MA 2034 Project Part I and II Fall 2023
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MA 2034
Project Part I and II
Fall 2023
Instructions
This Project has two Parts, Part I and Part II. You can choose to submit only Part I or you can submit both parts. If you only do Part 1 your worksheet grades will count 18% of your final course grade and Part 1 of the project will count 4% of your final course grade. If you do both parts then your worksheet grades will count 14% of your final course grade and Parts 1 and 2 of the project combined will count 8% of your final course grade. You can work in a group of up to 4 members. If you work in a group you must use the group submission option on GS otherwise all team members will not receive credit. Parts I and II are both due on Gradescope by 11:59pm December 17.
Project PART I. Our Carbon Budget
A major driver of global climate change is the emissions of greenhouse gases such as CO2 into the atmosphere from human activities such as the burning of fossil fuels for transportation, electric power generation, manufacturing, agriculture, home heating, and so forth. Global average temper- atures are now slightly higher than 1。C above their pre-Industrial Revolution levels. Damaging sea level rise and extreme weather events (such as the recent wildfires in Canada that led to the unhealthy air in NYC last summer) will become an increasingly major problem around the world if global average temperature rises more than 1.5。 C. Scientists estimate that the total additional amount of CO2 that can be put into the atmosphere without reaching the 1.5。C level is about 51 parts per million (ppm) or approximately 400 Gigatons (Gt). Let’s call 51 ppm our carbon budget. In this project, we want to use linear systems of differential equations to analyze the predictions of various model scenarios regarding carbon dioxide emissions and investigate when we might exceed our carbon budget and how changes in the parameters affect the solutions.
Creating a linear system of first-order differential equations to model atmospheric CO2 and CO2 emissions involves setting up a system where each equation represents one of these two variables, and their interaction is characterized by four parameters. The system can be solved using the method of eigenvalues and eigenvectors. This is a highly simplified model of the complex interactions between CO2 and the environment.
Consider the linear first-order differential equation system with four parameters to model atmo- spheric CO2 and CO2 emissions. Let x1 (t) represent the concentration of atmospheric CO2 at time t in ppm, and x2 (t) represents the CO2 emissions from fossil fuel and other human sources at time t in ppm. The system of differential equations can be expressed as:
= ax1 + bx2
dt = cx1 + dx2
Here, a,b,c, and d are the parameters of the system, representing various factors:
. a: Natural absorption rate of CO2 by Earth’s ecological systems.
. b: Proportion of fossil fuel emissions contributing to the increase in atmospheric CO2 .
. c: This parameter could reflect feedback mechanisms or policy responses on CO2 emissions.
. d: Rate at which fossil fuel emissions change for reasons not due to atmospheric CO2 .
1 General Questions
1. What does x2 (t) represent in the system?
(a) Concentration of atmospheric CO2 at time t
(b) Natural absorption rate of CO2
(c) Impact of CO2 on fossil fuel emissions
(d) CO2 emissions at time t
2. If b > 0 and the value of b is increased while keeping other parameters constant, what is the likely impact on long-term atmospheric CO2 ?
(a) Increase
(b) Decrease
(c) No change
(d) Initial decrease followed by an increase
3. A decrease in the parameter a would most likely represent:
(a) Improved CO2 absorption by natural processes.
(b) Increased efficiency in emission reduction technologies.
(c) A reduction in the rate of industrial emissions.
(d) A decrease in the natural absorption rate of CO2.
4. What is the significance of eigenvalues in the solution of this system?
(a) They determine the initial state of the system.
(b) They indicate possible equilibrium points.
(c) They represent the rate of change of variables.
(d) They help understand the system’s behavior over time.
5. In the context of this system, what would an eigenvalue of zero imply?
(a) The system has no solution.
(b) There is exponential growth or decay.
(c) One solution has no time dependence.
(d) The system is inherently unstable.
6. If both eigenvalues are real and negative, what can be inferred about the long term behavior of the system?
(a) That both the atmospheric CO2 concentration and the CO2 emissions will exponentially increase over time.
(b) It indicates stability and potential decay.
(c) That both the atmospheric CO2 concentration and the CO2 emissions will exponentially decay over time.
(d) It suggests no change in CO2 and emissions over time.
Solving the First Order System
The system can also be written in matrix form as:
dt(d)（x2(x1)) =（ c(a)
d(b))（ x2(x1))
where the parameters are defined as follows:
. a = −0.004
. b = 1.4
. c = 0
. d = 0.01
The initial conditions are assumed to be:
. x1 (0) = 424 , ppm starting from atmospheric CO2 concentration in 2023 baseline.
. x2 (0) = 7 ppm, representing the initial global emission concentration.
2 Questions
Parameter Set I
1. (a) Use the above parameters and solve the system. You will need to solve the system numerically using software but show your set up and write the complete solution to the IVP ( initial value problem) showing the eigenvalues and vectors and the values of the constants that your obtain from the computer software you used.
(b) Use a numerical solver to graph the solution of the system.
(c) Using your graph approximate the atmospheric CO2 concentration 30 years from today and 100 years from today.
(d) Upload a graph of your solution to GS.
Parameter Set II- Changing b
2. Answer the questions that follow using the parameters shown below.
. a = −0.004
. b = 0.95
. c = 0
. d = 0.01
The initial conditions are assumed to be:
. x1 (0) = 424ppm,
. x2 (0) = 7 ppm.
(a) Use a numerical solver to graph the solution of the system.
(b) Using your graph approximate the atmospheric CO2 concentration 30 years from today and 100 years from today.
(c) What does the constant b represent in the model?
(d) Did changing b have any effect on the solution? If so how would you describe the effect? (e) Upload a graph of your solution to GS.
Parameter Set III-Changing a
3. Answer questions for the parameters shown below.
. a = −0.008
. b = 1.4
. c = 0
. d = 0.01
The initial conditions are assumed to be:
. x1 (0) = 424 , ppm starting from atmospheric CO2 concentration in 2023 baseline. . x2 (0) = 7 ppm, representing the initial global emission concentration.
(a) Use a numerical solver to graph the solution of the system.
(b) Using your graph approximate the atmospheric CO2 concentration 30 years from today and 100 years from today.
(c) What does the constant a represent in the model?
(d) Did changing a have any effect on the solution? If so how would you describe the effect? (e) Upload a graph of your solution to GS.
Parameter Set IV-Changing d
4. Answer questions for the parameters shown below.
. a = −0.004
. b = 1.4
. c = 0
. d = −0.01
The initial conditions are assumed to be:
. x1 (0) = 424 ,
. x2 (0) = 7 ppm.
(a) Use a numerical solver to graph the solution of the system.
(b) Using your graph approximate the atmospheric CO2 concentration 30 years from today and 100 years from today.
(c) What does the constant d represent in the model?
(d) Did changing d have any effect on the solution? If so how would you describe the effect? (e) Upload a graph of your solution to GS.
Summary
Answer the following questions.
5. (a) If the current CO2 budget is 51 ppm, when does this model, using Parameter Set I, indicate that we will reach 51 ppm and exhaust our Carbon budget? Show your reasoning. Use the eigenvalues and vectors from that solution to calculate your answer.
(b) If CO2 emissions were somehow reduced through conservation and the introduction of new technologies such as carbon sequestration, could the length of time to exhaust our CO2 budget be doubled? What parameters would you need to change and by how much? If you can’t double the length of time, what is the longest extension you can achieve. Show your work.
(c) Explain what the parameters you changed in part (b) mean in terms of the model.
(d) Graph the solution to the system with the new parameters from (b) that would double (or extend) the time to exhaust the Carbon Budget.
(e) Some scientist have advocated that we will need to return to an atmospheric concen- tration of of 350ppm to reestablish a climate conducive to human existence. Given the model we have been using, can you change the parameters to achieve 350 ppm atmospheric concentration of CO2 in 100 years? in 50 years? What parameters did you change. Upload one graph with the results of your experiment.
Project Part II
3 Bifurcations in Linear Systems
For a first order Linear System such as the one shown below, the qualitative behavior of the solutions and the phase diagram can be determined from the eigenvalues of the matrix A.
= ax + by
dt = cx + dy
In this exercise1 our goal is to better understand how different linear systems are related to each other. In a one-parameter linear first-order system of differential equations, one of the parameters in the coefficient matrix is varied, thus the eigenvalues change and this leads to a corresponding change in the phase plane of the system. If the eigenvalues change dramatically, for example from positive to negative or from real to complex, the phase plane will also change qualitatively. These changes correspond to bifurcations of the system.
The System
We will explore variations in parameter a and the qualatative effect on the solutions of the system.
= ax + 4y
dt = −4x
Here, x and y are the variables, and a is a parameter that can take on any real value, not just integers. The nature of the solutions and the bifurcation phenomena depend on the values of a.
Questions to be entered into the Gradescope Form.
1. Compute the eigenvalues of the matrix (they depend on a) and determine the bifurcation value(s) of a. Show your calculations.
2. Upload a phase plane portrait for each of the bifurcation values, and for values slightly on each side of each of the bifurcation values for a total of 7 phase plane portraits. You will need to upload your graphs into the corresponding question on Gradescope. There are three bifurcation values, we will label them in order of increasing size as b1 < b2 < b3 . For each bifurcation value, choose a value for a such that −12 ≤ a ≤ 12 and upload a phase plane portrait to Gradescope for:
(a) a < b1
(b) a = b1
(c) b1 < a < b2
(d) a = b2
(e) b2 < a < b3
(f) a = b3
(g) a > b3
3. For each bifurcation value, write a short paragraph that describe the transition that takes place. Be sure to include:
(a) How to the eigenvalues change, for example, from real to complex.
(b) The name of the new phase diagram that results.
(c) Describe the long term behavior of the solutions. For example, does one or both go to infinity? -infinity? Do the solutions oscillate?
4 The Harmonic Oscillator
The same type of analysis can be applied to the harmonic oscillator equation for a mass m attached to a spring of spring constant k. If the motion includes friction also called damping, and the coefficient of damping is d, then the equation motion can be described by the second order linear homogeneous differential equation with constant coefficients.
my′′ + md · y′ + k · y = 0
If we take m = k = 1, then the above equation can be written as
y′′ + d · y′ + y = 0
This differential equation can be written as a system
= x2
dt = −x1 − d · x2
and the system can be written in matrix form.
= 
1 
d) X(⃗)
We can explore different cases of qualatative motion of the harmonic oscillator by varying the parameter d from 0 to very large.
Questions to be uploaded to Gradescope:
1. Find the eigenvalues for this system (they depend on d.)
2. Find the bifurcation value(s) for d.
3. For each of the bifurcation values describe behavior of the solutions of the harmonic oscillator.
4. Depending on the value of d there are 4 types of behavior the system exhibits. Match each of the behaviors below with the corresponding value or range of d.
(a) Undamped Solutions that oscillate with a constant amplitude and period 2π .
(b) Underdamped Solutions that oscillate with a decreasing amplitude.
(c) Critically-damped Solutions that tend exponentially towards zero at the rate of te−t or e−t.
(d) Overdamped Solutions with no and all solution tend to the origin at an exponential rate.
2023-12-18