Unformatted text preview:

MATH 454 Quiz #1 Name: ________________________ Note: Show all of your work and explain what you are doing. For each (a)-(e), find an equation )(xfdtdx=with the stated properties, or if there are no examples, explain why not. (In all cases, assume that f is a smooth function.) (a) Every real number is a fixed point. (b) Every integer is a fixed point, and there are no others. (c) There are precisely three fixed points, and all of them are stable. (d) There are no fixed points. (e) There are precisely 100 fixed points.MATH 454 Quiz #2 Name: ________________________ Note: Show all of your work and explain what you are doing. In parts (a)-(b), let V(x) be the potential, in the sense that .dxdVdtdx−= Sketch the potential as a function of r. Be sure to show all the qualitatively different cases, including bifurcation values of r. (a) (Saddle-node) 2xrdtdx−= (b) (Transcritical) 2 xxrdtdx−=MATH 454 Quiz #3 Name: ________________________ Note: Show all of your work and explain what you are doing. Find the eigenvalues and eigenvectors of the following matrix .310242441⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−=BMATH 454 Quiz #4 Name: ________________________ Note: Show all of your work and explain what you are doing. Find the general solution of the following system of differential equations, .132463220 and RX ,3⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−−−=∈= CCXdtdX You may find the following information useful: • The eigenvalues of the matrix C are λ = 2 with multiplicity 2, and λ = 1 with multiplicity one. • The eigenspace for λ = 1 is spanned by .122⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−• The (genuine) eigenspace for λ = 2 is spanned by .110⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−MATH 454 Quiz #5 Name: ________________________ Note: Show all of your work and explain what you are doing. Plot the phase portrait and classify the fixed point of the following linear system. If the eigenvectors are real, indicate them in your sketch. ⎪⎩⎪⎨⎧−−=+=yxdtdyyxdtdx51725MATH 454 Quiz #6 Name: ________________________ Note: Show all of your work and explain what you are doing. Find the fixed points of the following system, classify them, sketch the neighboring trajectories, and try to fill in the rest of the phase portrait. ⎪⎩⎪⎨⎧−=−=42xdtdyyxdtdxMATH 454 Quiz #7 Name: ________________________ Note: Show all of your work and explain what you are doing. (a) Show that the Duffing equation, 0 322=ε++ xxdtxdhas a nonlinear center at the origin for all ε > 0. (b) If ε < 0, show that all trajectories near the origin are closed. What about trajectories that are far from the origin?MATH 454 Quiz #8 Name: ________________________ Note: Show all of your work and explain what you are doing. Consider the family of linear systems ⎪⎩⎪⎨⎧≡α+α=≡α−α=,),()cos()sin(),()sin()cos(yxgyxdtdyyxfyxdtdxwhere α is a parameter that runs over the range .0π≤α≤Let C be a simple closed curve that does not pass through the origin. (a) Using the formula , 2122∫+−π=CcgfdfgdgfIshow that IC is independent of α.(b) Let C be a circle centered at the origin. Compute IC explicitly by evaluating the integral for any convenient choice of α.MATH 454 Quiz #9 Name: ________________________ Note: Show all of your work and explain what you are doing. Consider the dynamical system ()()⎪⎩⎪⎨⎧+−+=+−−=. 52222yxyyxdtdyyxxyxdtdx (a) Using , yyxxrr&&&+= find the equation for r&in polar coordinates. (b) Determine the circle of maximum radius, r1, centered on the origin, such that all trajectories have a radially outward component on it. (c) Determine the circle of minimum radius, r2, centered on the origin, such that all trajectories have a radially inward component on it. (d) Given that there are no fixed points in the trapping region ,21rrr≤≤ prove that the system has a limit cycle somewhere in this region.MATH 454 Quiz #10 Name: ________________________ Note: Show all of your work and explain what you are doing. Show that the system ⎪⎩⎪⎨⎧−−=−= 33yxdtdyxydtdx has no closed orbits, by constructing a Liapunov function V = a x2 + b y2 with suitable a,b.MATH 454 Quiz #11 Name: ________________________ Note: Show all of your work and explain what you are doing. Odell (1980) considered the system [], )()1( ⎪⎩⎪⎨⎧−=−−=axydtdyyxxxdtdx where x ≥ 0 is the dimensionless population of the prey, y ≥ 0 is the dimensionless population of the predator, and a ≥ 0 is a control parameter. (a) Classify the fixed point (a, a – a2) for a < 1. (b) Show that a Hopf bifurcation occurs at ac = ½. Is it subcritical or supercritical? (c) Estimate the frequency of limit cycle oscillations for a near the


View Full Document

UA MATH 454 - Study Guide in Math

Documents in this Course
Load more
Download Study Guide in Math
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Study Guide in Math and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Study Guide in Math 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?