Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780813349107
Author: Steven H. Strogatz
Publisher: PERSEUS D
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Chapter 3.4, Problem 12E
Interpretation Introduction
Interpretation:
An example of quad-furcation is to be constructed. We pointed out that the pitchfork bifurcation could be called a trifurcation since three branches of fixed points appear for
Concept Introduction:
Bifurcation theory is used to study the stability of the dynamical systems.
Fixed points are the points where,
The phenomenon in which fixed points are created and destroyed by varying the control parameter is termed as saddle-node bifurcation.
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Chapter 3 Solutions
Nonlinear Dynamics and Chaos
Ch. 3.1 - Prob. 1ECh. 3.1 - Prob. 2ECh. 3.1 - Prob. 3ECh. 3.1 - Prob. 4ECh. 3.1 - Prob. 5ECh. 3.2 - Prob. 1ECh. 3.2 - Prob. 2ECh. 3.2 - Prob. 3ECh. 3.2 - Prob. 4ECh. 3.2 - Prob. 5E
Ch. 3.3 - Prob. 1ECh. 3.3 - Prob. 2ECh. 3.4 - Prob. 1ECh. 3.4 - Prob. 2ECh. 3.4 - Prob. 3ECh. 3.4 - Prob. 4ECh. 3.4 - Prob. 5ECh. 3.4 - Prob. 6ECh. 3.4 - Prob. 7ECh. 3.4 - Prob. 8ECh. 3.4 - Prob. 9ECh. 3.4 - Prob. 10ECh. 3.4 - Prob. 11ECh. 3.4 - Prob. 12ECh. 3.4 - Prob. 13ECh. 3.4 - Prob. 14ECh. 3.4 - Prob. 15ECh. 3.4 - Prob. 16ECh. 3.5 - Prob. 1ECh. 3.5 - Prob. 2ECh. 3.5 - Prob. 3ECh. 3.5 - Prob. 4ECh. 3.5 - Prob. 5ECh. 3.5 - Prob. 6ECh. 3.5 - Prob. 7ECh. 3.5 - Prob. 8ECh. 3.6 - Prob. 1ECh. 3.6 - Prob. 2ECh. 3.6 - Prob. 3ECh. 3.6 - Prob. 4ECh. 3.6 - Prob. 5ECh. 3.6 - Prob. 6ECh. 3.6 - Prob. 7ECh. 3.7 - Prob. 1ECh. 3.7 - Prob. 2ECh. 3.7 - Prob. 3ECh. 3.7 - Prob. 4ECh. 3.7 - Prob. 5ECh. 3.7 - Prob. 6ECh. 3.7 - Prob. 7ECh. 3.7 - Prob. 8E
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- Q7arrow_forwardFind the general solution of the given differential equation. y' + 4x3y = x3 y(x) = ? Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution.arrow_forwardQ2*) Consider the extremisation of the integral I[y] = √²² F(x,y,y', y") dx x1 when y and y' are prescribed only at x = x1. Derive the so-called 'natural boundary conditions' that must be satisfied at x = x2. Taking a specific example: The functional I [y] is defined by I[y] = √² ((y″)² + y) dx with y(0) = 0 and y'(0) = 0. Write down the fourth-order Euler-Lagrange equation for this problem, stating the four boundary conditions. Find the general solution of the Euler-Lagrange equation, and then impose the boundary conditions to find the extremal.arrow_forward
- 3 00 By changing to circular coordinates, evaluate foo √²²+v³ dx dy.arrow_forward3. Z e2 n dz, n = 1, 2,. ..arrow_forwardQ/ By using polar Coordinates show that the system below has a limit cycle and show the stability of + his limit cycle: X² = x + x(x² + y² -1) y* = −x + y (x² + y²-1) -xarrow_forward
- xy Q/Given H (X,Y) = ex-XX+1 be a first integral find the corresponding system and study the Stability of of critical point of this system.arrow_forwardQ/ show that H (X,Y) = x²-4x-x² is 2 first integral of the system Y° = y 0 y° = 2x + x 3 then study the stability of critical point and draw phase portrait.arrow_forwardQ/Given the function H (X,Y) = H (X,Y) = y 2 X2 2 2 ²** 3 as a first integral, find the correspoding for this function and draw the phase portrait-arrow_forward
- Q/ show that the system has alimit cycle and draw phase portrait x = y + x ( 2-x²-y²)/(x² + y²) ½ 2 y = -x+y ( 2-x² - y²) / (x² + y²) ½/2arrow_forwardA sequence X = (xn) is said to be a contractive sequence if there is a constant 0 < C < 1 so that for all n = N. - |Xn+1 − xn| ≤ C|Xn — Xn−1| -arrow_forwardPlease explain this theorem and proofarrow_forward
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