Ordinary Differential Equations: Homework 1

Transcription

1 Orinary Differential Equations: Homework 1 M. Gameiro, J.-P. Lessar, J.D. Mireles James, K. Mischaikow January 12, 2017

2 2

3 Chapter 1 Motivation 1.1 Exercises Exercise (Frictionless spring) Consier a time varying population of organisms P (t) living in a close environment (possibly a petri ish). The initial population at time t = 0 is P (0) = p 0. The population in the ish at future times is governe by the logistic moel P (t) = kp (t) k K P (t)2, where k > 0 is the reprouctive rate of the species an K > 0 is the carrying capacity of the environment. Solve the equation analytically. What is the limiting value of P (t)? If the initial population in the ish is p 0 = 10, the reprouctive rate is k = 2 an the carrying capacity is K = 1000, at what time t > 0 oes the population ouble? Tripple? Ten times its initial value? Suppose that the initial population is very small compare to the carrying capacity. Derive a simplifie moel of the population growth. Explain what happens to the population size as time goes on (i.e. why was the Reveren Dr. Malthus so worrie after he forgot to inclue the P 2 term in his analysis?) Exercise (Frictionless spring) Recall Hook s law, which escribes the force exerte on a mass attache to the en of a stretche spring F (x) = Kx, where K is the stiffness constant of the spring, an x is the signe isplacement from the equilibrium position of the spring. 3

4 4 CHAPTER 1. MOTIVATION Let the object have mass m > 0. Use Newton s laws to erive the equation of motion for the system consisting of a mass connecte to a spring, moving freely on a frictionless table top (perhaps a surface of ice or perhaps the mass is attache to well oile wheels/bearings). Suppose that the mass is hel motionless at an initial position x(0) = a, an that at time t = 0 the mass is release. Describe the motion of the system for all future time. What is the frequency of the systems oscillation? Suppose that the mass is initially at rest in the equilibrium position x = 0, an that at time t = 0 the system is kicke, imbuing the mass with an initial velocity of x (0) = b. Describe the motion of the system for all future time. What is the frequency of the systems oscillation? Exercise (Spring with friction) Consier the same setup as in the previous problem, but now suppose that we account for friction between the table an the mass (or friction in the wheels). A typical moel of friction is to assume that the force of friction is proportional to the magnitue of the velocity, acting against the irection of motion. (So the faster the object moves the more friction tries to slow it own ). Let the object have mass m > 0. Use Newton s laws to erive the equation of motion for the system consisting of a mass connecte to a spring, moving on a table top, subject to the force of friction escribe above. Suppose that the proportionality coefficient of the friction is C > 0. Suppose that the mass is hel motionless at an initial position x(0) = a, an that at time t = 0 the mass is release. Describe the motion of the system for all future time. Suppose that the mass is initially at rest in the equilibrium position x = 0, an that at time t = 0 the system is kicke, imbuing the mass with an initial velocity of x (0) = b. Describe the motion of the system for all future time. Exercise Derive Kepler s laws from Newton s laws an the law of universal graviton. Consier only two boies. (Try this problems for a few minutes/a few hours before you look up the solution online). Exercise (The penulum) Use Newton s laws to erive the equations of motion for a frictionless penulum. Suppose that the penulum moves close to the surface of the Earth, so that the acceleration of the graational fiel can be assume constant.

5 1.1. EXERCISES 5 Rewrite the equation as a system of first orer ifferential equation, i.e. as a vector fiel. Sketch the vector fiel, first by han, an then using a computer program. What kins of behavior o you expect from the penulum? Compare this to what you see in the vector file rawings. Repeat the problem assuming that the penulum is subject to the force of friction. Suppose that the initial isplacement of the penulum is small compare to the length of the penulum. Simplify the equations of motion accounting this the smallness assumption. Now justify Galileo s observation that the perio of a penulum epens only on the length of the arm (an for example not the angle of maximum isplacement). Exercise (Meet the van er Pol equation) The Van er Pol equation is given by the secon-orer ifferential equation 2 2 x(t) µ(1 x(t)2 ) x(t) + x(t) = 0, an escribes the voltage versus time behavior in a certain electrical circuit consisting of resistors, inuctors, capacitors, a vacuum tube (trioe), an a constant external voltage source. Re-write the system as a vector fiel. Take µ = 1 an (x 0, y 0 ) = ( 1, 1). Use Euler s metho, a han hel calculator, an pen an paper to computer 10 steps on numerical integration with h = 0.1, Write a computer program which can numerically solve the equation (you can use or moify one of mine). Take µ = 0.1. Pick 25 initial conitions in the plane an integrate them for a long time. Describe what happens. Same problem as above but with µ = Now repeat with µ = 0.5,..., µ = 10, taking at least 5 ifferent values of µ. Describe what happens. Exercise (Meet the Lorenz equations) Consier the system of three nonlinear couple first orer orinary ifferential equations x = σ(y x) y = x(ρ z) y z = xy βz

6 6 CHAPTER 1. MOTIVATION These are calle the Lorenz equations, an they were erive by the Meteorologist Ewar Lorenz in the early 1960 s as a simple moel of weather (more precisely as a moel of a a pair of couple convection cells in the atmosphere). In the moel σ, β, ρ > 0 are positive real parameters. Take σ = 10, β = 8/3 an ρ = 0.75, an (x 0, y 0, z 0 ) = (0, 0, 1). Use Euler s metho, a han hel calculator, an pen an paper to computer 10 steps on numerical integration with h = 0.1, Write a program (or use one of mine) which can simulate solutions for the Lorenz equations. Take σ = 10, β = 8/3 an ρ = Choose 25 ifferent initial conitions an integrate them for a long time. Describe what happens. Take σ = 10, β = 8/3 an ρ = 2. Choose 25 ifferent initial conitions an integrate them for a long time. Describe what happens. Repeat this experiment with ρ = 5, 8, 10, 12. Repeat with ρ = 14, 18, 20, 22, 24. Repeat with ρ = 28. This is the so calle classic parameter for the Lorenz system. Explore what happens for ten ifferent values of ρ between ρ = an ρ = Exercise (Strange attractor scavenger hunt) Write computer programs (by moifying the Lorenz coe I gave you) which reprouce all 7 of the strange attractors foun on the posters in the secon floor hall way, outsie the main math office in the FAU Science Builing.

7 Bibliography [1] Carmen Chicone. Orinary ifferential equations with applications, volume 34 of Texts in Applie Mathematics. Springer, New York, secon eition, [2] Lawrence C. Evans. Partial ifferential equations, volume 19 of Grauate Stuies in Mathematics. American Mathematical Society, Provience, RI, secon eition, [3] Geral B. Follan. Introuction to partial ifferential equations. Princeton University Press, Princeton, NJ, secon eition, [4] Jack K. Hale. Orinary ifferential equations. Robert E. Krieger Publishing Co. Inc., Huntington, N.Y., secon eition, [5] James M. Ortega. The Newton-Kantorovich theorem. Amer. Math. Monthly, 75: , [6] Clark Robinson. Dynamical systems. Stuies in Avance Mathematics. CRC Press, Boca Raton, FL, secon eition, Stability, symbolic ynamics, an chaos. 7