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.
Want to see the full answer?
Check out a sample textbook solutionChapter 12 Solutions
Fundamentals of Differential Equations and Boundary Value Problems
- 3. Discuss the nature and stability of the simple critical points of the system corresponding to the following nonlinear equation +6² +420arrow_forwardQuestion 2. Solve the problem of time-optimal control to the origin for the system i1 = 2x2, i2 = – -2.x1 + 4u, where |u| < 1.arrow_forward5. Solve the following linear system: dX dt with the initial condition = [83] X (0) = -3 2 X Garrow_forward
- Denote 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_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_forwardFind all the critical point(s) of each system given. Then determine the type and stability -3 2 4arrow_forward
- 1. Find steady states of the equation: 2xn (а) хр+1 Xn+1 K (b) Xn+1 where k1, k2 and K are constants. ki + k2/Xnarrow_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_forward1. Find the critical points and determine their nature for the system x = 2y + xy, y=x+y. Hence sketch a possible phase diagram.arrow_forward
- Graph 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_forwardFor each of the phase portraits shown below, give a specific example of the possible general solution for the corresponding 2 x 2linear system, and classify the origin as a type of equilibrium point. Explain your process and answer. (Note: There isn't just one correct answer for each phase portrait. Answers will vary, so make sure you explain your choices.) (a) (b) 0- 大 元 (c)arrow_forwardIn Problem 18, find all the critical points for the given system and discuss the stability of the corresponding equilibrium solutions. Please don't answer in handwritten..thankuarrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning