Introduction to Systems and Phase Plane Analysis

Determine the equations of motion for the two masses described in Problem \({\bf{1}}\)if\({{\bf{m}}_{\bf{1}}}{\bf{ = 1\;kg,}}{{\bf{m}}_{\bf{2}}}{\bf{ = 1\;kg}}\),\({{\bf{k}}_{\bf{1}}}{\bf{ = 3\;N/m}}\), and\({{\bf{k}}_{\bf{2}}}{\bf{ = 2\;N/m}}\).

Four springs with the same spring constant and three equal masses are attached in a straight line on a horizontal frictionless surface as illustrated in Figure\(5.32\). Determine the normal frequencies for the system and describe the three normal modes of vibration.

Two springs, two masses, and a dashpot are attached in a straight line on a horizontal frictionless surface as shown in Figure\(5.33\). The dashpot provides a damping force on mass\({{\bf{m}}_{\bf{2}}}\), given by\({\bf{F =  - by'}}\). Derive the system of differential equations for the displacements \({\bf{x}}\) and\({\bf{y}}\).

Referring to the coupled mass-spring system discussed in Example , suppose an external force $\mathbf{E}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{37}\mathbf{cos}\mathbf{3}\mathbf{t}$  is applied to the second object of mass $\mathbf{1}\mathbf{ }\mathbf{kg}$. The displacement functions $\mathbf{x}\left(\mathbf{t}\right)$ and  $\mathbf{y}\left(\mathbf{t}\right)$ now satisfy the system

$$\begin{gathered} \mathbf{\left(}\mathbf{16}\mathbf{\right)}\mathbf{(}\mathbf{2}\mathbf{x}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{(}\mathbf{t}\mathbf{)}\mathbf{+}\mathbf{6}\mathbf{x}\mathbf{(}\mathbf{t}\mathbf{)}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{(}\mathbf{t}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{,} \\ \mathbf{\left(}\mathbf{17}\mathbf{\right)}\mathbf{(}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{(}\mathbf{t}\mathbf{)}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{(}\mathbf{t}\mathbf{)}\mathbf{-}\mathbf{2}\mathbf{x}\mathbf{(}\mathbf{t}\mathbf{)}\mathbf{=}\mathbf{37}\mathbf{cos}\mathbf{3}\mathbf{t} \end{gathered}$$ 

(a) Show that  $\mathbf{x}\left(\mathbf{t}\right)$ satisfies the equation $\mathbf{\left(}\mathbf{18}\mathbf{\right)}{\mathbf{x}}^{\mathbf{\left(}\mathbf{4}\mathbf{\right)}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{5}\mathbf{x}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{4}\mathbf{x}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{37}\mathbf{cos}\mathbf{3}\mathbf{t}$

(b) Find a general solution $\mathbf{x}\left(\mathbf{t}\right)$ to the equation (18). [Hint: Use undetermined coefficients with ${\mathbf{x}}_{\mathbf{p}}\mathbf{=}\mathbf{Acos}\mathbf{3}\mathbf{t}\mathbf{+}\mathbf{Bsin}\mathbf{3}\mathbf{t}$.]

(c) Substitute $\mathbf{x}\left(\mathbf{t}\right)$  back into (16) to obtain a formula for $\mathbf{y}\left(\mathbf{t}\right)$.

(d) If both masses are displaced 2m to the right of their equilibrium positions and then released, find the displacement functions $\mathbf{x}\left(\mathbf{t}\right)$ and $\mathbf{y}\left(\mathbf{t}\right)$.

Suppose the displacement functions  $\mathbf{x}\left(\mathbf{t}\right)$ and $\mathbf{y}\left(\mathbf{t}\right)$  for a coupled mass-spring system (similar to the one discussed in Problem 6) satisfy the initial value problem

$\begin{array}{rcl}& & \mathbf{x}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{5}\mathbf{x}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{0}\\ & & \mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{-}\mathbf{2}\mathbf{x}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{3}\mathbf{sin}\mathbf{2}\mathbf{t}\\ & & \mathbf{x}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{x}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\\ & & \mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\end{array}$ 

Solve for $\mathbf{x}\left(\mathbf{t}\right)$ and $\mathbf{y}\left(\mathbf{t}\right)$

A double pendulum swinging in a vertical plane under the influence of gravity (see Figure 5.35) satisfies the system

$$\begin{gathered} \left({\mathbf{m}}_{\mathbf{1}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}\right){\mathbf{l}}_{\mathbf{1}}^{\mathbf{2}}{\mathbf{\theta }}_{\mathbf{1}}^{}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}{\mathbf{l}}_{\mathbf{1}}{\mathbf{l}}_{\mathbf{2}}{\mathbf{\theta }}_{\mathbf{2}}^{}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\left({\mathbf{m}}_{\mathbf{1}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}\right){\mathbf{l}}_{\mathbf{1}}{\mathbf{g\theta }}_{\mathbf{1}}\mathbf{=}\mathbf{0} \\ {\mathbf{m}}_{\mathbf{2}}{\mathbf{l}}_{\mathbf{2}}^{\mathbf{2}}{\mathbf{\theta }}_{\mathbf{2}}^{}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}{\mathbf{l}}_{\mathbf{1}}{\mathbf{l}}_{\mathbf{2}}{\mathbf{\theta }}_{\mathbf{1}}^{}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}{\mathbf{l}}_{\mathbf{2}}{\mathbf{g\theta }}_{\mathbf{2}}\mathbf{=}\mathbf{0} \end{gathered}$$ 

When  ${\mathit{\theta }}_{\mathbf{1}}$ and ${\mathit{\theta }}_{\mathbf{2}}$ are small angles. Solve the system when 

${\mathbf{m}}_{\mathbf{1}}\mathbf{=}\mathbf{3}\mathbf{kg}\mathbf{,}{\mathbf{m}}_{\mathbf{2}}\mathbf{=}\mathbf{2}\mathbf{kg}\mathbf{,}{\mathbf{l}}_{\mathbf{1}}\mathbf{=}{\mathbf{l}}_{\mathbf{2}}\mathbf{=}\mathbf{5}\mathbf{m}\mathbf{,}{\mathbf{\theta }}_{\mathbf{1}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{\pi }\mathbf{/}\mathbf{6}\mathbf{,}{\mathbf{\theta }}_{\mathbf{2}}{\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{\theta }}_{\mathbf{1}}^{}{\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{\theta }}_{\mathbf{2}}^{}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$.

The motion of a pair of identical pendulums coupled with a spring is modeled by the system 

${\mathbf{mx}}_{\mathbf{1}}^{}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{=}\mathbf{-}\frac{\mathbf{mg}}{\mathbf{l}}{\mathbf{x}}_{\mathbf{1}}\mathbf{-}\mathbf{k}\left({\mathbf{x}}_{\mathbf{1}}\mathbf{-}{\mathbf{x}}_{\mathbf{2}}\right){\mathbf{,}\mathbf{mx}}_{\mathbf{2}}^{}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{=}\mathbf{-}\frac{\mathbf{mg}}{\mathbf{l}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{k}\left({\mathbf{x}}_{\mathbf{1}}\mathbf{-}{\mathbf{x}}_{\mathbf{2}}\right)$

for small displacements (see Figure 5.36). Determine the two normal frequencies for the system.

Suppose the coupled mass-spring system of Problem  (Figure 5.31) is hung vertically from support (with mass ${\mathbf{m}}_{\mathbf{2}}$ above ${\mathbf{m}}_{\mathbf{1}}$), as in Section 4.10, page 226.

(a) Argue that at equilibrium, the lower spring is stretched a distance ${\mathbf{l}}_1$ from its natural length ${\mathbf{l}}_2$, given by ${\mathbf{l}}_{\mathbf{1}}\mathbf{=}{\mathbf{m}}_{\mathbf{1}}\mathbf{g}\mathbf{/}{\mathbf{k}}_{\mathbf{1}}$.

(b) Argue that at equilibrium, the upper spring is stretched a distance ${\mathbf{l}}_{\mathbf{2}}\mathbf{=}\left({\mathbf{m}}_{\mathbf{1}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}\right)\mathbf{g}\mathbf{/}{\mathbf{k}}_{\mathbf{2}}$.

(c) Show that if ${\mathbf{x}}_{\mathbf{1}}$ and ${\mathbf{x}}_{\mathbf{2}}$ are redefined to be displacements from the equilibrium positions of the masses ${\mathbf{m}}_{\mathbf{1}}$ and ${\mathbf{m}}_{\mathbf{2}}$, then the equations of motion are identical with those derived in Problem 1.

Two springs, two masses, and a dashpot are attached in a straight line on a horizontal frictionless surface as shown in Figure \(5.34\). The system is set in motion by holding the mass \({{\bf{m}}_{\bf{2}}}\) at its equilibrium position and pushing the mass \({{\bf{m}}_1}\) to the left of its equilibrium position a distance \({\bf{2\;m}}\) and then releasing both masses. Determine the equations of motion for the two masses if \({{\bf{m}}_{\bf{1}}}{\bf{ = }}{{\bf{m}}_{\bf{2}}}{\bf{ = 1\;kg,}}{{\bf{k}}_{\bf{1}}}{\bf{ = }}{{\bf{k}}_{\bf{2}}}{\bf{ = 1\;N/m,}}\)and\({\bf{b = 1\;N - sec/m}}\). (Hint: The dashpot damps both \({{\bf{m}}_1}\) and \({{\bf{m}}_{\bf{2}}}\) with a force whose magnitude equals\({\bf{b}}\left| {{\bf{y' - x'}}} \right|\).)

An\({\bf{R L C}}\) series circuit has a voltage source given by \({\bf{E(t) = 20\;V}}\), a resistor of \({\bf{100\Omega }}\), an inductor of \({\bf{4H}}\), and a capacitor of \({\bf{0}}{\bf{.01\;F}}\). If the initial current is zero and the initial charge on the capacitor is \({\bf{4C}}\), determine the current in the circuit for \({\bf{t > 0}}\).

An \({\bf{R L C}}\) series circuit has a voltage source given by \({\bf{E(t) = 40cos2t\;V}}\), a resistor of \({\bf{2\Omega }}\), an inductor of \({\bf{1/4H}}\), and a capacitor of \({\bf{1/13\;F}}\). If the initial current is zero and the initial charge on the capacitor is \({\bf{3}}{\bf{.5C}}\), determine the charge on the capacitor for \({\bf{t > 0}}\).

An \({\bf{R L C}}\) series circuit has a voltage source given by \({\bf{E(t) = 10cos20t\;V}}\), a resistor of \({\bf{120\Omega }}\), an inductor of \({\bf{4H}}\), and a capacitor of \({{\bf{(2200)}}^{{\bf{ - 1}}}}{\bf{\;F}}\). Find the steady-state current (solution) for this circuit. What is the resonance frequency of the circuit?

An \({\bf{L}}{\rm{ }}{\bf{C}}\) series circuit has a voltage source given by\({\bf{E(t) = 30sin50t\;V}}\), an inductor of\({\bf{2H}}\), and a capacitor of \({\bf{0}}{\bf{.02\;F}}\) (but no resistor). What is the current in this circuit for \({\bf{t > 0}}\) if at\({\bf{t = 0, I}}\left( {\bf{0}} \right){\bf{ = q}}\left( {\bf{0}} \right){\bf{ = 0}}\)?

An RLC series circuit has a voltage source of form\({\bf{E(t) = }}{{\bf{E}}_{\bf{o}}}{\bf{cos\gamma t}}\)V, a resistor of 10 Ω, an inductor of 4 H, and a capacitor of 0.01 F. Sketch the frequency response curve for this circuit.

Show that when the voltage source in \(\left( {\bf{4}} \right)\) is of the form\({\bf{E(t) = }}{{\bf{E}}_{\bf{0}}}{\bf{sin}}\gamma {\bf{t}}\), then the steady-state solution \({{\bf{I}}_{\bf{p}}}\) is as given in the equation\(\left( {{\bf{10}}} \right)\).

A mass-spring system with damping consists of a \({\bf{7 - kg}}\) mass, a spring with spring constant \(\frac{{{\bf{3N}}}}{{\bf{m}}}\), a frictional component with damping constant \(2N{\bf{ - }}\frac{{{\bf{sec}}}}{{\bf{m}}}\), and an external force given by \({\bf{f(t) = 10cos10t\;N}}\). Using a \({\bf{10 - \Omega }}\) resistor, construct a \({\bf{R L C}}\) series circuit that is the analog of this mechanical system in the sense that the two systems are governed by the same differential equation.

A mass-spring system with damping consists of a \({\bf{16 - lb}}\) weight, a spring with spring constant\({\bf{64}}\frac{{{\bf{lb}}}}{{{\bf{ft}}}}\), a frictional component with damping constant\({\bf{10lb - }}\frac{{\sec }}{{{\bf{ft}}}}\), and an external force given by\({\bf{f(t) = 20cos8tlb}}\). Using an inductor of\({\bf{0}}{\bf{.01H}}\), construct, and \({\bf{R L C}}\) series circuit that is the analog of this mechanical system.

Because of Euler's formula, \({e^{{\bf{i}}\theta }}{\bf{ = cos}}\theta {\bf{ + isin}}\theta \), it is often convenient to treat the voltage sources \({E_{\bf{0}}}\cos \gamma t\)and \({E_{\bf{0}}}{\bf{sin}}\gamma {\bf{t}}\)simultaneously, use them\(E(t){\bf{ = }}{E_0}{e^{i\gamma t}}\). In this case, the equation \(\left( {\bf{3}} \right)\)becomes\({\bf{L}}\frac{{{{\bf{d}}^{\bf{2}}}{\bf{q}}}}{{{\bf{d}}{{\bf{t}}^{\bf{2}}}}}{\bf{ + R}}\frac{{{\bf{dq}}}}{{{\bf{dt}}}}{\bf{ + }}\frac{{\bf{1}}}{{\bf{C}}}{\bf{q = }}{{\bf{E}}_{\bf{0}}}{e^{i\gamma {\bf{t}}}}\), where \({\bf{q}}\) is now complex (recall\(I{\bf{ = q',I' = }}q''\)).

1. Use the method of undetermined coefficients to show that the steady-state solution to \(\left( {{\bf{22}}} \right)\)is\({q_p}(t) = \frac{{{E_0}}}{{\frac{{\bf{1}}}{{\bf{C}}} - {\gamma ^2}L{\bf{ + }}i\gamma R}}{e^{i\gamma t}}\). The technique is discussed in detail in Project \({\bf{F}}\) of Chapter\({\bf{4}}\), page\({\bf{237}}\).
2. Now show that the steady-state current is\({I_p}(t) = \frac{{{E_0}}}{{R{\bf{ + }}i(\gamma L{\bf{ - }}\frac{1}{{(\gamma C)}})}}{e^{i\gamma t}}\)
3. Use the relation\(\alpha  + i\beta {\bf{ = }}\sqrt {{\alpha ^2}{\bf{ + }}{\beta ^2}} {e^{i\theta }}\), where \(tan\theta {\bf{ = }}\frac{\beta }{\alpha }\), to show that \({I_p}\)can be expressed in the form\({I_p}(t) = \frac{{{E_0}}}{{\sqrt {{R^2}{\bf{ + }}{{(\gamma L{\bf{ - }}\frac{1}{{(\gamma C)}})}^2}} }}{e^{i(\gamma t{\bf{ + }}\theta )}}\), where \(\tan \theta {\bf{ = }}\frac{{\left( {\frac{1}{C} - L{\gamma ^2}} \right)}}{{(\gamma R)}}\).
4. The imaginary part of \({e^{i\gamma t}}\)is \(\sin \gamma t\), so the imaginary part of the solution to \(\left( {{\bf{22}}} \right)\) must be the solution to the equation \(\left( {\bf{3}} \right)\) for \(E(t){\bf{ = }}{E_0}\sin \gamma t\). Verify that this is also the case for the current by showing that the imaginary part of \({I_p}\) in part \(\left( {\bf{c}} \right)\) is the same as that given in the equation \(\left( {{\bf{10}}} \right)\).

Find a system of differential equations and initial conditions for the currents in the networks given in the schematic diagrams (Figures \({\bf{5}}{\bf{.39 - 5}}{\bf{.42}}\) on pages\({\bf{294 - 295}}\)). Assume that all initial currents are zero. Solve for the currents in each branch of the network.

Find a system of differential equations and initial conditions for the currents in the networks given in the schematic diagrams (Figures \({\bf{5}}{\bf{.39 - 5}}{\bf{.42}}\)on pages \({\bf{294 - 295}}\)). Assume that all initial currents are zero. Solve for the currents in each branch of the network.

Find a system of differential equations and initial conditions for the currents in the networks given in the schematic diagrams (Figures \({\bf{5}}.{\bf{39}} - {\bf{5}}.{\bf{42}}\) on pages\({\bf{294}} - {\bf{295}}\)). Assume that all initial currents are zero. Solve for the currents in each branch of the network.

Compute and graph the points of the Poincare map with \({\bf{t = 2\pi n,n = 0,1, \ldots ,20}}\)for equation (1), taking \({\bf{A = F = 1,f = 0,\omega  = }}\frac{3}{2}\). Repeat, taking \({\bf{\omega   = }}\frac{3}{5}\). Do you think the equation has a \({\bf{2\pi }}\)-periodic solution for either choice of \({\bf{\omega }}\)? A subharmonic solution?

Compute and graph the points of the Poincare map with \(t{\bf{ = }}2\pi n,n{\bf{ = }}0,1, \ldots ,20\) for equation\(\left( {\bf{1}} \right)\), taking\(A{\bf{ = F = }}1,\phi {\bf{ = }}0\), and\(\omega {\bf{ = }}\frac{1}{{\sqrt 3 }}\). Describe the limit set for this system.

Compute and graph the points of the Poincare map with\({\bf{t = 2\pi n,n = 0,1, \ldots ,20}}\)for equation (4), taking \({\bf{A = F = 1,f = 0,\omega  = 1,}}\) and\({\bf{b =  - 0}}{\bf{.1}}\). What is happening to these points as\({\bf{n}} \to \infty \)?

Compute and graph the Poincare map with \(t{\bf{ = }}2\pi n,n{\bf{ = }}0,1, \ldots ,20\) for equation\(\left( 4 \right)\), taking\(A{\bf{ = F = }}1,\phi {\bf{ = }}0,\omega {\bf{ = }}1,{\bf{b}} = {\bf{0}}.{\bf{1}}\). Describe the attractor for this system.

Compute and graph the Poincare map with \(t{\bf{ = }}2\pi n,n{\bf{ = }}0,1, \ldots ,20\) for equation\(\left( 4 \right)\), taking \(A{\bf{ = F = }}1,\phi {\bf{ = }}0,\omega {\bf{ = }}\frac{1}{3}\) and\({\bf{b}} = {\bf{0}}.22\). Describe the attractor for this system.

Show that for\({\bf{b > 0}}\), the Poincare map for the equation \(\left( 4 \right)\) is not chaotic by showing that as \({\bf{t}}\) gets large

\(\begin{aligned}{c}{x_n}{\bf{ = }}x(2\pi n) \approx \frac{F}{{\sqrt {{{\left( {{\omega ^2}{\bf{ - }}1} \right)}^2}{\bf{ + }}{b^2}} }}\sin (2\pi n{\bf{ + }}\theta )\\{v_n}{\bf{ = x}}'(2\pi n) \approx \frac{F}{{\sqrt {{{\left( {{\omega ^2}{\bf{ - }}1} \right)}^2}{\bf{ + }}{b^2}} }}\cos (2\pi n{\bf{ + }}\theta )\end{aligned}\)

Independent of the initial values\({{\bf{x}}_{\bf{0}}}{\bf{ = x(0),}}{{\bf{v}}_{\bf{0}}}{\bf{ = x}}'{\bf{(0)}}\).

Show that the Poincare map for equation (1) is not chaoticby showing that if\({\bf{(}}{{\bf{x}}_{\bf{o}}}{\bf{,}}{{\bf{\nu }}_{\bf{o}}}{\bf{)}}\)and\({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{o}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{o}}{\bf{)}}\)are two initial values that define the Poincare maps\({\bf{(}}{{\bf{x}}_{\bf{n}}}{\bf{,}}{{\bf{\nu }}_{\bf{n}}}{\bf{)}}\) and\({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{n}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{n}}{\bf{)}}\), respectively, using the recursive formulas in (3), then one can make the distance between\({\bf{(}}{{\bf{x}}_{\bf{n}}}{\bf{,}}{{\bf{\nu }}_{\bf{n}}}{\bf{)}}\)and\({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{n}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{n}}{\bf{)}}\)small by making the distance between\({\bf{(}}{{\bf{x}}_{\bf{o}}}{\bf{,}}{{\bf{\nu }}_{\bf{o}}}{\bf{)}}\) and \({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{o}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{o}}{\bf{)}}\)small. (Hint: Let \({\bf{(A,}}\phi {\bf{)}}\)and \({\bf{(}}{{\bf{A}}^{\bf{*}}}{\bf{,}}{\phi ^ * }{\bf{)}}\) be the polar coordinates of two points in the plane. From the law of cosines, it follows that the distance d between them is given by\({{\bf{d}}^{\bf{2}}}{\bf{ = (A - }}{{\bf{A}}^{\bf{*}}}{{\bf{)}}^{\bf{2}}}{\bf{ + 2A}}{{\bf{A}}^{\bf{*}}}{\bf{(1 - cos(}}\phi {\bf{ - }}{\phi ^ * }{\bf{))}}\).)

Consider the Poincare maps defined in \(\left( {\bf{3}} \right)\) with \({\bf{\omega  = }}\sqrt {\bf{2}} {\bf{,A = F = 1,}}\) and\(\phi   = 0\). If this map were ever to repeat, then for two distinct positive integers \({\bf{n}}\)and\({\bf{m}}\), \({\bf{sin(2}}\sqrt {\bf{2}} {\bf{\pi n) = sin(2}}\sqrt {\bf{2}} {\bf{\pi m)}}{\bf{.}}\)Using basic properties of the sine function, show that this would imply that \(\sqrt {\bf{2}} \) is rational. It follows from this contradiction that the points of the Poincare map do not repeat.

The doubling modulo \({\bf{1}}\) map defined by the equation \(\left( {\bf{9}} \right)\)exhibits some fascinating behavior. Compute the sequence obtained when

1. \({{\bf{x}}_0}{\bf{ = k / 7}}\)for\({\bf{k = 1,2, \ldots ,6}}\).
2. \({{\bf{x}}_0}{\bf{ = k / 15}}\)for\({\bf{k = 1,2, \ldots ,14}}\).
3. \({{\bf{x}}_{\bf{0}}}{\bf{ = k/}}{{\bf{2}}^{\bf{j}}}\), where \({\bf{j}}\)is a positive integer and \({\bf{k = 1,2, \ldots ,}}{{\bf{2}}^{\bf{j}}}{\bf{ - 1}}{\bf{.}}\)

Numbers of the form \({\bf{k/}}{{\bf{2}}^{\bf{j}}}\) are called dyadic numbers and are dense in \(\left( {{\bf{0,1}}} \right){\bf{.}}\)That is, there is a dyadic number arbitrarily close to any real number (rational or irrational).

Find a system of differential equations and initial conditions for the currents in the networks given in the schematic diagrams (Figures \({\bf{5}}{\bf{.39 - 5}}{\bf{.42}}\) on pages \({\bf{294 - 295}}\)). Assume that all initial currents are zero. Solve for the currents in each branch of the network.

To show that the limit set of the Poincare map given in \(\left( {\bf{3}} \right)\) depends on the initial values, do the following:

(a)  Show that when\({\bf{\omega   = 2or3}}\), the Poincare map consists of the single point\({\bf{(x,v) = }}\left( {{\bf{Asin}}\phi {\bf{ + }}\frac{{\bf{F}}}{{{{\bf{\omega }}^{\bf{2}}}{\bf{ - 1}}}}{\bf{,\omega Acos}}\phi } \right)\).

(b)  Show that when \({\bf{\omega   = }}\frac{{\bf{1}}}{{\bf{2}}}{\bf{,}}\) the Poincare map alternates between the two points\(\left( {\frac{{\bf{F}}}{{{{\bf{\omega }}^{\bf{2}}}{\bf{ - 1}}}}{\bf{ \pm Asin}}\phi {\bf{, \pm \omega Acos}}\phi } \right)\).

(c)  Use the results of parts \(\left( {\bf{a}} \right)\)and \(\left( {\bf{b}} \right)\)to show that when\({\bf{\omega   = 2,3,or}}\frac{1}{2}\), the Poincare map \(\left( {\bf{3}} \right)\)depends on the initial values \(\left( {{{\bf{x}}_{\bf{0}}}{\bf{,}}{{\bf{v}}_{\bf{0}}}} \right){\bf{.}}\)

To show that the limit set for the Poincare map \({{\bf{x}}_{\bf{n}}}{\bf{ = x(2\pi n),}}{{\bf{v}}_{\bf{n}}}{\bf{ = x'(2\pi n)}}\) where x(t) is a solution to equation (6), is an ellipse and that this ellipse is the same for any initial values\({\bf{(}}{{\bf{x}}_{\bf{o}}}{\bf{,}}{{\bf{\nu }}_{\bf{o}}}{\bf{)}}\), do the following:

(a) Argue that since the initial values affect only the transient solution to (6), the limit set for the Poincare map is independent of the initial values.

(b) Now show that for n large\({{\bf{x}}_{\bf{n}}} \approx {\bf{asin(2}}\sqrt {\bf{2}} {\bf{\pi n + }}\psi {\bf{)}},{{\bf{v}}_{\bf{n}}} \approx {\bf{c + }}\sqrt {\bf{2}} {\bf{acos(2}}\sqrt {\bf{2}} {\bf{\pi n + }}\psi {\bf{)}}\),

Where\({\bf{a = (1 + 2(0}}{\bf{.22}}{{\bf{)}}^{\bf{2}}}{{\bf{)}}^{\frac{{{\bf{ - 1}}}}{{\bf{2}}}}}{\bf{,c = (0}}{\bf{.22}}{{\bf{)}}^{{\bf{ - 1}}}}\), and\(\psi {\bf{ = arctan}}\left\{ {{\bf{ - [(0}}{\bf{.22)}}\sqrt {\bf{2}} {{\bf{]}}^{{\bf{ - 1}}}}} \right\}\).

(c) Use the result of part (b) to conclude that the ellipse\({{\bf{x}}^{\bf{2}}}{\bf{ + }}\frac{{{{{\bf{(v - c)}}}^{\bf{2}}}}}{{\bf{2}}}{\bf{ = }}{{\bf{a}}^{\bf{2}}}\)contains the limit set of the Poincare map.

Using a numerical scheme such as Runge–Kutta or a software package, calculate the Poincare map for equation (7) when b = 0.3, \({\bf{\gamma }}\)= 1.2, and F = 0.2. (Notice that the closer you start to the limiting point, the sooner the transient part will die out.) Compare your map with Figure 5.46(a) on page 300. Redo for F = 0.28.

Redo Problem 12 with F = 0.31. What kind of behavior does the solution exhibit?

Redo Problem 12 with F = 0.65. What kind of behavior does the solution exhibit?

Find a general solution\({\bf{x}}\left( {\bf{t}} \right){\bf{, y}}\left( {\bf{t}} \right)\)for the given system.

\(\begin{array}{c}{\bf{x' + y'' + y = 0}}\\{\bf{x'' + y' = 0}}\end{array}\)

Find a general solution \({\bf{x}}\left( {\bf{t}} \right){\bf{, y}}\left( {\bf{t}} \right)\) for the given system.

\(\begin{array}{l}{\bf{x' = x + 2y }}\\{\bf{y' =  - 4x - 3y}}\end{array}\)

Find a general solution \({\bf{x}}\left( {\bf{t}} \right)\),\({\bf{y}}\left( {\bf{t}} \right)\)for the given system.

\(\begin{array}{l}{\bf{2x' - y' = y + 3x + }}{{\bf{e}}^{\bf{t}}}{\bf{,}}\\{\bf{3y' - 4x' = y - 15x + }}{{\bf{e}}^{{\bf{ - t}}}}\end{array}\)

Find a general solution  $\mathbf{x}\left(\mathbf{t}\right)\mathbf{,}\mathbf{y}\left(\mathbf{t}\right)$ for the given system.

 $$\begin{gathered} \mathbf{x}\text{'}\text{'}\mathbf{+}\mathbf{x}\mathbf{-}\mathbf{y}\text{'}\text{'}\mathbf{=}\mathbf{2}{\mathbf{e}}^{\mathbf{-}\mathbf{t}} \\ \mathbf{x}\text{'}\text{'}\mathbf{-}\mathbf{x}\mathbf{+}\mathbf{y}\text{'}\text{'}\mathbf{=}\mathbf{0} \end{gathered}$$

Solve the given initial value problem.

 $$\begin{gathered} \mathit{x}\mathbf{\text{'}}\mathbf{=}\mathit{z}\mathbf{-}\mathit{y}\mathbf{;}\mathbf{ }\mathbf{ }\mathit{x}\left(0\right)\mathbf{=}\mathbf{0} \\ \mathit{y}\mathbf{\text{'}}\mathbf{=}\mathit{z}\mathbf{;}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathit{y}\left(0\right)\mathbf{=}\mathbf{0} \\ \mathit{z}\mathbf{\text{'}}\mathbf{=}\mathit{z}\mathbf{-}\mathit{x}\mathbf{;}\mathbf{ }\mathbf{ }\mathbf{ }\mathit{z}\left(0\right)\mathbf{=}\mathbf{2} \end{gathered}$$

Solve the given initial value problem. 

 $$\begin{gathered} \mathbf{x}\mathbf{\text{'}}\mathbf{=}\mathbf{y}\mathbf{+}\mathbf{z}\mathbf{;}\mathbf{x}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{2} \\ \mathbf{y}\mathbf{\text{'}}\mathbf{=}\mathbf{x}\mathbf{+}\mathbf{z}\mathbf{;}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{2} \\ \mathbf{z}\mathbf{\text{'}}\mathbf{=}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{;}\mathbf{z}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{1} \end{gathered}$$

For the interconnected tanks problem of Section  5.1, page  241 , suppose that instead of pure water being fed into the tank A, a brine solution with concentration $\mathbf{0}\mathbf{.}\mathbf{2}\mathbf{ }\mathbf{kg}\mathbf{/}\mathbf{L}$ is used; all other data remain the same. Determine the mass of salt in each tank at time  $\mathit{t}$ if the initial masses are   and ${\mathbf{y}}_{\mathbf{0}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{3}\mathbf{ }\mathbf{kg}$.

For the interconnected tanks problem of Section \({\bf{5}}{\bf{.1}}\), page \({\bf{241}}\) , suppose that instead of pure water being fed into the tank\({\bf{A}}\), a brine solution with concentration \({\bf{0}}{\bf{.2\;kg/L}}\)is used; all other data remain the same. Determine the mass of salt in each tank at time \({\bf{t}}\) if the initial masses are \({{\bf{x}}_{\bf{0}}}{\bf{ = 0}}{\bf{.1\;kg}}\) and\({{\bf{y}}_{\bf{0}}}{\bf{ = 0}}{\bf{.3\;kg}}\).

write the given higher-order equation or system in an equivalent normal form (compare Section\({\bf{5}}.{\bf{3}}\)).

\(2y'' - ty' + 8y = \sin t\)

Write the given higher-order equation or system in an equivalent normal form (compare Section\({\bf{5}}{\bf{.3}}\)).

\({\bf{3y''' + 2y' - }}{{\bf{e}}^{\bf{t}}}{\bf{y = 5}}\)

Write the given higher-order equation or system in an equivalent normal form (compare Section\({\bf{5}}{\bf{.3}}\)).

\(\begin{array}{c}{\bf{x'' - x + y = 0}}\\{\bf{x' - y + y'' = 0}}\end{array}\)

Write the given higher-order equation or system in an equivalent normal form (compare Section \(5.3\)).

\(\begin{array}{l}{\bf{x''' + y' + y'' = t }}\\{\bf{x'' - x' + y''' = 0}}\end{array}\)

In Problems 12 and 13, solve the related phase plane equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows) and describe the stability of the critical points (i.e., compare with figure 5.12, page 267).

\({\bf{x' = y - 2,y' = 2 - x}}\)

Solve the related phase plane equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows) and describe the stability of the critical points (i.e., compare with Figure \({\bf{5}}{\bf{.12}}\), page \({\bf{267}}\)).

\(\begin{array}{c}{\bf{x' = 4 - 4y}}\\{\bf{y' =  - 4x}}\end{array}\)