Consider the nonlinear planar systemx'=3y-6y'=3x-x²a. Find all of its stationary solutions.b. Find a nonconstant function H(x,y) such that every trajectory of the systemsatisfies H(x,y) =c for some constant c.c. Classify and give the stability of each stationary solution.d. Sketch the phase plane portrait of the system near each critical point. Marksketched solutions with arrows to indicate direction of motion.

In mathematics and science, a system can be a system within which the change of the output isn't…

Answered: Consider the nonlinear planar system…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

5. Consider the nonlinear system. Sx = -2y + xy ly = x + 4xy (a) Find all the equilibrium points of the nonlinear system, then linearize and classify the linearized systems. (b) Find the solution trajectories as implicit functions of y and x. Use Desmos to graph a bunch of them for different initial conditions. What kind of equilibrium point does the nonlinear system have at (0,0)?
2. For the following coupled system x=y y = x(1-x²)+y find all equilibrium points and determine the nature of the linear system around each of the equilibrium points. Draw the null-clines, directions of the trajectories on each null-clines and hence draw the approximate phase plots the the system.
1)Find the x- and y-Nullclines of the system and sketch them.How many equilibria are there? Where are they located? b)The Nullclines separate the plane into regions where solutionsare generally moving in the same direction. For each region in your plot,indicate the general direction that solutions are moving (e.g. “Northeast”or “Down-and-right”).
3. Consider the nonlinear system S a' = y y = 4 – a2 + 2y %3D a) Find all critical points. b) For each critical point find the linearized system and determine its type and stability. Then determine type and stability of each critical point for the nonlinear system. c) Sketch the phase portrait near each critical point. Carefully mark sketched solutions with arrows.
For each of the nonlinear systems below, use technology to draw phase portraits (attach to the solution), determine equillibrium solutions and visually estimate whether they are stable or unstable. (a) x' = In(y + 2) x² + y (b) I' y' = y-x² - 8y² X-
Consider the system of equations dx dt dy dt a (1-33 = x 1 - y) - Y Y y (1 - 12/1 - x), 4 taking (x, y) > 0. (a) Write an equation for the (non-zero) vertical (x-)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 = (Enter the points as comma-separated (x,y) pairs, e.g., (1,2), (3,4).) (c) Use your nullclines to estimate trajectories in the phase plane, completing the following sentence: If we start at the initial position (2, 3), trajectories ? the point (Enter the point as an (x,y) pair, e.g., (1,2).)
Consider the following system of equations. dx dt dy =y y=x-x²-y dt (a) Find the equilibrium points of the system. smallest x-value (x, y) = largest x-value (x, y) = (x, y) = 5 (b) Using technology, as directed by your instructor, sketch the direction field and phase portrait of the system. (c) Briefly describe the behavior of typical solutions, As t increases, typical solutions spiral in the Select direction Select one of the outside two solutions depending on the initial condition.
8. For the system below, find all the critical points, find the linearization at each crit- ical point and classify it. Also indicate if they are stable, asymptotically stable, or unstable. a' = (0.6 – 0.01y)x, y = (0.002r – 0.4)y
C. Consider the nonlinear system x' = sin(x) cos(y) y = – cos(x) sin(y) () T/2 i) Verify that and are equilibrium solutions. T/2 ii) Linearize the system at the equilibrium points in part (i). Determine whether the equilibrium point is hyperbolic or not. If it is hyperbolic, deter- mine the whether the equilibrium is a source, sink or saddle. iii) Plot the vector field using a vector field plotter.
3 please

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS