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5. Consider the system r' = r - ;= sin @+a a. For which values of...
Converting to linear system for three different cases: ii) y 0 For each cases provide general solutions, the phase portrait and the value of gamma at which there is a bifurcation, Converting...
Converting to linear system for three different cases: ii) y 0 For each cases provide general solutions, the phase portrait and the value of gamma at which there is a bifurcation,
Converting to linear system for three different cases: ii) y 0 For each cases provide general solutions, the phase portrait and the value of gamma at which there is a bifurcation,
Consider the following system: dx/dt=y(x^2+y^2-1) dy/dt= -x(x^2 +y^2-1) Find the equilibrium solution. 13. Consider the following system dx dy (e) Find the equilibrium solutions (0 Use Maple to sk...
Consider the following system:
dx/dt=y(x^2+y^2-1)
dy/dt= -x(x^2 +y^2-1)
Find the equilibrium solution.
13. Consider the following system dx dy (e) Find the equilibrium solutions (0 Use Maple to sketch a phase portrait (me to understand the qualitative behavior of
13. Consider the following system dx dy (e) Find the equilibrium solutions (0 Use Maple to sketch a phase portrait (me to understand the qualitative behavior of
Consider the nonlinear System of differential equations di dt dt (a) Determine all critical points of the system (b) For each critical point with nonnegative x value (20) i. Determine the linearised...
Consider the nonlinear System of differential equations di dt dt (a) Determine all critical points of the system (b) For each critical point with nonnegative x value (20) i. Determine the linearised system and discuss whether it can be used to approximate the ii. For each critical point where the approximation is valid, determine the general solution of iii. Sketch by hand the phase portrait of each linearised system where the approximation behaviour of the non-linear system the linearised system...
Consider the homogeneous system 0 ' = Až = a) T, a ER -4 a (a)...
Consider the homogeneous system 0 ' = Až = a) T, a ER -4 a (a) Determine the eigenvalues of A in terms of a. (if any) where the qualitative nature of phase portrait for (b) Identify the values of a system changes. (c) For each value of a that you listed in part (b), sketch the phase portrait for a value of a that is slightly less than the value(s) that you identified. For example, if you identified only...
4. Below is a piecewise function, determine -5 lim,f(x)= c, lim f(x)= e. lim f(x)- d. y = sin x (not drawn to scale) explain Consider the piece of f(x) in the first quadrant resembling e. Determi...
4. Below is a piecewise function, determine -5 lim,f(x)= c, lim f(x)= e. lim f(x)- d. y = sin x (not drawn to scale) explain Consider the piece of f(x) in the first quadrant resembling e. Determine lim sinand the behavior of the graph near zero. 5. Using your graphing calculator sketch h(x)-x4-2x3 over [-2,2] below, find the critical values and on the graph label the coordinates of any local, global(absolute) minima, maxima or point of inflection on the sketch,...
1. The populations of two competing species x(t) and y(t) are governed by the non-linear system...
1. The populations of two competing species x(t) and y(t) are governed by the non-linear system of differential equations dx dt 10x – x2 – 2xy, dy dt 5Y – 3y2 + xy. (a) Determine all of the critical points for the population model. (b) Determine the linearised system for each critical point in part (a) and discuss whether it can be used to approximate the behaviour of the non-linear system. (c) For the critical point at the origin: (i)...
2. Consider the systems: dz =2x(1-5)-yr dt dy dt a) Which system corresponds to a predator-prey o...
2. Consider the systems: dz =2x(1-5)-yr dt dy dt a) Which system corresponds to a predator-prey one? Which is the predator and which the prey? Briefly justify your answer. b) Find the equilibrium solutions only for the predator-prey one. c) Sketch its phase plane showing the equilibrium solutions and the behavior on the r- and y-axis (only for the predator-prey one) d) Describe briefly what kind of situation could the other system represent.
2. Consider the systems: dz =2x(1-5)-yr dt...
Consider the system * =y j= +1. Find the fixed points and the linearisation of the...
Consider the system * =y j= +1. Find the fixed points and the linearisation of the system at each. Identify the type and sketch a local phase portrait (i.e. a sketch of the orbits just around the fixed points) at each fixed point. Show that the system has a time reversal symmetry. Draw a sketch showing isoclines and the directions of orbits in all parts of phase space. Use this information, together with the symmetry, to show there exists a...
1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each...
1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each of the regions b. Linearize the system around the equilibrium point, and use your result to classify the type of the c. Use the information from parts a and b to sketch the phase portrait of the system. 2. Sketch the phase portraits for the following systems...
1. Consider the system 2(t)--3i(t) +z2(t) +3() (a) (i) Find the linearised system at the equilibrium point (0, 0)....
1. Consider the system 2(t)--3i(t) +z2(t) +3() (a) (i) Find the linearised system at the equilibrium point (0, 0). (ii) What type of equilibrium point is (0,0)? (State your reasons fully.) (ii) Sketch the phase portrait for the linearised system near (0,0). (b) Repeat part (a) for the equilibrium point at (1,0). (c) (i) Are there any other equilibria? (ii) Read the Grobman-Hartman theorem and confirm that it applies to the above equilibria.
1. Consider the system 2(t)--3i(t) +z2(t) +3()...
Converting to linear system for three different cases: ii) y 0 For each cases provide general solutions, the phase portrait and the value of gamma at which there is a bifurcation,
Converting to linear system for three different cases: ii) y 0 For each cases provide general solutions, the phase portrait and the value of gamma at which there is a bifurcation,
Consider the following system:
dx/dt=y(x^2+y^2-1)
dy/dt= -x(x^2 +y^2-1)
Find the equilibrium solution.
13. Consider the following system dx dy (e) Find the equilibrium solutions (0 Use Maple to sketch a phase portrait (me to understand the qualitative behavior of
13. Consider the following system dx dy (e) Find the equilibrium solutions (0 Use Maple to sketch a phase portrait (me to understand the qualitative behavior of
Consider the nonlinear System of differential equations di dt dt (a) Determine all critical points of the system (b) For each critical point with nonnegative x value (20) i. Determine the linearised system and discuss whether it can be used to approximate the ii. For each critical point where the approximation is valid, determine the general solution of iii. Sketch by hand the phase portrait of each linearised system where the approximation behaviour of the non-linear system the linearised system...
Consider the homogeneous system 0 ' = Až = a) T, a ER -4 a (a) Determine the eigenvalues of A in terms of a. (if any) where the qualitative nature of phase portrait for (b) Identify the values of a system changes. (c) For each value of a that you listed in part (b), sketch the phase portrait for a value of a that is slightly less than the value(s) that you identified. For example, if you identified only...
4. Below is a piecewise function, determine -5 lim,f(x)= c, lim f(x)= e. lim f(x)- d. y = sin x (not drawn to scale) explain Consider the piece of f(x) in the first quadrant resembling e. Determine lim sinand the behavior of the graph near zero. 5. Using your graphing calculator sketch h(x)-x4-2x3 over [-2,2] below, find the critical values and on the graph label the coordinates of any local, global(absolute) minima, maxima or point of inflection on the sketch,...
1. The populations of two competing species x(t) and y(t) are governed by the non-linear system of differential equations dx dt 10x – x2 – 2xy, dy dt 5Y – 3y2 + xy. (a) Determine all of the critical points for the population model. (b) Determine the linearised system for each critical point in part (a) and discuss whether it can be used to approximate the behaviour of the non-linear system. (c) For the critical point at the origin: (i)...
2. Consider the systems: dz =2x(1-5)-yr dt dy dt a) Which system corresponds to a predator-prey one? Which is the predator and which the prey? Briefly justify your answer. b) Find the equilibrium solutions only for the predator-prey one. c) Sketch its phase plane showing the equilibrium solutions and the behavior on the r- and y-axis (only for the predator-prey one) d) Describe briefly what kind of situation could the other system represent.
2. Consider the systems: dz =2x(1-5)-yr dt...
Consider the system * =y j= +1. Find the fixed points and the linearisation of the system at each. Identify the type and sketch a local phase portrait (i.e. a sketch of the orbits just around the fixed points) at each fixed point. Show that the system has a time reversal symmetry. Draw a sketch showing isoclines and the directions of orbits in all parts of phase space. Use this information, together with the symmetry, to show there exists a...
1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each of the regions b. Linearize the system around the equilibrium point, and use your result to classify the type of the c. Use the information from parts a and b to sketch the phase portrait of the system. 2. Sketch the phase portraits for the following systems...
1. Consider the system 2(t)--3i(t) +z2(t) +3() (a) (i) Find the linearised system at the equilibrium point (0, 0). (ii) What type of equilibrium point is (0,0)? (State your reasons fully.) (ii) Sketch the phase portrait for the linearised system near (0,0). (b) Repeat part (a) for the equilibrium point at (1,0). (c) (i) Are there any other equilibria? (ii) Read the Grobman-Hartman theorem and confirm that it applies to the above equilibria.
1. Consider the system 2(t)--3i(t) +z2(t) +3()...
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