**3. Periodic solution of typical Hamiltonian systems**

### **3.1 Nonlinear simple pendulum**

The simple pendulum is arguably the most investigated physical system and provides very interesting insights into nonlinear phenomena. Butikov [57] calls it "an antique but evergreen physical model." The undamped oscillation of a simple pendulum is a Hamiltonian system governed by the well-known nonlinear ODE as shown:

$$
\ddot{u} + \Omega\_0^2 \sin u = 0,\tag{13}
$$

are determined using the expression for *u*\_, and all the other constants for the solution of each discretization are determined based on *Krs*. A plot of the frequencyamplitude response for the simple pendulum when Ω<sup>0</sup> ¼ 1*:*0 is given in **Figure 2a** and the corresponding error of the CPLM solution in comparison with the exact solution (Eq. (15)) is shown in **Figure 2b**. We see that for *A* ≤ 178°, the maximum error in the CPLM estimate is less than 0.45% for *n* ¼ 50 and 0.14% for *n* ¼ 100. Also, the oscillation history for moderate-amplitude (*A* ¼ 45°) and large-amplitude (*A* ¼ 135°) is shown in **Figure 3** and an excellent agreement between the CPLM solution for *n* ¼ 100 and the exact solution (Eq. (14)) is observed. We noted that trigonometric nonlinearity is usually difficult to deal with and that is why the CPLM shows a relatively slow convergence to accurate results. Hence, many discretizations (e.g., *n* ¼ 50 � 100) are required to get an accuracy that is within 1.0% of the exact solution during large-amplitude oscillations (90°< *A* < 180°) of

*Oscillation history of the simple pendulum for (a) A* ¼ *45*° *and (b) A* ¼ *135*°*. CPLM estimate—Lines; exact*

*(a) Frequency-amplitude response of the simple pendulum for 0*°< *A* <*180*°*. (b) CPLM error analysis.*

*Periodic Solution of Nonlinear Conservative Systems DOI: http://dx.doi.org/10.5772/intechopen.90282*

Consider the motion of a satellite along a path that is equidistant from two identical massive stars with mutually interacting gravitational fields. If the distance between the two stars is 2*d* and the coordinate of the satellite motion is *u*, then the

2*Mu*

*<sup>d</sup>*<sup>2</sup> <sup>þ</sup> *<sup>u</sup>*<sup>2</sup> <sup>3</sup>*=*<sup>2</sup> <sup>¼</sup> 0, (17)

the simple pendulum.

**243**

**Figure 2.**

**Figure 3.**

*solution—Markers.*

**3.2 Motion of satellite equidistant from twin stars**

equation of motion of the satellite is given as [3]:

*u*€ þ

where *<sup>u</sup>* is the angular displacement, <sup>Ω</sup><sup>0</sup> <sup>¼</sup> ffiffiffiffiffiffi *g=l* p , *l* is the length of the pendulum, and *g* ¼ 9*:*8 *m=s* 2. The initial conditions are given as *<sup>u</sup>*ð Þ¼ <sup>0</sup> *<sup>A</sup>* and *<sup>u</sup>*\_ð Þ¼ <sup>0</sup> 0. The same initial conditions are applicable to all other oscillators discussed subsequently. The exact solution to Eq. (13) is expressed in terms of elliptic functions. The displacement and natural frequency are given as [58]:

$$u\_{\rm ex}(t) = 2\sin^{-1}\left[k\text{sn}\left(\Omega\_0 t + K\left(k^2\right); k\right)\right] \tag{14}$$

$$
\omega\_{\rm ex} = \frac{\pi \Omega\_0}{2K(k^2)},
\tag{15}
$$

Where sn is the Jacobi elliptic sine function, *<sup>k</sup>* <sup>¼</sup> sin ð Þ *<sup>A</sup>=*<sup>2</sup> , and *K k*<sup>2</sup> � � is the complete elliptic integral of the first kind given as:

$$K(k^2) = \int\_0^{\pi/2} \frac{1}{\sqrt{1 - k^2 \sin^2 \phi}} d\phi. \tag{16}$$

From Eq. (13), the restoring force for the pendulum is *f u*ð Þ¼ <sup>Ω</sup><sup>2</sup> <sup>0</sup> sin *u* and looks like the plot in **Figure 1**. This means that *Krs* <sup>¼</sup> <sup>Ω</sup><sup>2</sup> <sup>0</sup>ð Þ sin *us* � sin *ur =*ð Þ *us* � *ur* and *u*\_ ¼ �Ω<sup>0</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cos ð Þ *<sup>u</sup>* � cos *<sup>A</sup>* <sup>p</sup> . The initial and final velocities for each discretization

*Periodic Solution of Nonlinear Conservative Systems DOI: http://dx.doi.org/10.5772/intechopen.90282*

**Figure 2.**

transformation only requires a simple algebraic manipulation as demonstrated

(displacement and velocity) and can be derived exactly in closed-form for all conservative systems. The CPLM extends this bilateral relationship into a tripartite one by finding the value of the corresponding independent variable (i.e., time) that matches the values of the state variables in each discretization.

7.The phase equation gives the relationship between the state variables

8.For few discretization, say *n* ≤20, it would be necessary to extract values within each discretization in order to obtain a smooth plot of the oscillation history. The values can be extracted using the approximate solution of the displacement. However, for many discretizations, say *n*≥ 50, there is no need

9.The usual initial conditions investigated for nonlinear conservative oscillators are nonzero displacement and zero velocity. However, the CPLM algorithm can comfortably handle nonzero initial conditions for displacement and velocity.

The simple pendulum is arguably the most investigated physical system and provides very interesting insights into nonlinear phenomena. Butikov [57] calls it "an antique but evergreen physical model." The undamped oscillation of a simple pendulum is a Hamiltonian system governed by the well-known nonlinear ODE as

same initial conditions are applicable to all other oscillators discussed subsequently. The exact solution to Eq. (13) is expressed in terms of elliptic functions. The

*<sup>ω</sup>ex* <sup>¼</sup> *<sup>π</sup>*Ω<sup>0</sup>

Where sn is the Jacobi elliptic sine function, *<sup>k</sup>* <sup>¼</sup> sin ð Þ *<sup>A</sup>=*<sup>2</sup> , and *K k*<sup>2</sup> � � is the

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>k</sup>*<sup>2</sup> sin <sup>2</sup>

2 cos ð Þ *<sup>u</sup>* � cos *<sup>A</sup>* <sup>p</sup> . The initial and final velocities for each discretization

ð*<sup>π</sup>=*<sup>2</sup> 0

From Eq. (13), the restoring force for the pendulum is *f u*ð Þ¼ <sup>Ω</sup><sup>2</sup>

2. The initial conditions are given as *<sup>u</sup>*ð Þ¼ <sup>0</sup> *<sup>A</sup>* and *<sup>u</sup>*\_ð Þ¼ <sup>0</sup> 0. The

*uex*ðÞ¼ *<sup>t</sup>* 2 sin �<sup>1</sup> *<sup>k</sup>*sn <sup>Ω</sup>0*<sup>t</sup>* <sup>þ</sup> *K k*<sup>2</sup> � �; *<sup>k</sup>* � � � � (14)

*ϕ*

<sup>0</sup> sin *u* ¼ 0, (13)

<sup>2</sup>*K k*<sup>2</sup> � � , (15)

<sup>q</sup> *<sup>d</sup>ϕ:* (16)

<sup>0</sup>ð Þ sin *us* � sin *ur =*ð Þ *us* � *ur* and

<sup>0</sup> sin *u* and looks

*g=l* p , *l* is the length of the pendulum,

*<sup>u</sup>*€ <sup>þ</sup> <sup>Ω</sup><sup>2</sup>

to extract values from any discretization.

**3.1 Nonlinear simple pendulum**

shown:

and *g* ¼ 9*:*8 *m=s*

*u*\_ ¼ �Ω<sup>0</sup>

**242**

**3. Periodic solution of typical Hamiltonian systems**

where *<sup>u</sup>* is the angular displacement, <sup>Ω</sup><sup>0</sup> <sup>¼</sup> ffiffiffiffiffiffi

displacement and natural frequency are given as [58]:

complete elliptic integral of the first kind given as:

like the plot in **Figure 1**. This means that *Krs* <sup>¼</sup> <sup>Ω</sup><sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*K k*<sup>2</sup> � � <sup>¼</sup>

in Section 4.

*Progress in Relativity*

*(a) Frequency-amplitude response of the simple pendulum for 0*°< *A* <*180*°*. (b) CPLM error analysis.*

**Figure 3.** *Oscillation history of the simple pendulum for (a) A* ¼ *45*° *and (b) A* ¼ *135*°*. CPLM estimate—Lines; exact solution—Markers.*

are determined using the expression for *u*\_, and all the other constants for the solution of each discretization are determined based on *Krs*. A plot of the frequencyamplitude response for the simple pendulum when Ω<sup>0</sup> ¼ 1*:*0 is given in **Figure 2a** and the corresponding error of the CPLM solution in comparison with the exact solution (Eq. (15)) is shown in **Figure 2b**. We see that for *A* ≤ 178°, the maximum error in the CPLM estimate is less than 0.45% for *n* ¼ 50 and 0.14% for *n* ¼ 100. Also, the oscillation history for moderate-amplitude (*A* ¼ 45°) and large-amplitude (*A* ¼ 135°) is shown in **Figure 3** and an excellent agreement between the CPLM solution for *n* ¼ 100 and the exact solution (Eq. (14)) is observed. We noted that trigonometric nonlinearity is usually difficult to deal with and that is why the CPLM shows a relatively slow convergence to accurate results. Hence, many discretizations (e.g., *n* ¼ 50 � 100) are required to get an accuracy that is within 1.0% of the exact solution during large-amplitude oscillations (90°< *A* < 180°) of the simple pendulum.

#### **3.2 Motion of satellite equidistant from twin stars**

Consider the motion of a satellite along a path that is equidistant from two identical massive stars with mutually interacting gravitational fields. If the distance between the two stars is 2*d* and the coordinate of the satellite motion is *u*, then the equation of motion of the satellite is given as [3]:

$$
\ddot{u} + \frac{2Mu}{\left(d^2 + u^2\right)^{3/2}} = 0,\tag{17}
$$

where *<sup>M</sup>* is the mass of a star and the restoring force is *f u*ð Þ¼ <sup>2</sup>*Mu<sup>=</sup> <sup>d</sup>*<sup>2</sup> <sup>þ</sup> *<sup>u</sup>*<sup>2</sup> � �3*=*<sup>2</sup> . Eq. (17) shows that the satellite-star interaction results in a conservative oscillation of the satellite. **Figure 4** shows the nonlinear restoring force, which is an irrational force because of the bottom square root. The restoring force spikes on both sides of the vertical axis close to the origin. The spikes indicate the point when the satellite is most influenced by the mutual gravitational field of the stars. Away from the origin, the restoring force decreases gradually and approaches the horizontal axis asymptotically. This means that the satellite is far away from the stars and experiences a much smaller gravitational force. This problem was discussed qualitatively by Jordan and Smith [3] and Arnold [59], but here, the periodic solution was investigated.

The main CPLM constant is *Krs* <sup>¼</sup> <sup>2</sup>*M d*<sup>2</sup> <sup>þ</sup> *<sup>u</sup>*<sup>2</sup> *s* � ��3*=*<sup>2</sup> � *<sup>d</sup>*<sup>2</sup> <sup>þ</sup> *<sup>u</sup>*<sup>2</sup> *r* � ��3*=*<sup>2</sup> h i and the velocity was derived as: *u*\_ ¼ �2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *M d*<sup>2</sup> <sup>þ</sup> *<sup>u</sup>*<sup>2</sup> � ��1*=*<sup>2</sup> � *<sup>d</sup>*<sup>2</sup> <sup>þ</sup> *<sup>A</sup>*<sup>2</sup> � ��1*=*<sup>2</sup> <sup>r</sup> h i. The periodic

solutions obtained by the CPLM and exact numerical solution are shown in **Figures 5** and **6**. The numerical solution was obtained by solving Eq. (17) using the NDSolve function in Mathematica™. The NDSolve function is a Mathematica subroutine for solving ordinary, partial, and algebraic differential equations numerically. In its basic form, it automatically selects the numerical method to use from a list of standard methods such as implicit Runge–Kutta, explicit Runge– Kutta, symplectic partitioned Runge–Kutta, predictor–corrector Adams, and backward difference methods. In some cases, the NDSolve function can combine two or

more methods to obtain the required solution. This basic form is preferable because the NDSolve function uses the method(s) that best solves the differential equation considering accuracy and solution time. Hence, the NDSolve function was used in

*Oscillation history of satellite for (a) A* ¼ *50 and (b) A* ¼ *1500. CPLM estimate—Lines; exact solution—*

The input values used for simulation are *<sup>M</sup>* <sup>¼</sup> <sup>10</sup><sup>5</sup> ½ � *kg* and *<sup>d</sup>* <sup>¼</sup> <sup>1000</sup>½ � *<sup>m</sup>* . In contrast to the simple pendulum, the oscillation of the satellite requires less discretization for accurate results because there is no trigonometric nonlinearity. The maximum error of the CPLM estimate for the frequency-amplitude response is less than 0.55% for *n* ¼ 10 and 0.20% for *n* ¼ 20. Significantly higher accuracies can be achieved by increasing *n*, but the results show that *n* ¼ 10 gives reasonably

On the other hand, **Figure 6** shows the oscillation history of the satellite during small-amplitude (*A* ¼ 50) and large-amplitude (*A* ¼ 1500) oscillations. The former gives a simple harmonic response with a natural frequency that is independent of

an *anharmonic* response with a natural frequency that depends strongly on the

An interesting oscillator that has been the subject of several studies [60–65] is the Hamiltonian oscillator with odd fractional nonlinearity. For the purpose of the present investigation, we consider an oscillator that is characterized by a general

where the restoring force, *f u*ð Þ¼ *<sup>u</sup>*<sup>1</sup>*=*ð Þ <sup>2</sup>*m*þ<sup>1</sup> , has a fractional index for all

ð Þ *us* � *ur* and the velocity as *<sup>u</sup>*\_ ¼ �½ � ð Þ <sup>2</sup>*<sup>m</sup>* <sup>þ</sup> <sup>1</sup> *<sup>=</sup>*ð Þ *<sup>m</sup>* <sup>þ</sup> <sup>1</sup> <sup>1</sup>*=*<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

The periodic solution for the case of *<sup>m</sup>* <sup>¼</sup> 1, i.e., *<sup>u</sup>*<sup>1</sup>*=*<sup>3</sup> oscillator, is shown in **Figures 7** and **8**. The exact frequency-amplitude response used for the verification of the CPLM

<sup>p</sup> *<sup>Γ</sup>*ð Þ <sup>3</sup>*=*<sup>4</sup> *<sup>Γ</sup>*ð Þ <sup>1</sup>*=*<sup>2</sup> *<sup>A</sup>*<sup>1</sup>*=*<sup>3</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>070451</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2*M=d*<sup>3</sup> q

, while the latter exhibits

<sup>1</sup>*=*ð Þ <sup>2</sup>*m*þ<sup>1</sup> *<sup>s</sup>* � *<sup>u</sup>*

h i*<sup>=</sup>*

*<sup>A</sup>*ð Þ <sup>2</sup>*m*þ<sup>2</sup> *<sup>=</sup>*ð Þ <sup>2</sup>*m*þ<sup>1</sup> � *<sup>u</sup>*ð Þ <sup>2</sup>*m*þ<sup>2</sup> *<sup>=</sup>*ð Þ <sup>2</sup>*m*þ<sup>1</sup>

*<sup>A</sup>*<sup>1</sup>*=*<sup>3</sup> *:* (19)

<sup>1</sup>*=*ð Þ <sup>2</sup>*m*þ<sup>1</sup> *<sup>r</sup>*

.

*<sup>u</sup>*€ <sup>þ</sup> *<sup>u</sup>*<sup>1</sup>*=*ð Þ <sup>2</sup>*m*þ<sup>1</sup> <sup>¼</sup> 0, (18)

p

its basic form for all numerical solutions obtained in this chapter.

accurate estimates.

amplitude.

**Figure 6.**

*Markers.*

solution is [63]:

**245**

the amplitude and approximately equal to

*Periodic Solution of Nonlinear Conservative Systems DOI: http://dx.doi.org/10.5772/intechopen.90282*

fractional nonlinearity as follows [65]:

**3.3 Mass-spring oscillator with fractional nonlinearity**

*m ϵ ℤ*<sup>þ</sup> f g. The main CPLM constant is evaluated as *Krs* ¼ *u*

*<sup>ω</sup>ex* <sup>¼</sup> <sup>2</sup>*πΓ*ð Þ <sup>5</sup>*=*<sup>4</sup> ffiffiffi 6

**Figure 4.** *Restoring force for Eq. (18): M* <sup>¼</sup> *105*½ � *kg ; d* <sup>¼</sup> *100 m*½ �*; A* <sup>¼</sup> *500.*

**Figure 5.** *(a) Frequency-amplitude response for satellite. (b) CPLM error analysis.*

*Periodic Solution of Nonlinear Conservative Systems DOI: http://dx.doi.org/10.5772/intechopen.90282*

**Figure 6.**

where *<sup>M</sup>* is the mass of a star and the restoring force is *f u*ð Þ¼ <sup>2</sup>*Mu<sup>=</sup> <sup>d</sup>*<sup>2</sup> <sup>þ</sup> *<sup>u</sup>*<sup>2</sup> � �3*=*<sup>2</sup>

*s* � ��3*=*<sup>2</sup>

r h i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� *<sup>d</sup>*<sup>2</sup> <sup>þ</sup> *<sup>u</sup>*<sup>2</sup>

� ��3*=*<sup>2</sup> h i

� *<sup>d</sup>*<sup>2</sup> <sup>þ</sup> *<sup>A</sup>*<sup>2</sup> � ��1*=*<sup>2</sup>

*r*

and the

. The periodic

Eq. (17) shows that the satellite-star interaction results in a conservative oscillation of the satellite. **Figure 4** shows the nonlinear restoring force, which is an irrational force because of the bottom square root. The restoring force spikes on both sides of the vertical axis close to the origin. The spikes indicate the point when the satellite is most influenced by the mutual gravitational field of the stars. Away from the origin, the restoring force decreases gradually and approaches the horizontal axis asymptotically. This means that the satellite is far away from the stars and experiences a much smaller gravitational force. This problem was discussed qualitatively by Jordan and Smith [3] and Arnold [59], but here, the periodic solution was investigated.

*M d*<sup>2</sup> <sup>þ</sup> *<sup>u</sup>*<sup>2</sup> � ��1*=*<sup>2</sup>

solutions obtained by the CPLM and exact numerical solution are shown in **Figures 5** and **6**. The numerical solution was obtained by solving Eq. (17) using the NDSolve function in Mathematica™. The NDSolve function is a Mathematica subroutine for solving ordinary, partial, and algebraic differential equations numerically. In its basic form, it automatically selects the numerical method to use from a list of standard methods such as implicit Runge–Kutta, explicit Runge– Kutta, symplectic partitioned Runge–Kutta, predictor–corrector Adams, and backward difference methods. In some cases, the NDSolve function can combine two or

The main CPLM constant is *Krs* <sup>¼</sup> <sup>2</sup>*M d*<sup>2</sup> <sup>þ</sup> *<sup>u</sup>*<sup>2</sup>

*Restoring force for Eq. (18): M* <sup>¼</sup> *105*½ � *kg ; d* <sup>¼</sup> *100 m*½ �*; A* <sup>¼</sup> *500.*

*(a) Frequency-amplitude response for satellite. (b) CPLM error analysis.*

velocity was derived as: *u*\_ ¼ �2

*Progress in Relativity*

**Figure 4.**

**Figure 5.**

**244**

.

*Oscillation history of satellite for (a) A* ¼ *50 and (b) A* ¼ *1500. CPLM estimate—Lines; exact solution— Markers.*

more methods to obtain the required solution. This basic form is preferable because the NDSolve function uses the method(s) that best solves the differential equation considering accuracy and solution time. Hence, the NDSolve function was used in its basic form for all numerical solutions obtained in this chapter.

The input values used for simulation are *<sup>M</sup>* <sup>¼</sup> <sup>10</sup><sup>5</sup> ½ � *kg* and *<sup>d</sup>* <sup>¼</sup> <sup>1000</sup>½ � *<sup>m</sup>* . In contrast to the simple pendulum, the oscillation of the satellite requires less discretization for accurate results because there is no trigonometric nonlinearity. The maximum error of the CPLM estimate for the frequency-amplitude response is less than 0.55% for *n* ¼ 10 and 0.20% for *n* ¼ 20. Significantly higher accuracies can be achieved by increasing *n*, but the results show that *n* ¼ 10 gives reasonably accurate estimates.

On the other hand, **Figure 6** shows the oscillation history of the satellite during small-amplitude (*A* ¼ 50) and large-amplitude (*A* ¼ 1500) oscillations. The former gives a simple harmonic response with a natural frequency that is independent of the amplitude and approximately equal to ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2*M=d*<sup>3</sup> q , while the latter exhibits an *anharmonic* response with a natural frequency that depends strongly on the amplitude.

#### **3.3 Mass-spring oscillator with fractional nonlinearity**

An interesting oscillator that has been the subject of several studies [60–65] is the Hamiltonian oscillator with odd fractional nonlinearity. For the purpose of the present investigation, we consider an oscillator that is characterized by a general fractional nonlinearity as follows [65]:

$$
\ddot{u} + u^{1/(2m+1)} = \mathbf{0},\tag{18}
$$

where the restoring force, *f u*ð Þ¼ *<sup>u</sup>*<sup>1</sup>*=*ð Þ <sup>2</sup>*m*þ<sup>1</sup> , has a fractional index for all *m ϵ ℤ*<sup>þ</sup> f g. The main CPLM constant is evaluated as *Krs* ¼ *u* <sup>1</sup>*=*ð Þ <sup>2</sup>*m*þ<sup>1</sup> *<sup>s</sup>* � *<sup>u</sup>* <sup>1</sup>*=*ð Þ <sup>2</sup>*m*þ<sup>1</sup> *<sup>r</sup>* h i*<sup>=</sup>* ð Þ *us* � *ur* and the velocity as *<sup>u</sup>*\_ ¼ �½ � ð Þ <sup>2</sup>*<sup>m</sup>* <sup>þ</sup> <sup>1</sup> *<sup>=</sup>*ð Þ *<sup>m</sup>* <sup>þ</sup> <sup>1</sup> <sup>1</sup>*=*<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>A</sup>*ð Þ <sup>2</sup>*m*þ<sup>2</sup> *<sup>=</sup>*ð Þ <sup>2</sup>*m*þ<sup>1</sup> � *<sup>u</sup>*ð Þ <sup>2</sup>*m*þ<sup>2</sup> *<sup>=</sup>*ð Þ <sup>2</sup>*m*þ<sup>1</sup> p . The periodic solution for the case of *<sup>m</sup>* <sup>¼</sup> 1, i.e., *<sup>u</sup>*<sup>1</sup>*=*<sup>3</sup> oscillator, is shown in **Figures 7** and **8**. The exact frequency-amplitude response used for the verification of the CPLM solution is [63]:

$$
\omega\_{\rm ex} = \frac{2\pi \Gamma(5/4)}{\sqrt{6} \Gamma(3/4) \Gamma(1/2) A^{1/3}} = \frac{1.070451}{A^{1/3}}.\tag{19}
$$

**Figure 7.** *(a) Frequency-amplitude response for u1=<sup>3</sup> oscillator. (b) CPLM error analysis.*

and rotating at a constant speed, Ω, about the *y*-axis (**Figure 9a**). The *u*-axis represents the perpendicular displacement of the mass from the *y*-axis. The kinetic

*(a) Schematic of particle on a rotating parabola. (b) Restoring force when q* ¼ 1*:*0*; Λ* ¼ *10:0; A* ¼ *2:0.*

*<sup>m</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*<sup>2</sup> *u*<sup>2</sup> *u*\_

> *∂L ∂u*\_

Therefore, using Eqs. (21) and (22), the motion of a particle on a rotating

where *<sup>Λ</sup>* <sup>¼</sup> <sup>2</sup>*gq* � <sup>Ω</sup><sup>2</sup> and the initial conditions are: *<sup>u</sup>*ð Þ¼ <sup>0</sup> *<sup>A</sup>* and *<sup>u</sup>*\_ð Þ¼ <sup>0</sup> 0. To solve Eq. (23) using the CPLM, it must be recast in the form of Eq. (1). The

where *h* is a constant representing the total energy in the system. Eq. (24)

*<sup>u</sup>*<sup>2</sup> , is not constant. Applying the initial conditions, we get *<sup>h</sup>* <sup>¼</sup> *<sup>Λ</sup>A*<sup>2</sup> so

*<sup>Λ</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*<sup>2</sup>*A*<sup>2</sup> *<sup>u</sup>*

Next, we substitute Eq. (21) into the Euler–Lagrange equation to derive the

� *∂L*

*uu*\_

<sup>2</sup> <sup>þ</sup> <sup>Ω</sup><sup>2</sup> *<sup>u</sup>*<sup>2</sup> ;*<sup>V</sup>* <sup>¼</sup> *mgqu*<sup>2</sup>

> <sup>2</sup> <sup>þ</sup> <sup>Ω</sup><sup>2</sup> *<sup>u</sup>*<sup>2</sup> � *mgqu*<sup>2</sup>

*:* (20)

*:* (21)

*<sup>∂</sup><sup>u</sup>* <sup>¼</sup> <sup>0</sup>*:* (22)

<sup>2</sup> <sup>þ</sup> *<sup>Λ</sup><sup>u</sup>* <sup>¼</sup> 0, (23)

<sup>2</sup> <sup>þ</sup> *<sup>Λ</sup>u*<sup>2</sup> <sup>¼</sup> *<sup>h</sup>*, (24)

<sup>2</sup> <sup>þ</sup> *<sup>Λ</sup>* <sup>þ</sup> <sup>2</sup>Ω<sup>2</sup> *<sup>u</sup>*<sup>2</sup> <sup>¼</sup>

<sup>2</sup> in Eq. (24) and

<sup>2</sup> *<sup>m</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*<sup>2</sup>*u*<sup>2</sup> ð Þ*u*\_

<sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*<sup>2</sup>*u*<sup>2</sup> ð Þ<sup>2</sup> <sup>¼</sup> <sup>0</sup>*:* (25)

and potential energies of the system are given as [4]:

*<sup>m</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*<sup>2</sup> *u*<sup>2</sup> *u*\_

2

equation of motion. The Euler–Lagrange equation can be written as:

*d dt*

*<sup>u</sup>*<sup>2</sup> *<sup>u</sup>*€ <sup>þ</sup> <sup>4</sup>*q*<sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*<sup>2</sup> *u*<sup>2</sup> *u*\_

<sup>2</sup> <sup>¼</sup> *<sup>Λ</sup> <sup>A</sup>*<sup>2</sup> � *<sup>u</sup>*<sup>2</sup> *<sup>=</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*<sup>2</sup>*u*<sup>2</sup> ð Þ. Substituting this expression for *<sup>u</sup>*\_

*u*€ þ

<sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*<sup>2</sup>

conservation of energy for Eq. (23) is given as [4]:

confirms that the Hamiltonian, *<sup>H</sup>* <sup>¼</sup> *<sup>T</sup>* <sup>þ</sup> *<sup>V</sup>* <sup>¼</sup> <sup>1</sup>

after algebraic simplification, we get:

*<sup>T</sup>* <sup>¼</sup> <sup>1</sup> 2

*Periodic Solution of Nonlinear Conservative Systems DOI: http://dx.doi.org/10.5772/intechopen.90282*

*<sup>L</sup>* <sup>¼</sup> *<sup>T</sup>* � *<sup>V</sup>* <sup>¼</sup> <sup>1</sup>

Hence, the Lagrangian is:

**Figure 9.**

parabola is governed by:

1

that *u*\_

**247**

<sup>2</sup> *m h* <sup>þ</sup> <sup>2</sup>Ω<sup>2</sup>

**Figure 8.** *Oscillation history of u1=<sup>3</sup> oscillator for (a) A* <sup>¼</sup> *<sup>0</sup>:01 and (b) A* <sup>¼</sup> *10. CPLM estimate—lines; exact solution markers.*

while the exact oscillation history was obtained by the numerical solution of Eq. (18) using the NDSolve function in Mathematica™. The CPLM solution demonstrates an excellent agreement with the exact solution.

**Figure 7** compares the CPLM estimates of the frequency-amplitude response with Eq. (19), and the maximum error of the CPLM solution is 0.22% for *n* ¼ 10 and 0.076% for *n* ¼ 20. The error in the CPLM estimate is approximately constant for all amplitudes, and the maximum error is well below 1.0% for *n* ¼ 10. The results also reveal that the frequency approaches zero as *A* ! ∞. In **Figure 8**, the oscillation history for small-amplitude (*A* ¼ 0*:*01) and large-amplitude (*A* ¼ 10*:*0) oscillations is shown to exhibit similar anharmonic response, which is an indication of strong nonlinearity. Therefore, it can be concluded that the *u*<sup>1</sup>*=*<sup>3</sup> oscillator is highly nonlinear for all amplitudes. This quality of possessing strong nonlinearity for all amplitudes is in contrast to most Hamiltonian oscillators that are linear for small amplitudes, e.g., the oscillators considered in Sections 3.1 and 3.2 above. Another Hamiltonian oscillator that possesses strong nonlinearity for all amplitudes is the geometrically nonlinear crank [1].

#### **4. Periodic solution of non-Hamiltonian conservative systems**

The non-Hamiltonian conservative systems are generally more complex and difficult to solve compared with the Hamiltonian systems. In order to demonstrate the application of the CPLM algorithm to non-Hamiltonian conservative systems, we consider the motion of a particle on a rotating parabola. This system consists of a frictionless mass sliding on a vertical parabolic wire described by *<sup>y</sup>* <sup>¼</sup> *qu*<sup>2</sup> for *<sup>q</sup>* <sup>&</sup>gt;<sup>0</sup>

*Periodic Solution of Nonlinear Conservative Systems DOI: http://dx.doi.org/10.5772/intechopen.90282*

**Figure 9.** *(a) Schematic of particle on a rotating parabola. (b) Restoring force when q* ¼ 1*:*0*; Λ* ¼ *10:0; A* ¼ *2:0.*

and rotating at a constant speed, Ω, about the *y*-axis (**Figure 9a**). The *u*-axis represents the perpendicular displacement of the mass from the *y*-axis. The kinetic and potential energies of the system are given as [4]:

$$T = \frac{1}{2}m\left[\left(\mathbf{1} + 4q^2u^2\right)\dot{u}^2 + \Omega^2u^2\right]; V = mgqu^2. \tag{20}$$

Hence, the Lagrangian is:

$$L = T - V = \frac{1}{2}m\left[ (\mathbf{1} + 4q^2 u^2)\dot{u}^2 + \Omega^2 u^2 \right] - mgqu^2. \tag{21}$$

Next, we substitute Eq. (21) into the Euler–Lagrange equation to derive the equation of motion. The Euler–Lagrange equation can be written as:

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{u}}\right) - \frac{\partial L}{\partial u} = 0.\tag{22}$$

Therefore, using Eqs. (21) and (22), the motion of a particle on a rotating parabola is governed by:

$$(\mathbf{1} + 4q^2 u^2)\ddot{u} + 4q^2 u \dot{u}^2 + \Lambda u = \mathbf{0},\tag{23}$$

where *<sup>Λ</sup>* <sup>¼</sup> <sup>2</sup>*gq* � <sup>Ω</sup><sup>2</sup> and the initial conditions are: *<sup>u</sup>*ð Þ¼ <sup>0</sup> *<sup>A</sup>* and *<sup>u</sup>*\_ð Þ¼ <sup>0</sup> 0.

To solve Eq. (23) using the CPLM, it must be recast in the form of Eq. (1). The conservation of energy for Eq. (23) is given as [4]:

$$(\mathbf{1} + 4q^2 u^2)\dot{u}^2 + \Lambda u^2 = h,\tag{24}$$

where *h* is a constant representing the total energy in the system. Eq. (24) confirms that the Hamiltonian, *<sup>H</sup>* <sup>¼</sup> *<sup>T</sup>* <sup>þ</sup> *<sup>V</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> *<sup>m</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*<sup>2</sup>*u*<sup>2</sup> ð Þ*u*\_ <sup>2</sup> <sup>þ</sup> *<sup>Λ</sup>* <sup>þ</sup> <sup>2</sup>Ω<sup>2</sup> *<sup>u</sup>*<sup>2</sup> <sup>¼</sup> 1 <sup>2</sup> *m h* <sup>þ</sup> <sup>2</sup>Ω<sup>2</sup> *<sup>u</sup>*<sup>2</sup> , is not constant. Applying the initial conditions, we get *<sup>h</sup>* <sup>¼</sup> *<sup>Λ</sup>A*<sup>2</sup> so that *u*\_ <sup>2</sup> <sup>¼</sup> *<sup>Λ</sup> <sup>A</sup>*<sup>2</sup> � *<sup>u</sup>*<sup>2</sup> *<sup>=</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*<sup>2</sup>*u*<sup>2</sup> ð Þ. Substituting this expression for *<sup>u</sup>*\_ <sup>2</sup> in Eq. (24) and after algebraic simplification, we get:

$$
\ddot{u} + \frac{\Lambda \left(1 + 4q^2 A^2\right) u}{\left(1 + 4q^2 u^2\right)^2} = 0.\tag{25}
$$

while the exact oscillation history was obtained by the numerical solution of Eq. (18) using the NDSolve function in Mathematica™. The CPLM solution dem-

*Oscillation history of u1=<sup>3</sup> oscillator for (a) A* <sup>¼</sup> *<sup>0</sup>:01 and (b) A* <sup>¼</sup> *10. CPLM estimate—lines; exact solution—*

**Figure 7** compares the CPLM estimates of the frequency-amplitude response with Eq. (19), and the maximum error of the CPLM solution is 0.22% for *n* ¼ 10 and 0.076% for *n* ¼ 20. The error in the CPLM estimate is approximately constant for all amplitudes, and the maximum error is well below 1.0% for *n* ¼ 10. The results also reveal that the frequency approaches zero as *A* ! ∞. In **Figure 8**, the oscillation history for small-amplitude (*A* ¼ 0*:*01) and large-amplitude (*A* ¼ 10*:*0) oscillations is shown to exhibit similar anharmonic response, which is an indication of strong nonlinearity. Therefore, it can be concluded that the *u*<sup>1</sup>*=*<sup>3</sup> oscillator is highly nonlinear for all amplitudes. This quality of possessing strong nonlinearity for all amplitudes is in contrast to most Hamiltonian oscillators that are linear for small amplitudes, e.g., the oscillators considered in Sections 3.1 and 3.2 above. Another Hamiltonian oscillator that possesses strong nonlinearity for all amplitudes

**4. Periodic solution of non-Hamiltonian conservative systems**

The non-Hamiltonian conservative systems are generally more complex and difficult to solve compared with the Hamiltonian systems. In order to demonstrate the application of the CPLM algorithm to non-Hamiltonian conservative systems, we consider the motion of a particle on a rotating parabola. This system consists of a frictionless mass sliding on a vertical parabolic wire described by *<sup>y</sup>* <sup>¼</sup> *qu*<sup>2</sup> for *<sup>q</sup>* <sup>&</sup>gt;<sup>0</sup>

onstrates an excellent agreement with the exact solution.

*(a) Frequency-amplitude response for u1=<sup>3</sup> oscillator. (b) CPLM error analysis.*

is the geometrically nonlinear crank [1].

**Figure 7.**

*Progress in Relativity*

**Figure 8.**

*markers.*

Therefore, the restoring force is *f u*ð Þ¼ *<sup>Λ</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*2*A*<sup>2</sup> � �*u<sup>=</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*2*u*<sup>2</sup> ð Þ<sup>2</sup> . **Figure 9b** shows that *f u*ð Þ is linear at small amplitudes and strongly nonlinear at large amplitudes. The main CPLM constant was calculated as *Krs* <sup>¼</sup> *<sup>Λ</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*<sup>2</sup>*A*<sup>2</sup> � � <sup>1</sup>*<sup>=</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*<sup>2</sup>*u*<sup>2</sup> *s* � �<sup>2</sup> � h <sup>1</sup>*<sup>=</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*<sup>2</sup>*u*<sup>2</sup> *r* � �<sup>2</sup> �, and the velocity was evaluated as *u*\_ ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>Λ</sup> <sup>A</sup>*<sup>2</sup> � *<sup>u</sup>*<sup>2</sup> � �*<sup>=</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*2*u*<sup>2</sup> ð Þ <sup>q</sup> . The exact time period for this oscillator can be derived in terms of elliptic function as follows [4]:

$$T\_{ex} = 4\left[\left(\mathbf{1} + 4q^2 \mathbf{A}^2\right) / \Lambda\right]^{1/2} E(\mathbf{k}^2),\tag{26}$$

**5. Concluding remarks**

**Acknowledgements**

University of Port Harcourt.

Akuro Big-Alabo\* and Chinwuba Victor Ossia

provided the original work is properly cited.

Harcourt, Port Harcourt, Nigeria

**Author details**

**249**

derived in terms of special functions.

*Periodic Solution of Nonlinear Conservative Systems DOI: http://dx.doi.org/10.5772/intechopen.90282*

Conservative oscillators generally exhibit nonlinear response, and they form a large class of natural and artificially vibrating systems. Hence, the study of the dynamic response of nonlinear conservative systems is important for understanding many physical phenomena and the design of systems. The main challenge in the theoretical analysis of nonlinear conservative systems is that exact solutions are normally not available except for a few special cases where exact solutions are

To date, many approximate analytical methods have been formulated for the periodic solution of nonlinear conservative oscillators. This chapter provides a brief survey of the recent advances in the formulation of approximate analytical schemes and then introduced a recent approximate analytical algorithm called the continuous piecewise linearization method. The CPLM has been shown to overcome the challenges of solution accuracy and simplicity usually encountered in using most of the existing approximate analytical methods. The CPLM combines major desirable features of solution schemes such as inherent stability, accuracy, and simplicity. It is simple enough to be introduced at the undergraduate level and is capable of handling conservative oscillators with very complex nonlinearity. Conservative systems of broad interest were used to demonstrate the wide applicability of the CPLM algorithm. As demonstrated above, an accuracy of less than 1.0% relative error can be achieved for most oscillators using few discretizations, say *n*≤20, except for oscillators with trigonometric nonlinearity where such accuracy is achieved with many discretizations. This chapter has been designed to stimulate interest in the use of CPLM for analyzing various types of

conservative systems, especially those with complex nonlinearity.

The authors are grateful to the Vice-Chancellor of the University of Port Harcourt, Prof. Ndowa E.S. Lale and the Dean of Engineering, Prof. Ogbonna Joel, for approving the publication grant for this chapter. The OAPF was funded by the African Centre of Excellence in Oilfield Chemicals Research (ACE-FOR),

Department of Mechanical Engineering, Faculty of Engineering, University of Port

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: akuro.big-alabo@uniport.edu.ng

where *E k*<sup>2</sup> � � <sup>¼</sup> <sup>Ð</sup> *<sup>π</sup>=*<sup>2</sup> <sup>0</sup> <sup>1</sup> � *<sup>k</sup>*<sup>2</sup> sin <sup>2</sup> *ϕ* � �1*=*<sup>2</sup> *dϕ* is the complete elliptic integral of the second kind and *<sup>k</sup>*<sup>2</sup> <sup>¼</sup> <sup>4</sup>*q*2*A*<sup>2</sup> *<sup>=</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*2*A*<sup>2</sup> � �. Then, the exact frequency was computed as *ωex* ¼ 2*π=Tex*, while the exact oscillation history was obtained by solving Eq. (25) numerically using the NDSolve function in Mathematica™.

A comparison of CPLM frequency estimate and the exact frequency is shown in **Figure 10**, while the oscillation histories for *A* ¼ 0*:*50 and *A* ¼ 2*:*0 are shown in **Figure 11**. As demonstrated in [4], periodic solutions for this system exist only for *Λ*> 0. Hence, the simulations in **Figures 10** and **11** were conducted for *Λ* ¼ 10 and *q* ¼ 1*:*0. An excellent agreement is observed between the CPLM estimates and the exact results. For 0< *A* ≤20, the maximum error in the CPLM estimate of the frequency-amplitude response is 0.642% for *n* ¼ 10 and 0.101% for *n* ¼ 20, both of which are well below 1.0%. Also, the CPLM solution gives an accurate prediction of the strong anharmonic response in the oscillation history as shown in **Figure 11**.

#### **Figure 11.**

*Oscillation history of particle on a rotating parabola for (a) A* ¼ *0:50 and (b) A* ¼ *2:0. CPLM estimate— Lines; exact solution—Markers.*
