Student Solutions Manual for Nonlinear Dynamics and Chaos, 2nd edition

Mitchal Dichter

• Éditeur: CRC Press
• Année: 2018
• ISBN:
  9780429972638
• Édition: 1
• Sujet: Mathematics
• Sous-sujet: Mathematics General

This official Student Solutions Manual includes solutions to the odd-numbered exercises featured in the second edition of Steven Strogatz's classic text Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. The textbook and accompanying Student Solutions Manual are aimed at newcomers to nonlinear dynamics and chaos, especially students taking a first course in the subject. Complete with graphs and worked-out solutions, this manual demonstrates techniques for students to analyze differential equations, bifurcations, chaos, fractals, and other subjects Strogatz explores in his popular book.

7

Limit Cycles

7.1 Examples

7.1.1

$\dot{r}=r^3-4r\dot{\theta }=1$

7.1.3

$\dot{r}=r\left(1-r^2\right)\left(4-r^2\right)\dot{\theta }=2-r^2$

7.1.5

$\dot{r}=r\left(1-r^2\right)\dot{\theta }=1$

$\begin{array}{ll}\dot{x}\hfill & =\frac{\text{d}}{\text{d}t}\left(r\cos \left(\theta \right)\right)\hfill \\ \hfill & =\dot{r}\cos \left(\theta \right)-r\sin \left(\theta \right)\dot{\theta }\hfill \\ \hfill & =r\left(1-r^2\right)\cos \left(\theta \right)-r\sin \left(\theta \right)\hfill \\ \hfill & =\left(1-r^2\right)r\cos \left(\theta \right)-r\sin \left(\theta \right)\hfill \\ \hfill & =\left(1-x^2-y^2\right)x-y\hfill \\ \hfill & =x-y-x\left(x^2+y^2\right)\hfill \end{array}$

$\begin{array}{ll}\dot{y}\hfill & =\frac{\text{d}}{\text{d}t}\left(r\sin \left(\theta \right)\right)\hfill \\ \hfill & =\dot{r}\sin \left(\theta \right)+r\cos \left(\theta \right)\dot{\theta }\hfill \\ \hfill & =r\left(1-r^2\right)\sin \left(\theta \right)+r\cos \left(\theta \right)\hfill \\ \hfill & =\left(1-r^2\right)r\sin \left(\theta \right)+r\cos \left(\theta \right)\hfill \\ \hfill & =\left(1-x^2-y^2\right)y+x\hfill \\ \hfill & =x+y-y\left(x^2+y^2\right)\hfill \end{array}$

7.1.7

$\dot{r}=r\left(4-r^2\right)\dot{\theta }=1$

The initial condition of (r, θ) = (0.1, 0) in Cartesian coordinat...