Homework Answers
Nullcline is the point in the phase where dx/dt = 0 and also it is the point where vector are either straight up or down.
Nullcline is calculated by solving f (x,y) = 0.
So, in attached solution, I have taken two given euqations and calculated the value of x1 and x2 by equating the equations as zero.
i.e, f1 = 0 and f2 = 0.
1. (20 points) Let (a) Determine and plot the equilibrium points and nullclines of the system....
1. (20 points) Let
(a) Determine and plot the equilibrium points and nullclines of
the system.
(b) Show the direction of the vector field between the
nullclines
(c) Sketch some solution curves starting near, but not on, the
equilibrium point(s).
(d) Label each equilibrium point as stable or unstable depending on
the behavior of the
solutions nearby, and describe the long-term behavior of all of the
solutions.
2. Use the nonlinear system below to answer each part. 5.2 x = y-1, y =...
2. Use the nonlinear system below to answer each part. 5.2 x = y-1, y = -y + 1 -2 (a) Plot the nullclines and indicate the direction of the orbits in each region sepa- rated by the nullclines (do not use software, do this by hand). (b) Find all equilibria for the system. (c) Determine the Jacobian matrix for the system and use it to classify the local stability properties of each equilibrium in part (b).
Consider the non-linear system y-y(1-x-y). (a) Find equations for all of the x- and y-nullclines....
Consider the non-linear system y-y(1-x-y). (a) Find equations for all of the x- and y-nullclines. (b) Find the coordinates of each equilibrium point of the system. (c) Sketch the nullclines in the phase plane. Clearly mark the equilibrium points. Also indicate the direction of flow on the nullclines.
Consider the non-linear system y-y(1-x-y). (a) Find equations for all of the x- and y-nullclines. (b) Find the coordinates of each equilibrium point of the system. (c) Sketch the nullclines in the...
2. (28 marks) This questions is about the following system of equations x = (2-x)(y-1) (a) Find all equilibrium solutio...
2. (28 marks) This questions is about the following system of equations x = (2-x)(y-1) (a) Find all equilibrium solutions and determine their type (e.g., spiral source, saddle) Hint: you should find three equilibria. b) For each of the equilibria you found in part (a), draw a phase portrait showing the behaviour of solutions near that equilibrium. -2 (c) Find the nullclines for the system and sketch them on the answer sheet provided. Show the direction of the vector field...
Can somebody please explain to me how to get each arrow of the nullclines step by...
Can somebody please explain to me how to get each arrow of the
nullclines step by step?
dx x(-x-3y150) dt dy=y(-2x - y + 100) dt 5. (a) y 100 150
dx x(-x-3y150) dt dy=y(-2x - y + 100) dt
5. (a) y 100 150
5. For the system =1-1-y = 1 - 12 - y2 In a single figure, show...
5. For the system =1-1-y = 1 - 12 - y2 In a single figure, show the following: (a) Determine and plot the equilibrium points and nullclines. (b) Show the direction of the vector field between the nullclines as illustrated in Example 2 and Figure 2.6.3 in the textbook. (c) Sketch some solution curves starting near, but not on, the equilibrium points. (d) Label each equilibrium as stable or unstable depending on the behavior of the solutions that start nearby.
Consider the system of equations dxdt=x(3−x−4y) dydt=y(1−3x), taking (x,y)>0. (1 point) Consider the system of equations...
Consider the system of equations dxdt=x(3−x−4y) dydt=y(1−3x),
taking (x,y)>0.
(1 point) Consider the system of equations de = 2(3 – 2 – 49) = y(1 - 33), taking (2,y) > 0. (a) Write an equation for the (non-zero) vertical (-)nullcline of this system: (Enter your equation, e.g., y=x.) And for the (non-zero) horizontal (y-)nullcline: (Enter your equation, e.g. y=x.) (Note that there are also nullclines lying along the axes.) (b) What are the equilibrium points for the system? Equilibria =...
Draw a hypothetical plot of how you think the strength of alloys in an eutectic system...
Draw a hypothetical plot of how you think the strength of alloys in an eutectic system changes with composition. What are the principal mechanisms in operation that govern the strength and how it changes?
Consider the following system of equations:
Consider the following system of equations:(a) Suppose a = 1. Sketch the nullclines of these equations. (b) Determine what happens if a is increased to large values (a >> 1). How does this influence the number of steady states? (c) Determine what happens when a is decreased (0 < a < 1). How does this influence the number of steady states? (d) Find a condition necessary for the intersection of the nullclines to occur exactly at the minimum of the x-nullcline.
Find the equilibria points of the following system. dx/dt = (x-1)[r-1-(x-1)^2]. Here r is a real...
Find the equilibria points of the following system. dx/dt = (x-1)[r-1-(x-1)^2]. Here r is a real number. Just need the equilibria points not the individual cases for r to check my solution.
1. (20 points) Let
(a) Determine and plot the equilibrium points and nullclines of
the system.
(b) Show the direction of the vector field between the
nullclines
(c) Sketch some solution curves starting near, but not on, the
equilibrium point(s).
(d) Label each equilibrium point as stable or unstable depending on
the behavior of the
solutions nearby, and describe the long-term behavior of all of the
solutions.
2. Use the nonlinear system below to answer each part. 5.2 x = y-1, y = -y + 1 -2 (a) Plot the nullclines and indicate the direction of the orbits in each region sepa- rated by the nullclines (do not use software, do this by hand). (b) Find all equilibria for the system. (c) Determine the Jacobian matrix for the system and use it to classify the local stability properties of each equilibrium in part (b).
Consider the non-linear system y-y(1-x-y). (a) Find equations for all of the x- and y-nullclines. (b) Find the coordinates of each equilibrium point of the system. (c) Sketch the nullclines in the phase plane. Clearly mark the equilibrium points. Also indicate the direction of flow on the nullclines.
Consider the non-linear system y-y(1-x-y). (a) Find equations for all of the x- and y-nullclines. (b) Find the coordinates of each equilibrium point of the system. (c) Sketch the nullclines in the...
2. (28 marks) This questions is about the following system of equations x = (2-x)(y-1) (a) Find all equilibrium solutions and determine their type (e.g., spiral source, saddle) Hint: you should find three equilibria. b) For each of the equilibria you found in part (a), draw a phase portrait showing the behaviour of solutions near that equilibrium. -2 (c) Find the nullclines for the system and sketch them on the answer sheet provided. Show the direction of the vector field...
Can somebody please explain to me how to get each arrow of the
nullclines step by step?
dx x(-x-3y150) dt dy=y(-2x - y + 100) dt 5. (a) y 100 150
dx x(-x-3y150) dt dy=y(-2x - y + 100) dt
5. (a) y 100 150
5. For the system =1-1-y = 1 - 12 - y2 In a single figure, show the following: (a) Determine and plot the equilibrium points and nullclines. (b) Show the direction of the vector field between the nullclines as illustrated in Example 2 and Figure 2.6.3 in the textbook. (c) Sketch some solution curves starting near, but not on, the equilibrium points. (d) Label each equilibrium as stable or unstable depending on the behavior of the solutions that start nearby.
Consider the system of equations dxdt=x(3−x−4y) dydt=y(1−3x),
taking (x,y)>0.
(1 point) Consider the system of equations de = 2(3 – 2 – 49) = y(1 - 33), taking (2,y) > 0. (a) Write an equation for the (non-zero) vertical (-)nullcline of this system: (Enter your equation, e.g., y=x.) And for the (non-zero) horizontal (y-)nullcline: (Enter your equation, e.g. y=x.) (Note that there are also nullclines lying along the axes.) (b) What are the equilibrium points for the system? Equilibria =...
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