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1. Discuss lyapunov function theory and how it can be used to prove global assmptotic stability of solutions.(Give an example form natural and engineering sciences.)
--- Draw level sets of Lyapunov function.
2. Discuss Perron-Frobenius theorem and how it can be used to prove global stability of solutions(Give an example) .
3. Discuss the stability of solutions of non-autonomous system dx/dt=f(x,t).
4. Discuss bifurcation theory, and show how to (theoretically) characterise varies, bifurcation types apriori (saddle node, pitchfork, hopf, transcritical). Show to charactrise super and sub-critical pitchfork and hopf.(Give an example of each)
1) transcritical.
2)pitchfork(super and sub-critical)
3)Hopf(super and sub-critical)
4)saddle-node
Prove that if f and g are functions, then f intersect g is a function by showing f intersect g = glA A={x:g(x)=f(x)}
A parcel of land, value $250,000 is sold to an investor who signs a contract agreeing to pay a deposit of $25,000 followed by equal quarterly payments for as long as necessary, wit
Parametric objective-function problems
Find out the linear approximation for at x =8 . Utilizes the linear approximation to approximate the value of and Solution Since it is just the tangent line there
Write down the system of differential equations for the population of both predators and prey by using the assumptions above. Solution We will start off through letting that
Consider the following linear programming problem: Min (12x 1 +18x 2 ) X 1 + 2x 2 ≤ 40 X 1 ≤ 50 X 1 + X 2 = 40 X
Let's take a look at one more example to ensure that we've got all the ideas about limits down that we've looked at in the last couple of sections. Example: Given the below gr
The frequency of oscillation of an object suspended on a spring depends on the stiffness k of the spring (called the spring constant) and the mass m of the object. If the spring is
Steps for Alternating Series Test Suppose that we have a series ∑a n and either a n = (-1) n b n or a n = (-1) n+1 b n where b n > 0 for all n. Then if, 1.
Urn A contains 1 white,2 black and 3 red balls;Urn B contains 2 white,1 black and 1 red balls;and Urn C contains 4 white,5 black and 3 red balls.One urn is chosen at random and two
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