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Brown, A. (2019). Dynamical Systems and Applications. Wiley.

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Chen, H. (2022). Advanced Topics in Dynamical Systems: Chaos and Fractals. MIT Press.

PROBLEM SET 1.1
Let $x=\left(x_1, x_2, x_3\right)^{\mathrm{T}}=(x, y, z)^{\mathrm{T}}$ and $x(0)=\left(x_0 \cdot y_0, z_0\right)^{\mathrm{T}}$.

(a) $x(t)=x_0 e^t, y(t)=y_0 e^t$, and solution curves lie on the straight lines $y=\left(y_0 / x_0\right) x$ or on the $y$-axis. The phase portrait is given in Problem 3 below with $a=1$.
(b) $x(t)=x_0 e^t \cdot y(t)=y_0 e^{2 t}$, and solution curves. other than those on the $x$ and $y$ axes, lie on the parabolas $y=\left(y_0 / x_0^2\right) x^2$. Cf. Problem 3 below with $\mathrm{a}=2$.
(c) $\mathrm{x}(\mathrm{t})=\mathrm{x}_0 \mathrm{e}^{\mathrm{t}} \cdot \mathrm{y}(\mathrm{t})=\mathrm{y}_0 \mathrm{e}^{3 \mathrm{3}}$, and solution curves lie on the curves $\mathrm{y}=\left(\mathrm{y}_0 / \mathrm{x}_0^3\right) \mathrm{x}^3$.
(d) $\dot{x}=-y, \dot{y}=x$ can be written as $\ddot{y}=\dot{x}=-y$ or $\ddot{y}+y=0$ which has the general solution $y(t)=c_1 \cos t+c_2 \sin t$; thus, $x(t)=\dot{y}(t)=-c_1 \sin t+c_2 \cos t ;$ or in terms of the initial conditions $x(t)=x_0 \cos t-y_0 \sin t$ and $y(t)=x_0 \sin t+y_0 \cos t$. It follows that for all $t \in \mathbf{R}$, $x^2(t)+y^2(t)=x_0^2+y_0^2$ and solution curves lie on these circles. Cf. Figure 4 in Section 1.5.
(e) $y(t)=c_2 \mathrm{e}^{-t}$ and then solving the first-order linear differential equation $\dot{x}+x=c_2 \mathrm{e}^{-t}$ leads to $\mathrm{x}(\mathrm{t})=\mathrm{c}_1 \mathrm{e}^{-1}+\mathrm{c}_2 \mathrm{te}^{-1}$ with $\mathrm{c}_1=\mathrm{x}_0$ and $\mathrm{c}_2=\mathrm{y}_0$. Cf. Figure 2 with $\lambda<0$ in Section 1.5.

(a) $x(t)=x_0 e^t, y(t)=y_0 e^t, z(t)=z_0 e^t$. and $E^u=R^3$.
(b) $x(t)=x_0 \mathrm{e}^{-1}, y(t)=y_0 \mathrm{e}^{-1}, \mathrm{z}(\mathrm{t})=\mathrm{z}_0 \mathrm{e}^{\mathrm{t}}, \mathrm{E}^{\mathrm{s}}=\operatorname{Span}\left{(1,0,0)^{\mathrm{T}},(0,1,0)^{\mathrm{T}}\right}$, and $E^u=\operatorname{Span}{(0,0,1)}$. Cf. Figure 3 with the arrows reversed.
(c) $\mathrm{x}(\mathrm{t})=\mathrm{x}_0 \cos \mathrm{t}-\mathrm{y}_0 \sin \mathrm{t}, \mathrm{y}(\mathrm{t})=\mathrm{x}_0 \sin \mathrm{t}+\mathrm{y}_0 \cos t, \mathrm{z}(\mathrm{t})=\mathrm{z}_0 \mathrm{e}^{-t}$; solution curves lie on the cylinders $\mathrm{x}^2+\mathrm{y}^2=\mathrm{c}^2$ and approach circular periodic orbits in the $\mathrm{x}, \mathrm{y}$ plane as $\mathrm{t} \rightarrow \infty$ : $\mathrm{E}^{\mathrm{c}}=\operatorname{Span}\left{(1,0,0)^{\mathrm{T}},(0,1 \cdot 0)^{\mathrm{T}}\right} \cdot \mathrm{E}^{\mathrm{s}}=\operatorname{Span}\left{(0,0,1)^{\mathrm{T}}\right}$.

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