3. Consider the system in R² given in polar coordinates byr = r(1 — r²)0 = 1.(a) Find all limit cycles and all equilibria.(b) Determine stability of all limit cycles and equilibria.

Given system of equations:To find:a) The limit cycles and all equilibria.b) Stability of the limit…

Answered: Consider the system in R² given in…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

3. Consider the system in R² given in polar coordinates by
r = r(1 — r²)
0 = 1.
(a) Find all limit cycles and all equilibria.
(b) Determine stability of all limit cycles and equilibria.

Expert Solution

Step 1: Analysis and Introduction

Given system of equations:

$r with dot on top equals r open parentheses 1 minus r squared close parentheses theta with dot on top equals 1$

To find:

a) The limit cycles and all equilibria.

b) Stability of the limit cycle and all equilibria.

Solution process:

Limit cycle can be identified by solving the differential equations.

Stability of the limit cycles can be identified using the limit of solution on r approaches to infinity.

Equilibrium solutions can be identified by equating the derivative to zero.

Stability of the equilibria can be identified by finding the eigenvalues of the Jacobian matrix.

Concept used:

If the eigenvalues are negative, then the node is stable node at equilibria, otherwise unstable.

If the eigenvalues are alternative with one as negative, then the node as saddle points.

If the eigenvalues are non-negative and Lyopunov function exists, then the point is stable node. Otherwise it is unstable.

Lyopunov function exists only if there exists a finite number of limit cycles.

Integration formula:

$table row cell integral fraction numerator f apostrophe open parentheses x close parentheses over denominator f open parentheses x close parentheses end fraction d x end cell equals cell ln open vertical bar f open parentheses x close parentheses close vertical bar plus c end cell row cell integral d x end cell equals cell x plus c end cell end table$

Here, $c$ is an integrating constant.

Differentiation formula:

$fraction numerator d over denominator d x end fraction open parentheses x to the power of n close parentheses equals n x to the power of n minus 1 end exponent$ , for any real n.

Logarithmic law:

$table row cell ln open parentheses m n close parentheses end cell equals cell ln open parentheses m close parentheses plus ln open parentheses n close parentheses end cell row cell ln open parentheses m over n close parentheses end cell equals cell ln open parentheses m close parentheses minus ln open parentheses n close parentheses end cell row cell ln open parentheses m to the power of n close parentheses end cell equals cell n ln open parentheses m close parentheses end cell row cell ln open parentheses e to the power of x close parentheses end cell equals cell e to the power of ln open parentheses x close parentheses end exponent equals x end cell end table$