlab help





2





Problem Set 5












1





Problem Set 6












C 1

W = V (1)L


1

E0I0V C

W

L

i

I0

i=φt(v)

E0 v




1 V = 0

C LC


x+ ε(x 2 minus 1) x + x = 0





1 x = x + y2 y = sin x + 4y




2





Problem Set 7









x+ ax(x 2 + x 2 minus 1) + x = 0



1

Strogatz 832





2





Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3




















1


Strogatz 234


















2








1


Reading

MATLAB help



Problem Set 2



Strogatz 5110






















2





Problem Set 3






1

Strogatz 6519

Strogatz 6514







2

Strogatz 657





Problem Set 4














3x(e) x = rx + 1+x2







1







For model (c)







Matlab help





2





Problem Set 5












1





Problem Set 6












C 1

W = V (1)L


1

E0I0V C

W

L

i

I0

i=φt(v)

E0 v




1 V = 0

C LC


x+ ε(x 2 minus 1) x + x = 0





1 x = x + y2 y = sin x + 4y




2





Problem Set 7









x+ ax(x 2 + x 2 minus 1) + x = 0



1

Strogatz 832





2





Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3





















2








1


Reading

MATLAB help



Problem Set 2



Strogatz 5110






















2





Problem Set 3






1

Strogatz 6519

Strogatz 6514







2

Strogatz 657





Problem Set 4














3x(e) x = rx + 1+x2







1







For model (c)







Matlab help





2





Problem Set 5












1





Problem Set 6












C 1

W = V (1)L


1

E0I0V C

W

L

i

I0

i=φt(v)

E0 v




1 V = 0

C LC


x+ ε(x 2 minus 1) x + x = 0





1 x = x + y2 y = sin x + 4y




2





Problem Set 7









x+ ax(x 2 + x 2 minus 1) + x = 0



1

Strogatz 832





2





Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3











1


Reading

MATLAB help



Problem Set 2



Strogatz 5110






















2





Problem Set 3






1

Strogatz 6519

Strogatz 6514







2

Strogatz 657





Problem Set 4














3x(e) x = rx + 1+x2







1







For model (c)







Matlab help





2





Problem Set 5












1





Problem Set 6












C 1

W = V (1)L


1

E0I0V C

W

L

i

I0

i=φt(v)

E0 v




1 V = 0

C LC


x+ ε(x 2 minus 1) x + x = 0





1 x = x + y2 y = sin x + 4y




2





Problem Set 7









x+ ax(x 2 + x 2 minus 1) + x = 0



1

Strogatz 832





2





Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3

























2





Problem Set 3






1

Strogatz 6519

Strogatz 6514







2

Strogatz 657





Problem Set 4














3x(e) x = rx + 1+x2







1







For model (c)







Matlab help





2





Problem Set 5












1





Problem Set 6












C 1

W = V (1)L


1

E0I0V C

W

L

i

I0

i=φt(v)

E0 v




1 V = 0

C LC


x+ ε(x 2 minus 1) x + x = 0





1 x = x + y2 y = sin x + 4y




2





Problem Set 7









x+ ax(x 2 + x 2 minus 1) + x = 0



1

Strogatz 832





2





Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3








Problem Set 3






1

Strogatz 6519

Strogatz 6514







2

Strogatz 657





Problem Set 4














3x(e) x = rx + 1+x2







1







For model (c)







Matlab help





2





Problem Set 5












1





Problem Set 6












C 1

W = V (1)L


1

E0I0V C

W

L

i

I0

i=φt(v)

E0 v




1 V = 0

C LC


x+ ε(x 2 minus 1) x + x = 0





1 x = x + y2 y = sin x + 4y




2





Problem Set 7









x+ ax(x 2 + x 2 minus 1) + x = 0



1

Strogatz 832





2





Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3










2

Strogatz 657





Problem Set 4














3x(e) x = rx + 1+x2







1







For model (c)







Matlab help





2





Problem Set 5












1





Problem Set 6












C 1

W = V (1)L


1

E0I0V C

W

L

i

I0

i=φt(v)

E0 v




1 V = 0

C LC


x+ ε(x 2 minus 1) x + x = 0





1 x = x + y2 y = sin x + 4y




2





Problem Set 7









x+ ax(x 2 + x 2 minus 1) + x = 0



1

Strogatz 832





2





Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3








Problem Set 4














3x(e) x = rx + 1+x2







1







For model (c)







Matlab help





2





Problem Set 5












1





Problem Set 6












C 1

W = V (1)L


1

E0I0V C

W

L

i

I0

i=φt(v)

E0 v




1 V = 0

C LC


x+ ε(x 2 minus 1) x + x = 0





1 x = x + y2 y = sin x + 4y




2





Problem Set 7









x+ ax(x 2 + x 2 minus 1) + x = 0



1

Strogatz 832





2





Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3










For model (c)







Matlab help





2





Problem Set 5












1





Problem Set 6












C 1

W = V (1)L


1

E0I0V C

W

L

i

I0

i=φt(v)

E0 v




1 V = 0

C LC


x+ ε(x 2 minus 1) x + x = 0





1 x = x + y2 y = sin x + 4y




2





Problem Set 7









x+ ax(x 2 + x 2 minus 1) + x = 0



1

Strogatz 832





2





Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3








Problem Set 5












1





Problem Set 6












C 1

W = V (1)L


1

E0I0V C

W

L

i

I0

i=φt(v)

E0 v




1 V = 0

C LC


x+ ε(x 2 minus 1) x + x = 0





1 x = x + y2 y = sin x + 4y




2





Problem Set 7









x+ ax(x 2 + x 2 minus 1) + x = 0



1

Strogatz 832





2





Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3








Problem Set 6












C 1

W = V (1)L


1

E0I0V C

W

L

i

I0

i=φt(v)

E0 v




1 V = 0

C LC


x+ ε(x 2 minus 1) x + x = 0





1 x = x + y2 y = sin x + 4y




2





Problem Set 7









x+ ax(x 2 + x 2 minus 1) + x = 0



1

Strogatz 832





2





Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3






1 V = 0

C LC


x+ ε(x 2 minus 1) x + x = 0





1 x = x + y2 y = sin x + 4y




2





Problem Set 7









x+ ax(x 2 + x 2 minus 1) + x = 0



1

Strogatz 832





2





Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3








Problem Set 7









x+ ax(x 2 + x 2 minus 1) + x = 0



1

Strogatz 832





2





Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3








2





Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3








Problem Set 8













1

Strogatz 841











2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3














2





Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3








Problem Set 9





θ + ω02(1 + h cos(2ωt))θ = 0 h ge 0 (1)








n=1





1



ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3






ω h gt 2

1 minus

(3)ω0






2









k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3












k 2


k24 + g2



dx

dT= sminus rx+

x2 r gt 0 s

1 + x2ge 0 (2)










1




Consider a small




2








Topics


















1




Sample problems










2




Matlab











3







Consider a small




2








Topics


















1




Sample problems










2




Matlab











3











Topics


















1




Sample problems










2




Matlab











3







Sample problems










2




Matlab











3







Matlab











3




18353spring 2001

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