Question

1 and 2

1. Construct a table of the possible linear systems as follows: (a) The first column contains the type of the system (sink, spiral sink, source, if it has a name. (b) The second column contains the condition on the eigenvalues that corresponds (c) The third column contains a small picture of two or more possible phase por- (d) The fourth column contains x(1)-and y(a)-graphs of typical solutions indicated Hint: The most complete table contains 14 cases. Don't forget the double eigen- In Exercises 2-7, we consider the one-parameter families of linear systems depending to this case. traits for this system, and in your phase portraits. value and zero eigenvalue cases.] on the parameter a. Each family therefore determines a curve in the trace-determinant plane. For each family (a) sketch the corresponding curve in the trace-determinant plane; (b) in a brief essay, discuss different types of behaviors exhibited by the system as a increases along the real line (unless otherwise noted); and (c) identify the values of a where the type of the system changes. These are the bifur- cation values of a dt

Answer 1

ANSWER 1):

Construct a table of the possible linear system: (a)Tye Eigenvalues (c) Phase portrait Complex eigenvalues with Spiranegative sink real part i.e Imaginary eigenvalues .e. tbi b+0 Center Two distinct Source positive eigenvalues -2

Two distinct negative eigenvalues Sink -1 -2 Complex eigenvalues with positive Spia real part -3-2 atbi. a

one positive, Saddle and one -32 negative eigenvalue -2 -3 Sink. Repeated coficentnegative matrix is nota egenvalues negativeEgenvectors multiple along one line of I

Sink coefficient negative matrix iseigenvalues. negativeEvery vector multipleis an Repeated egenvector n of the above table, Source, coefficient matrix iS not a Repeated positive eigenvalues. Eigenvectors positivealong one multiple of I -3

Source, Repeated coefficient positive matrix iseigenvalues. positive Every vector multipleis an of I 4 -31 egenvector -2 One eigenvalue is zero, and the other is NA negative. -2 One line of equilibrium points

One eigenvalue s zero, and the other is positive One line of equilibrium NA -2 Repeated eigenvalues. Eigenvectors3 along one line NA Repeated eigenvalues. Every vector Every point is an equilibrium point NA s an egenvector

the wasi is atora ace O 0 Ta and D- a The uue trace- deleminaritplane 3 072 -43 -2-0 and he eienvaluesse

ve egev System Tf a, too positive eigenvaues Suetem s a Sounce (seclion H Snk(seckion A of e cuenue ste IS a Spira Sink (section c o We cuove havea complet ees eeigenv uith postive veal pat, and he system is a. Spiral Sorte (sechion F e ceve) 1aa biucdion point 8 o Cunve tea e Sustem is a Sink with repeaked eipenvalues T a22 , we have a b on e cusve oncation point

sylem is asoote athvepeae eigenvalues inal, aso, ue also haue (pint E on e cve). t s point,te System sa Centen