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In Problems 1-6, find all the critical points for the given system, discuss the type and stability of each critical point, and sketch the phase plane diagrams near each of the critical points.
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Fundamentals of Differential Equations and Boundary Value Problems
- Question 2. Solve the problem of time-optimal control to the origin for the system i1 = 2x2, i2 = – -2.x1 + 4u, where |u| < 1.arrow_forwardGraph the following Discrete Dynamical Systems. Explain their long-term behavior. Try to find realistic scenarios that these DDS might explain. 7. a(n + 1) = -1.3 a(n) + 20, a(0) = 9arrow_forwardDenote the owl and wood rat populations at time k by xk Ok Rk and R is the number of rats (in thousands). Suppose Ok and RK satisfy the equations below. Determine the evolution of the dynamical system. (Give a formula for xx.) As time passes, what happens to the sizes of the owl and wood rat populations? The system tends toward what is sometimes called an unstable equilibrium. What might happen to the system if some aspect of the model (such as birth rates or the predation rate) were to change slightly? Ok+ 1 = (0.1)0k + (0.6)RK Rk+1=(-0.15)0k +(1.1)Rk Give a formula for XK- = XK C +0₂ , where k is in months, Ok is the number of owls,arrow_forward
- Find all the critical point(s) of each system given. Then determine the type and stability -3 2 4arrow_forwardIn Example 1 we used Lotka-Volterra equations to model populations of rabbits and wolves. Let's modify those equations as follows: dR 0.09R(1 0.0001R) - 0.002RW dt dW = = -0.02W+0.00001RW dt Find all of the equilibrium solutions. Enter your answer as a list of ordered pairs (R, W), where R is the number of rabbits and W the number of wolves. For example, if you found three equilibri solutions, one with 100 rabbits and 10 wolves, one with 200 rabbits and 20 wolves, and one with 300 rabbits and 30 wolves, you would enter (100, 10), (200, 20), (300, 30). Do not round fractional answers to the nearest integer. Answer = (0,0)arrow_forwardIn Example 1 we used Lotka-Volterra equations to model populations of rabbits and wolves. Let's modify those equations as follows: dR = 0.1R(1 – 0.0001R) – 0.003RW dt dW -0.01W + 0.00004RW dt Find all of the equilibrium solutions. Enter your answer as a list of ordered pairs (R, W), where Ris the number of rabbits and W the number of wolves. For example, if you found three equilibrium solutions, one with 100 rabbits and 10 wolves, one with 200 rabbits and 20 wolves, and one with 300 rabbits and 30 wolves, you would enter (100, 10), (200, 20), (300, 30). Do not round fractional answers to the nearest integer. Answer =|arrow_forward
- 1. Find the critical points and determine their nature for the system x = 2y + xy, y=x+y. Hence sketch a possible phase diagram.arrow_forwardIn Example 1 we used Lotka-Volterra equations to model populations of rabbits and wolves. Let's modify those equations as follows: 0.09R(1 – 0.0001R)5 0.003RW dt MP = -0.01W +0.00001RW dt Find all of the equilibrium solutions. Enter your answer as a list of ordered pairs (R, W), where Ris the number of rabbits and W the number of wolves. For example, ir you found three equilibrium solutions, one with 100 rabbits and 10 wolves, one with 200 rabbits and 20 wolves, and one with 300 rabbits and 30 wolves, you would enter (100, 10), (200, 20), (300, 30). Do not round tractional answers to the nearest integer. Answer (0.0)(10000,0)8(2000,36)arrow_forward5. Solve the following linear system: dX dt with the initial condition = [83] X (0) = -3 2 X Garrow_forward
- Love Affairs (Strogatz, Nonlinear Dynamics and Chaos, 1994) Let R(t) = Romeo's love/hate for Juliet at time t J(t) = Juliet's love/hate for Romeo at time t What happens when romantic opposites get together? A model for their romance is R = aJ j=-bR+aJ, a and b positive e. If 4b = a i. Write the general solution. ii. Classify the origin. iii. Summarize what happens in their relationship. (Hint: Think about the eigenvectors. Be sure to consider all the qualitatively different possibilities.)arrow_forward5. A particular rocket taking off from the Earth's surface uses fuel at a constant rate of 12.5 gallons per minute. The rocket initially contains 225 gallons of fuel. (b) Below is a general sketch of what the graph of your model should look like. Using your calculator, determine the x and y intercepts of this model and label them on the graph at points A and B respectively. (a) Determine a linear model, in y= ax + b form, for the amount of fuel the rocket has remaining, y, as a function of the number of minutes, x. (c) The rocket must still contain 50 gallons of fuel when it hits the stratosphere. What is the maximum number of minutes the rocket can take to hit the stratosphere? Show this point on your graph by also graphing the horizontal line y= 50 and showing the intersection point.arrow_forward13.In the Lotka–Volterra equations, the interaction between the two species is modeled by terms proportional to the product xy of the respective populations. If the prey population is much larger than the predator population, this may overstate the interaction; for example, a predator may hunt only when it is hungry and ignore the prey at other times. In this problem we consider an alternative model proposed by Rosenzweig and MacArthur.10 a.Consider the system x′=x(1−x5−2yx+6),y′=y(−14+xx+6).x′=x1−x5−2yx+6,y′=y−14+xx+6. Find all of the critical points of this system. b.Determine the type and stability characteristics of each critical point.arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning