MAT 331: Project 2

Project description

In this project we will use Mathematica to solve several problems involving differential equations. Differential equations are currently used to model a wide range of phenomena from biology, physics, chemistry, computer science, economic analysis, etc. The theory of differential equations has become an essential tool in all areas of science, particularly since computers became commonly available.

Problem 1

The fish and game department in a certain state is planning to issue hunting permits to control the deer population (one deer per permit). It is known that if the deer population falls below a certain level m, the deer will become extinct. It is also known that if the deer population rises above the carrying capacity M, the population will decrease back to M through disease and malnutrition. Consider the following model for the growth rate of the deer population P as a function of time t:

(dP/dt)=r P(M-P)(P-m)

where P is the deer population and r=0.00003 is a constant of proportionality. The values of the other parameters are M=100 and m=55.

Plot the vector field of the differential equation using VectorPlot[..]. Then plot the vector field using StreamPlot[..]; identify all the constant solutions and color them in red using StreamPlot[..]. Try a window size of 100x130 (in the txP plane). What happens to the solutions as time t increases, t→∞? Color a couple of trajectories to illustrate the possible behaviors, then explain.

Solve the system of differential equations for the initial condition P(20)=110. If DSolve[...] does not work (and most likely it won’t), then a numeric approximation may be the next best thing to hope for, so use Mathematica’s NDSolve[..] function to find a numerical approximation of the true solution, then plot it.

If the initial deer population is 140, about how many hunting permits should be issued so that the deer population does not become extinct?

Write a small interactive model using Manipulate[..] and Locator[..], that initially plots the StreamPlot[..] from part 1, and then on click, colors the solution curve that passes through the point where the user clicked on.

Solve the differential equation given in the problem. It is a separable differential equation, so you can separate the variables and integrate to obtain an implicit equation involving P.  When integrating, you may need to use partial fraction decomposition (the Mathematica function Apart[..] might come in handy). Plot a couple of relevant level curves of the implicit equation you have obtained, and compare them to the trajectories of the PhasePortrait from part 1.

Problem 2

In this problem, we will visualize a famous phenomenon in dynamics of nonlinear differential equations, called the Poincaré-Andronov-Hopf bifurcation, or in short a Hopf bifurcation. It exhibits the birth of a limit cycle through a change in the stability of the equilibrium point. This type of bifurcation is widely present in physical systems. “Hopf bifurcation underlies many ‘spontaneous’ oscillations such as airfoil flutter and other wind-induced oscillations (e.g., those that caused the Tacoma-Narrows bridge collapse) in structural engineering systems, vortex shedding in fluid flow around a solid body at sufficiently high stream velocity, LCR oscillations in electrical circuits, relaxation oscillations (e.g., as described by the Van der Pol oscillator), the periodic firing of neurons in nervous systems (e.g., in the FitzHugh-Nagumo equation modelling these phenomena), oscillations in autocatalytic chemical reactions (e.g., the Belousov-Zhabotinsky reaction) as described by the Brusselator and similar models, oscillations in fish populations (as described by predator-prey models), periodic fluctuations in the number of individuals suffering from an infectious disease (as described by epidemic models)” Gert van der Heijden. Hopf bifurcations are also one of the most studied types of bifurcations in economics (for example, debt-financed investment-led growth models, optimal growth model with several capital goods, etc.) Weather prediction models and climate change models also exhibit Hopf bifurcations (for example, the El Niño-Southern Oscillation is a quasi-periodic climate pattern that occurs across the equatorial Pacific Ocean roughly every three to seven years. It is characterized by a change in sea surface temperatures in the eastern Pacific off the coast of Peru and accompanying changes in the air pressure difference between the central and western Pacific Ocean).

In our project, we will first analyze an example of a system of differential equations which exhibits a Hopf bifurcation. 

Part I: Consider the nonlinear system 𝒮 of differential equations given below, where α  is a real parameter:

({{{, ({{x'= y-x(x^2+y^2-α), (1)}, {y'= -x-y(x^2+y^2-α), (2)}})}}) .    Its linearization at (0,0) is denoted by ℒ and given by({{{, ({{x'= αx + y, (1)}, {y'= -x+αy, (2)}})}})

Find the equilibrium points of the nonlinear system. Find the eigenvalues of the coefficient matrix of the linearized system at the equilibrium points in terms of the parameter α.

Look at the linear system ℒ and assume that the value of the parameter α is -1/4.

Assume that the system satisfies the initial condition x(0) = -10 and y(0) = 20. Use DSolve[..] to find the solution of the corresponding initial value problem, then plot the corresponding trajectory in the xy plane.

For your trajectory in part 2.0.1, draw the graphs of x versus t and y versus t on the same graph.

For your trajectory in part 2.0.1, draw the corresponding graph in the three-dimensional txy-space.

In parts 2.0.1 - 2.0.3, it is important that you choose appropriate ranges for x,y,t and appropriate scales for your plot.

By using Manipulate[...] and StreamPlot[..], do an interactive model of the vector field of the system, with the parameter α  taking values in the closed interval [-2, 2]. The interactive model will show the Phase Portrait of the nonlinear system 𝒮 , the Phase Portrait of the linear system ℒ and the eigenvalues at the equilibrium points.

Analyze the type and stability of the equilibrium points of the nonlinear system 𝒮 for all values of the parameter α.  Use the interactive model to find the values of the parameter α where the qualitative nature of the solutions for the system changes.

Use the Interactive model to find the values of the parameter α for which the system develops limit cycles. Then prove mathematically the existence of the limit cycle by changing the system into polar coordinates x = r cos(θ), y = r sin(θ) and solving it.

Part II: For the second part of the project, browse the literature and find a concrete example from physical sciences which exhibits a Hopf bifurcation. The (non-exhaustive) list of applications given in the introduction should give you many starting points for your investigation. Once you find such an example, describe its mathematical model (the system of differential equations) and explain for what values of the parameters the system undergoes a Hopf bifurcation. No mathematical proof is necessary or required, but StreamPlot pictures are encouraged. Explain the relevance of all quantities modeled by the system of differential equations (for example, x is the debt at time t, y is the investment at time t, etc). Then describe the relevance of the birth of the limit cycle to the practical application.

General considerations:

No previous knowledge of differential equations is assumed is this project, except for the theory developed in the lecture notes posted on our course webpage. Please read the lab notes on Blackboard to find what Mathematica commands are used for solving differential equations and visualizing Phase Portraits. More examples and tutorials about solving differential equations and plotting vector fields can be found in the Mathematica documentation (Help → Wolfram Documentation). You are of course welcome to consult introductory courses in differential equations as well, if needed for the mathematical proofs.