Ask Question, Ask an Expert


Ask Math Expert

Home >> Math

Problem 1 Assume that f(x) has the following graph:

Consider the (1-dimensional) ODE:
                     x' = f(x):
(a) Find all the fi xed points, and study their stability.

(b) Draw the phase portrait of the system, as well as the graphs of the solutions in all relevant cases.

(c) Study the bifurcations of the ODE

x' = f(x) +α ; αε R -a parameter:

In particular, determine all the bifurcation values of α , and describe the change in behavior before during and after each bifurcation. Make sure to draw the appropriate graphs!

Problem 2 Consider the following matrix:

(a) Find a Jordan form of the matrix, as well as a basis that corresponds to that Jordan form.

(b) Compute the exponential matrix eA.

Problem 3 Consider the following linear system of ODE:

x' = -x - y
y' = x + 3y
z' = 4x + 6y - z

Note that he matrix of this system is exactly the same as in Problem 2.

(a) Study the stability of the fixed point (0,0): is it a source (all solutions diverge to ∞ from it),sink (all solutions converge to it), saddle (any solution is either convergent to the xed point or diverges to ∞), or neither?

(b) Determine stable and unstable subspaces of (0, 0). (The nal answer should be: the stable subspace is spanned by vectors ...., or the stable subspace does not exist.)

(c) Draw a phase portrait of your system in the unstable subspace.

(d) Briefly describe the behavior of solutions to this system. (e.g. "all the solutions except those in xy-plane will go to ∞ while rotating around z-axis; the solutions that start in xy-plane will stay in that plane and will rotate on the circle centered at the xed point (0; 0)

(e) prepare down the general solution of the system above using the initial data

x(0) = x0; y(0) = y0; z(0) = z0:

Problem 4 Consider the following non-linear system 
x' = x2 - ay
y' = y2 - y

(a) Find the xed points of this system. (depending on a, there may be di erent fi xed points!)

(b) Study stability of each xed point via linearization. In the case the linearization is inconclusive, use directions of vector- eld analysis (or any other information contained in the equations) to show stability/instability.

(c) Use the information above to determine the bifurcation values for a, and draw the phase portraits for the system before, at, and after each bifurcation. On phase portraits identify the fixed points as well as their stable/unstable manifolds (curves) where appropriate.

Math, Academics

  • Category:- Math
  • Reference No.:- M9807

Have any Question? 

Related Questions in Math

The graph shows the functions fx px and gxgraph of function

The graph shows the functions f(x), p(x), and g(x): Graph of function g of x is y is equal to 1.8 to the power of x. The straight line f of x joins ordered pairs 6, 2 and 2, minus 2 and is extended on both sides. The str ...

This assignment involves constructing a vision for the

This assignment involves constructing a vision for the future and envisioning the results of its realization. Enabling others to act in realizing the vision is a crucial aspect of leadership. Practice and repetition of t ...

What kind of professionals try to understand who is likely

What kind of professionals try to understand who is likely to be interested in working for their organization and where to find these people? is it recruiting, Supervising, leadership, sourcing?

Select a grade level ccrs related to measurement and

Select a grade level CCRS related to Measurement and Data.  Using the Class Profile, create one lesson plan with instructional activities that: Makes comparisons; Uses models of measuring units; Uses measuring instrument ...

A population of beetles are growing according to a linear

A population of beetles are growing according to a linear growth model. The initial population (week 0) is P0=5, and the population after 9 weeks is P9=104. Find an explicit formula for the beetle population aftern n wee ...

In 450 words with referencesdiscuss why productivity is

In 450 words with references: Discuss why productivity is important, particularly in a healthcare organization. How do staffing models and labor optimization algorithms contribute towards productivity? During project man ...

In this discussion you will solve quadratic equations by

In this discussion, you will solve quadratic equations by two main methods: factoring and using the quadratic formula. Read the following instructions in order and view the example to complete this discussion. Please com ...

Trigonometry and calculusquestions 1-20 select the one best

Trigonometry and Calculus Questions 1-20: Select the one best answer to each question. 1. As a lumberjack, you attach a 50-foot wire from the top of a lean¬ing pine tree to a stake in the ground, forming a 70 degree angl ...

On january 15 1990 anders brought an action against barnes

On January 15, 1990, Anders brought an action against Barnes for breach of a contract for the sale of electronic parts. The breach occurred on January 10, 1986, and Anders discovered the breach on January 20, 1986. The s ...

Let sc be a bargaining game and let td be the image of sc

Let (S,c) be a bargaining game, and let (T,d) be the image of (S,c) under the positive affine transformation x #→ ax + b on R2; that is, T = aS + b and d = ac + b. Prove that the set argmaxy∈T (y 1  - d 1 )(y 2  - d 2 ) ...

  • 4,153,160 Questions Asked
  • 13,132 Experts
  • 2,558,936 Questions Answered

Ask Experts for help!!

Looking for Assignment Help?

Start excelling in your Courses, Get help with Assignment

Write us your full requirement for evaluation and you will receive response within 20 minutes turnaround time.

Ask Now Help with Problems, Get a Best Answer

Section onea in an atwood machine suppose two objects of

SECTION ONE (a) In an Atwood Machine, suppose two objects of unequal mass are hung vertically over a frictionless

Part 1you work in hr for a company that operates a factory

Part 1: You work in HR for a company that operates a factory manufacturing fiberglass. There are several hundred empl

Details on advanced accounting paperthis paper is intended

DETAILS ON ADVANCED ACCOUNTING PAPER This paper is intended for students to apply the theoretical knowledge around ac

Create a provider database and related reports and queries

Create a provider database and related reports and queries to capture contact information for potential PC component pro

Describe what you learned about the impact of economic

Describe what you learned about the impact of economic, social, and demographic trends affecting the US labor environmen