3. Example 1

Consider system (8) which is equivalent to the equation [3]

$$\ddot{\mathbf{x}} + \mathbf{x} + \mathbf{x}^3 = \left(P\_1 + P\_2 \mathbf{x}^2 + P\_3 \mathbf{x} \sin(\nu t)\right) \dot{\mathbf{x}} + P\_4 \sin(\nu t), \tag{20}$$

the saddle point S, forming a contour (see Figure 4(b)). As P<sup>3</sup> increases further, two closed invariant curves appear, shown in Figure 5 for P<sup>3</sup> ¼ 0:15. The structural changes of the resonance zone observed in the experiment are in good agreement with the theoretical results for

Figure 4. Poincaré map for Eq. (20) with P<sup>1</sup> ¼ 0:0472, P<sup>2</sup> ¼ �0:008, P<sup>4</sup> ¼ 2, and ν ¼ 4 and (a) P<sup>3</sup> ¼ 0:018 and (b)

Periodic Perturbations: Parametric Systems http://dx.doi.org/10.5772/intechopen.79513 91

P<sup>3</sup> ¼ 0:0489755.

γ ¼ 0. The observations for γ 6¼ 0 are consistent with the theory, too.

Figure 5. Poincaré map for Eq. (20) with P<sup>1</sup> ¼ 0:0472, P<sup>2</sup> ¼ �0:008, P<sup>3</sup> ¼ 0:15, P<sup>4</sup> ¼ 2, and ν ¼ 4.

where Pi, ið Þ ¼ 1; 2; 3; 4 are parameters. Here, we focus only on the effects which are due to the nonlinear parametric term xx\_sinð Þ νt . Let us assume ν ¼ 4. Then, for small Pið Þ i ¼ 1; 2; 3; 4 system (20) can have only two "splittable" resonance levels: H xð Þ¼ ; y h11,H xð Þ¼ ; y h<sup>31</sup> and h<sup>31</sup> < h11. The corresponding autonomous system (P<sup>3</sup> ¼ P<sup>4</sup> ¼ 0) has at most one LC. The passage of this LC through the resonances under a change of parameter P<sup>2</sup> was considered in [2]. If this LC lies outside the neighborhoods of resonance levels H xð Þ¼ ; y h11,H xð Þ¼ ; y h31, then in the original nonautonomous system (20), there is a two-dimensional invariant torus T<sup>2</sup> corresponding to the cycle. There is a generating "Kolmogorov torus" in the Hamiltonian system ð Þ P<sup>1</sup> ¼ P<sup>2</sup> ¼ P<sup>3</sup> ¼ 0 .

A computer program was developed by the author for a simulation of Eq. (20). The results of such simulation are presented in Figures 4–6. In the numerical integration, the Runge-Kuttatype formulae are used with an error of order O h<sup>6</sup> per integration step h. In Figure 4(a) we present the Poincaré map for P<sup>1</sup> ¼ 0:0472, P<sup>2</sup> ¼ �0:008, and P<sup>3</sup> ¼ 0:018, which determines the structure of the main resonance zone ð Þ p ¼ 1; q ¼ 1 . Along with the separatrices of the saddle fixed point S, a closed invariant curve encircling the unstable fixed point O is shown, which corresponds to a stable LC in the oscillatory domain of Eq. (6). This closed invariant curve appears for P<sup>3</sup> ≈ 0:014 when the fixed point O loses its stability. As P<sup>3</sup> increases, so does the size of the closed invariant curve, and for P<sup>3</sup> ≈ 0:0487 the curve clings to the separatrix of

the OD and one stable non-contractible LC; and (7) when γ < γ�

90 Perturbation Methods with Applications in Science and Engineering

lower half-cylinder uð Þ < 0 ; (4) when γ ¼ γþ, a contour Γ<sup>þ</sup>

unstable LCs exist for u > 0; (8) when γ ¼ γþ, a contour Γ<sup>þ</sup>

Consider system (8) which is equivalent to the equation [3]

stable LC exists for u < 0.

unstable LC exists for u > 0.

when γ�

γ ¼ γ�

3. Example 1

system ð Þ P<sup>1</sup> ¼ P<sup>2</sup> ¼ P<sup>3</sup> ¼ 0 .

when γ ¼ γ<sup>þ</sup>

contractible LC which lies in the lower half-cylinder u < 0 and no more than p nð Þ � 1 in the OD.

γ<sup>þ</sup> < γ < γ�, one stable LC exists on the upper half-cylinder uð Þ > 0 and one stable LC on the

2. Let a∈ð Þ a∗; 0 . Then, in the OD there are p nð Þ � 1 LCs, and in the rotational domain, (1) when

3. Let a∈ ð Þ 0; 1 . Then, there are at most p nð Þ � 1 LCs in the OD, and in the rotational domain, (1)

when γ > γ� and u < 0, Eq. (14) has one stable LC; (2) when γ ¼ γ�, a contour Γ�

<sup>0</sup> , a semi-stable LC is formed for u > 0; (7) when γ<sup>þ</sup> < γ < γ<sup>þ</sup>

where Pi, ið Þ ¼ 1; 2; 3; 4 are parameters. Here, we focus only on the effects which are due to the nonlinear parametric term xx\_sinð Þ νt . Let us assume ν ¼ 4. Then, for small Pið Þ i ¼ 1; 2; 3; 4 system (20) can have only two "splittable" resonance levels: H xð Þ¼ ; y h11,H xð Þ¼ ; y h<sup>31</sup> and h<sup>31</sup> < h11. The corresponding autonomous system (P<sup>3</sup> ¼ P<sup>4</sup> ¼ 0) has at most one LC. The passage of this LC through the resonances under a change of parameter P<sup>2</sup> was considered in [2]. If this LC lies outside the neighborhoods of resonance levels H xð Þ¼ ; y h11,H xð Þ¼ ; y h31, then in the original nonautonomous system (20), there is a two-dimensional invariant torus T<sup>2</sup> corresponding to the cycle. There is a generating "Kolmogorov torus" in the Hamiltonian

A computer program was developed by the author for a simulation of Eq. (20). The results of such simulation are presented in Figures 4–6. In the numerical integration, the Runge-Kuttatype formulae are used with an error of order O h<sup>6</sup> per integration step h. In Figure 4(a) we present the Poincaré map for P<sup>1</sup> ¼ 0:0472, P<sup>2</sup> ¼ �0:008, and P<sup>3</sup> ¼ 0:018, which determines the structure of the main resonance zone ð Þ p ¼ 1; q ¼ 1 . Along with the separatrices of the saddle fixed point S, a closed invariant curve encircling the unstable fixed point O is shown, which corresponds to a stable LC in the oscillatory domain of Eq. (6). This closed invariant curve appears for P<sup>3</sup> ≈ 0:014 when the fixed point O loses its stability. As P<sup>3</sup> increases, so does the size of the closed invariant curve, and for P<sup>3</sup> ≈ 0:0487 the curve clings to the separatrix of

<sup>x</sup>€ <sup>þ</sup> <sup>x</sup> <sup>þ</sup> <sup>x</sup><sup>3</sup> <sup>¼</sup> <sup>P</sup><sup>1</sup> <sup>þ</sup> <sup>P</sup>2x<sup>2</sup> <sup>þ</sup> <sup>P</sup>3xsinð Þ <sup>ν</sup><sup>t</sup> <sup>x</sup>\_ <sup>þ</sup> <sup>P</sup>4sinð Þ <sup>ν</sup><sup>t</sup> , (20)

<sup>0</sup> < γ < γ� and u < 0, there is a stable LC born from Γ�

<sup>0</sup> , the stable and unstable LCs merge together; (5) when γ<sup>þ</sup>

γ > γ�, Eq. (14) has one stable LC for u > 0; (2) when γ ¼ γ�, a contour Γ�

<sup>1</sup> , Eq. (14) has one stable non-

<sup>p</sup> appears; and (5) when γ < γþ, one

<sup>p</sup> and an unstable LC; (4) when

<sup>p</sup> is formed; and (9) when γ < γþ, one

<sup>0</sup> < γ < γ�

<sup>p</sup> appears; (3) when

<sup>p</sup> appears; (3)

<sup>0</sup> , no LCs exist; (6)

<sup>0</sup> , one stable and one

Figure 4. Poincaré map for Eq. (20) with P<sup>1</sup> ¼ 0:0472, P<sup>2</sup> ¼ �0:008, P<sup>4</sup> ¼ 2, and ν ¼ 4 and (a) P<sup>3</sup> ¼ 0:018 and (b) P<sup>3</sup> ¼ 0:0489755.

Figure 5. Poincaré map for Eq. (20) with P<sup>1</sup> ¼ 0:0472, P<sup>2</sup> ¼ �0:008, P<sup>3</sup> ¼ 0:15, P<sup>4</sup> ¼ 2, and ν ¼ 4.

the saddle point S, forming a contour (see Figure 4(b)). As P<sup>3</sup> increases further, two closed invariant curves appear, shown in Figure 5 for P<sup>3</sup> ¼ 0:15. The structural changes of the resonance zone observed in the experiment are in good agreement with the theoretical results for γ ¼ 0. The observations for γ 6¼ 0 are consistent with the theory, too.

The motion of the pendulum with vertically oscillating suspension (under some simplifying

pendulum in which the force of resistance is created by a vertical plate perpendicular to the plane of oscillations. Consider Eq. (22) when it is close to integrable, i.e., for small values of

Eq. (23) in the conservative case, when C<sup>2</sup> ¼ C<sup>3</sup> ¼ 0, is considered in many publications. For instance, for small angles of the deviation x, the case β ffi 2 is studied in [8]. The criterion of resonance overlap is applied in [11] to estimating the width of the "ergodic layer." The existence of homoclinic solutions is discussed in [12] without the assumption on smallness of

Phase curves of the unperturbed mathematical pendulum equation are determined by the integral H xð Þ� ; <sup>x</sup>\_ <sup>x</sup>\_ <sup>2</sup> � cos<sup>x</sup> <sup>¼</sup> h, where <sup>h</sup><sup>∈</sup> ð Þ �1; <sup>1</sup> in the oscillatory domain and <sup>h</sup> <sup>&</sup>gt; 1 in the rotational domain. The peculiarity lies in the way period τ depends on h in the oscillatory

<sup>τ</sup>ð Þ¼ <sup>h</sup> <sup>2</sup>π=<sup>ω</sup> <sup>¼</sup> <sup>4</sup>Kð Þ<sup>k</sup> , k<sup>2</sup> <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>h</sup> <sup>=</sup>2, � <sup>1</sup> <sup>&</sup>lt; <sup>h</sup> <sup>&</sup>lt; <sup>1</sup>,

Here, K ¼ Kð Þk is the complete elliptic integral of the first kind, k being its modulus. From Eq. (24) it follows that the period τ changes noticeably only for h close to 1, i.e., in the neighborhood of the separatrix. Therefore, small intervals of period τ, which determines the

In the investigation of the perturbed equation, we first focus on the structure of resonance zones in domains <sup>G</sup><sup>1</sup> <sup>¼</sup> f g ð Þ <sup>x</sup>; <sup>x</sup>\_ : �<sup>1</sup> <sup>&</sup>lt; <sup>h</sup>� <sup>≤</sup> H xð Þ ; <sup>y</sup> <sup>≤</sup> <sup>h</sup><sup>þ</sup> <sup>&</sup>lt; <sup>1</sup> and <sup>G</sup><sup>2</sup> <sup>¼</sup> fð Þ <sup>x</sup>; <sup>x</sup>\_ : H xð Þ ; <sup>y</sup> <sup>≥</sup> <sup>h</sup><sup>∗</sup> <sup>&</sup>gt; <sup>1</sup>g. The resonance condition <sup>τ</sup> hpq <sup>¼</sup> ð Þ <sup>p</sup>=<sup>q</sup> <sup>2</sup>π=<sup>β</sup> , where p, q are relatively prime integers, deter-

width of resonance zones, correspond to fairly large intervals of variable x.

ð Þ i ¼ 1; 2; 3 . Denote pi ¼ εCi, where ε is a small parameter. Then, the original

Let us now complicate the model even more and consider the equation

x€ þ sinx þ p1cosβtsinx þ p2x\_ ¼ 0, (21)

Periodic Perturbations: Parametric Systems http://dx.doi.org/10.5772/intechopen.79513 93

<sup>x</sup>€ <sup>þ</sup> sin<sup>x</sup> <sup>þ</sup> <sup>p</sup>1cosβtsin<sup>x</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup> <sup>þ</sup> <sup>p</sup>3cos<sup>x</sup> <sup>x</sup>\_ <sup>¼</sup> <sup>0</sup>, (22)

<sup>x</sup>€ <sup>þ</sup> sin<sup>x</sup> <sup>¼</sup> <sup>ε</sup> <sup>C</sup>1cosβtsin<sup>x</sup> <sup>þ</sup> ð Þ <sup>C</sup><sup>2</sup> <sup>þ</sup> <sup>C</sup>3cos<sup>x</sup> <sup>x</sup>\_ , (23)

<sup>τ</sup>ð Þ¼ <sup>h</sup> <sup>2</sup>kK, k<sup>2</sup> <sup>¼</sup> <sup>1</sup>=ð Þ <sup>1</sup> <sup>þ</sup> <sup>h</sup> , h <sup>&</sup>gt; <sup>1</sup>: (24)

. The term p3x\_cosx appears, for example, in the case of the

assumptions) is described by the equation [13]

where p1, p2, β are parameters.

with the phase space <sup>R</sup><sup>1</sup> � <sup>S</sup><sup>1</sup> � <sup>S</sup><sup>1</sup>

4.1. Structure of resonant zones

mines the resonance levels of energy H xð Þ¼ ; y hpq.

parameters pi

parameter ε.

domain. We have

Eq. (22) takes the form

Figure 6. Poincaré map for Eq. (20) with P<sup>1</sup> ¼ 0:0472, P<sup>2</sup> ¼ �0:008, P<sup>3</sup> ¼ 0:0487, P<sup>4</sup> ¼ 8, and ν ¼ 4 (a) and quasi-attractor (b).

In the case presented in Figure 6, the transversal intersection of the separatrices of S cannot be detected visually. We, therefore, increased P<sup>4</sup> to obtain a better picture of the homoclinic structure. When P<sup>4</sup> ¼ 8, the structure can be seen clearly (Figure 6(a)). The corresponding quasi-attractor is the only attracting set (Figure 6(b)). Stable periodic points with long periods can exist inside the quasi-attractor itself. However, they are extremely difficult to detect numerically.

## 4. Example 2

As opposed to Example 1, this one pursues a different goal, namely, to study the transition from the classical parametric resonance to the nonlinear resonance. One of the problems for which this can be done is that of the pendulum with a vibrating suspension.

The pendulum with vibrating suspension is a classical example of a problem in which a parametric resonance can be observed. A large number of publications (see, e.g., [8, 9]) are devoted to this problem. Other problems of this sort include the bending oscillations of straight rod under a periodic longitudinal force [10], the motion of a charged particle (electron) in the field of two running waves [11], etc. The parametric resonance in this kind of systems appears when a fixed point of the corresponding Poincaré map loses its stability and is, therefore, usually described by the linearization near this point.

It is interesting to study the behavior of a parametric system when the ring-like resonance zone is contracted into a point, i.e., to describe the bifurcations which occur in the course of transition from the plain nonlinear resonance to the parametric one. This paragraph is devoted to the solution of this problem in the case of a nonconservative pendulum with a vertically oscillating suspension.

The motion of the pendulum with vertically oscillating suspension (under some simplifying assumptions) is described by the equation [13]

$$
\ddot{x} + \sin x + p\_1 \cos \beta t \sin x + p\_2 \dot{x} = 0,\tag{21}
$$

where p1, p2, β are parameters.

Let us now complicate the model even more and consider the equation

$$(\ddot{\mathbf{x}} + \sin \mathbf{x} + p\_1 \cos \beta t \sin \mathbf{x} + (p\_2 + p\_3 \cos \mathbf{x})\dot{\mathbf{x}} = \mathbf{0},\tag{22}$$

with the phase space <sup>R</sup><sup>1</sup> � <sup>S</sup><sup>1</sup> � <sup>S</sup><sup>1</sup> . The term p3x\_cosx appears, for example, in the case of the pendulum in which the force of resistance is created by a vertical plate perpendicular to the plane of oscillations. Consider Eq. (22) when it is close to integrable, i.e., for small values of parameters pi ð Þ i ¼ 1; 2; 3 . Denote pi ¼ εCi, where ε is a small parameter. Then, the original Eq. (22) takes the form

$$\ddot{\mathbf{x}} + \sin \mathbf{x} = \varepsilon \left[ \mathbf{C}\_1 \mathbf{cos} \beta t \sin \mathbf{x} + (\mathbf{C}\_2 + \mathbf{C}\_3 \cos \mathbf{x}) \dot{\mathbf{x}} \right],\tag{23}$$

Eq. (23) in the conservative case, when C<sup>2</sup> ¼ C<sup>3</sup> ¼ 0, is considered in many publications. For instance, for small angles of the deviation x, the case β ffi 2 is studied in [8]. The criterion of resonance overlap is applied in [11] to estimating the width of the "ergodic layer." The existence of homoclinic solutions is discussed in [12] without the assumption on smallness of parameter ε.

Phase curves of the unperturbed mathematical pendulum equation are determined by the integral H xð Þ� ; <sup>x</sup>\_ <sup>x</sup>\_ <sup>2</sup> � cos<sup>x</sup> <sup>¼</sup> h, where <sup>h</sup><sup>∈</sup> ð Þ �1; <sup>1</sup> in the oscillatory domain and <sup>h</sup> <sup>&</sup>gt; 1 in the rotational domain. The peculiarity lies in the way period τ depends on h in the oscillatory domain.

We have

In the case presented in Figure 6, the transversal intersection of the separatrices of S cannot be detected visually. We, therefore, increased P<sup>4</sup> to obtain a better picture of the homoclinic structure. When P<sup>4</sup> ¼ 8, the structure can be seen clearly (Figure 6(a)). The corresponding quasi-attractor is the only attracting set (Figure 6(b)). Stable periodic points with long periods can exist inside the quasi-attractor itself. However, they are extremely difficult to detect

Figure 6. Poincaré map for Eq. (20) with P<sup>1</sup> ¼ 0:0472, P<sup>2</sup> ¼ �0:008, P<sup>3</sup> ¼ 0:0487, P<sup>4</sup> ¼ 8, and ν ¼ 4 (a) and quasi-attractor (b).

As opposed to Example 1, this one pursues a different goal, namely, to study the transition from the classical parametric resonance to the nonlinear resonance. One of the problems for

The pendulum with vibrating suspension is a classical example of a problem in which a parametric resonance can be observed. A large number of publications (see, e.g., [8, 9]) are devoted to this problem. Other problems of this sort include the bending oscillations of straight rod under a periodic longitudinal force [10], the motion of a charged particle (electron) in the field of two running waves [11], etc. The parametric resonance in this kind of systems appears when a fixed point of the corresponding Poincaré map loses its stability and is,

It is interesting to study the behavior of a parametric system when the ring-like resonance zone is contracted into a point, i.e., to describe the bifurcations which occur in the course of transition from the plain nonlinear resonance to the parametric one. This paragraph is devoted to the solution of this problem in the case of a nonconservative pendulum with a vertically

which this can be done is that of the pendulum with a vibrating suspension.

therefore, usually described by the linearization near this point.

92 Perturbation Methods with Applications in Science and Engineering

numerically.

4. Example 2

oscillating suspension.

$$\begin{aligned} \pi(h) &= 2\pi/\omega = 4\mathbf{K}(k), k^2 = (1+h)/2, \ -1 < h < 1, \\ \pi(h) &= 2k\mathbf{K}, k^2 = 1/(1+h), h > 1. \end{aligned} \tag{24}$$

Here, K ¼ Kð Þk is the complete elliptic integral of the first kind, k being its modulus. From Eq. (24) it follows that the period τ changes noticeably only for h close to 1, i.e., in the neighborhood of the separatrix. Therefore, small intervals of period τ, which determines the width of resonance zones, correspond to fairly large intervals of variable x.

#### 4.1. Structure of resonant zones

In the investigation of the perturbed equation, we first focus on the structure of resonance zones in domains <sup>G</sup><sup>1</sup> <sup>¼</sup> f g ð Þ <sup>x</sup>; <sup>x</sup>\_ : �<sup>1</sup> <sup>&</sup>lt; <sup>h</sup>� <sup>≤</sup> H xð Þ ; <sup>y</sup> <sup>≤</sup> <sup>h</sup><sup>þ</sup> <sup>&</sup>lt; <sup>1</sup> and <sup>G</sup><sup>2</sup> <sup>¼</sup> fð Þ <sup>x</sup>; <sup>x</sup>\_ : H xð Þ ; <sup>y</sup> <sup>≥</sup> <sup>h</sup><sup>∗</sup> <sup>&</sup>gt; <sup>1</sup>g. The resonance condition <sup>τ</sup> hpq <sup>¼</sup> ð Þ <sup>p</sup>=<sup>q</sup> <sup>2</sup>π=<sup>β</sup> , where p, q are relatively prime integers, determines the resonance levels of energy H xð Þ¼ ; y hpq.

System (25) is defined in <sup>D</sup> � <sup>S</sup><sup>1</sup> where <sup>D</sup> is a certain domain in <sup>R</sup><sup>2</sup>

<sup>ξ</sup> <sup>¼</sup> <sup>η</sup>, <sup>η</sup>\_ ¼ �<sup>ξ</sup> <sup>þ</sup> <sup>ε</sup> <sup>C</sup><sup>1</sup> � <sup>ξ</sup> � cos <sup>β</sup><sup>t</sup> � � <sup>þ</sup> ð Þ <sup>C</sup><sup>2</sup> <sup>þ</sup> <sup>C</sup><sup>3</sup> <sup>η</sup> � � <sup>þ</sup> <sup>O</sup> <sup>ε</sup><sup>2</sup> � �: (26)

<sup>=</sup><sup>6</sup> � �: (27)

<sup>2</sup><sup>I</sup> <sup>p</sup> cosϑ. In terms of this variables,

ϑ:

By discarding in Eq. (26) the terms O ε<sup>2</sup> � �, we arrive at the Mathieu equation with the extra term resulting from the viscous friction. It is clear that in the framework of a linear equation one cannot observe the (nonlinear) effects which accompany the transition from the nonlinear resonance to the parametric one. So, let us consider a wider neighborhood U<sup>2</sup> ð Þ n ¼ 2 of the origin. In Eq. (25) we discard the terms O ε<sup>2</sup> � � and, for the resulting system, consider the resonance cases when ω ¼ 1 ¼ qβ=p (p and q being relatively prime integers). We then study the bifurcations pertaining to the transition from the parametric resonance to the ordinary one. We once again introduce the detuning 1 � qβ=p ¼ γ1ε. As a result, the system in question will

<sup>η</sup>\_ ¼ � <sup>q</sup>β=<sup>p</sup> � �<sup>ξ</sup> <sup>þ</sup> <sup>ε</sup> <sup>C</sup>1ξcosβ<sup>t</sup> <sup>þ</sup> ð Þ <sup>C</sup><sup>2</sup> <sup>þ</sup> <sup>C</sup><sup>3</sup> <sup>η</sup> � <sup>γ</sup>1<sup>ξ</sup> <sup>þ</sup> <sup>ξ</sup><sup>3</sup>

<sup>2</sup><sup>I</sup> <sup>p</sup> sinϑ, <sup>R</sup> <sup>¼</sup> <sup>G</sup>sin<sup>ϑ</sup> <sup>þ</sup> <sup>γ</sup>1cosϑ<sup>=</sup> ffiffiffiffi

<sup>G</sup> <sup>¼</sup> <sup>C</sup>1sinϑcos<sup>φ</sup> <sup>þ</sup> ð Þ <sup>C</sup><sup>2</sup> <sup>þ</sup> <sup>C</sup><sup>3</sup> cos<sup>ϑ</sup> � <sup>γ</sup>1sin<sup>ϑ</sup> <sup>þ</sup> ð Þ <sup>I</sup>=<sup>3</sup> sin<sup>3</sup>

Let us introduce in Eq. (28) the "resonance phase" ψ ¼ ϑ � qφ=p and average the resulting system over the "fast" variable φ. As a result, we arrive at the two-dimensional autonomous

u\_ ¼ ε½ Þ C1=2 usin2v þ ð Þ C<sup>2</sup> þ C<sup>3</sup> u�

u\_ ¼ εð Þ C<sup>2</sup> þ C<sup>3</sup>

when <sup>p</sup> 6¼ 2 and/or <sup>q</sup> <sup>&</sup>gt; 1. As we know, <sup>u</sup> <sup>¼</sup> <sup>I</sup> <sup>þ</sup> <sup>O</sup>ð Þ<sup>ε</sup> , v <sup>¼</sup> <sup>ψ</sup> <sup>þ</sup> <sup>O</sup> <sup>ε</sup><sup>2</sup> � �. From Eq. (29) and (30), it follows that (in our approximation) only one resonance with p ¼ 2, q ¼ 1 appears in the

Now, we introduce the action (I) – angle (ϑ) variables. Since the unperturbed system is linear,

<sup>2</sup><sup>I</sup> <sup>p</sup> sin<sup>ϑ</sup> and <sup>η</sup> <sup>¼</sup> ffiffiffiffi

<sup>I</sup> <sup>¼</sup> <sup>ε</sup>F Ið Þ ; <sup>ϑ</sup>;<sup>φ</sup> , <sup>ϑ</sup>\_ <sup>¼</sup> <sup>q</sup>β=<sup>p</sup> � <sup>ε</sup>R Ið Þ ; <sup>ϑ</sup>; <sup>φ</sup> , <sup>φ</sup>\_ <sup>¼</sup> <sup>β</sup>, (28)

<sup>v</sup>\_ <sup>¼</sup> <sup>ε</sup> ð Þ <sup>C</sup>1=<sup>4</sup> cos2<sup>v</sup> � <sup>u</sup>=<sup>8</sup> � <sup>γ</sup>1=<sup>2</sup> � � (29)

<sup>v</sup>\_ <sup>¼</sup> <sup>ε</sup> �u=<sup>8</sup> � <sup>γ</sup>1=<sup>2</sup> � � (30)

<sup>2</sup><sup>I</sup> <sup>p</sup>

U<sup>1</sup> ð Þ n ¼ 1 , system (25) assumes the form \_

<sup>ξ</sup>\_ <sup>¼</sup> <sup>q</sup>β=<sup>p</sup> � �<sup>η</sup> <sup>þ</sup> <sup>γ</sup>1<sup>ε</sup>

the substitution has the simple form <sup>ξ</sup> <sup>¼</sup> ffiffiffiffi

\_

when p ¼ 2 and q ¼ 1 and to the system

ffiffiffiffi

system (27) will be written as

where F ¼ 2IGcosϑ � γ<sup>1</sup>

system

neighborhood U2.

be rewritten as

. In the neighborhood

95

Periodic Perturbations: Parametric Systems http://dx.doi.org/10.5772/intechopen.79513

Figure 7. Invariant curves (separatrices) of Poincaré map for Eq. (22) with p<sup>1</sup> ¼ �0, 1, p<sup>3</sup> ¼ 0, 1, β ¼ 1:6, and p<sup>2</sup> ¼ �0, 07 (a) with p<sup>2</sup> ≃ � 1=30 (b).

The structure of individual resonance zones U<sup>μ</sup> is described (up to the terms O ε<sup>3</sup>=<sup>2</sup> ) by the pendulum-type Eq. (6). Since functions A<sup>0</sup> and σ have different forms in the oscillatory and rotational domains, we introduce the notations Að Þ<sup>s</sup> <sup>0</sup> <sup>v</sup>; hpq and <sup>σ</sup>ð Þ<sup>s</sup> <sup>v</sup>; hpq , where <sup>s</sup> <sup>¼</sup> 1 corresponds to the oscillatory domain and s ¼ 2 to the rotational one.

In our case the divergence of the vector field of Eq. (23) contains no terms explicitly depending on t; hence, σ does not depend on v, i.e., σ ¼ const.

The functions Að Þ<sup>s</sup> <sup>0</sup> and σð Þ<sup>s</sup> in an explicit form were obtained in [13]. It is also found that the width of the resonance zone decreases rapidly with the increase of p when q ¼ 1.

A computer-generated picture of invariant curves of the Poincaré map for Eq. (22), with β ¼ 1:6, is shown in Figure 7. In Eq. 21(a) a case of synchronization of oscillations in the subharmonic with p ¼ 2, q ¼ 1 (p<sup>1</sup> ¼ 0, 1, p<sup>2</sup> ¼ 0, 07, p<sup>3</sup> ¼ �0, 1Þ is shown, and in Figure 7(b), a partly passable resonance with <sup>p</sup> <sup>¼</sup> <sup>2</sup>, q <sup>¼</sup> <sup>1</sup> <sup>p</sup><sup>1</sup> <sup>¼</sup> <sup>0</sup>, <sup>1</sup> , <sup>p</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>=30, <sup>p</sup><sup>3</sup> ¼ �0, <sup>1</sup><sup>Þ</sup> is shown. In the domain <sup>G</sup><sup>2</sup> the synchronization of oscillations on the main resonance (<sup>p</sup> <sup>¼</sup> <sup>q</sup> <sup>¼</sup> 1) takes place.

#### 4.2. Neighborhood of the origin

Denote Un <sup>¼</sup> ð Þ <sup>x</sup>; <sup>y</sup> : <sup>0</sup> <sup>≤</sup> H xð Þ ; <sup>y</sup> <sup>≤</sup>Cε<sup>2</sup>=<sup>n</sup> and substitute in Eq. (23):

$$\mathfrak{x} = \mathfrak{e}^{1/n}\xi \lrcorner \mathfrak{z} = \dot{\mathfrak{x}} = \mathfrak{e}^{1/n}\mathfrak{y}$$

As a result, we arrive at the system

$$\begin{aligned} \dot{\xi} &= \eta\_{\prime} \quad \dot{\eta} = -\xi + \varepsilon \left[ \mathbf{C}\_{1} \xi \cos(\beta t) + (\mathbf{C}\_{2} + \mathbf{C}\_{3}) \eta \right] + \varepsilon^{2/n} \xi^{3}/6 - \\ &- \varepsilon^{1+2/n} \left( \mathbf{C}\_{1} \xi^{3} \cos(\beta t)/6 + \xi^{2} \eta \right) + \dots \end{aligned} \tag{25}$$

System (25) is defined in <sup>D</sup> � <sup>S</sup><sup>1</sup> where <sup>D</sup> is a certain domain in <sup>R</sup><sup>2</sup> . In the neighborhood U<sup>1</sup> ð Þ n ¼ 1 , system (25) assumes the form

$$\dot{\xi} = \eta,\ \dot{\eta} = -\xi + \varepsilon \left[ \mathbb{C}\_1 \cdot \xi \cdot \cos(\beta t) + (\mathbb{C}\_2 + \mathbb{C}\_3)\eta \right] + O(\varepsilon^2). \tag{26}$$

By discarding in Eq. (26) the terms O ε<sup>2</sup> � �, we arrive at the Mathieu equation with the extra term resulting from the viscous friction. It is clear that in the framework of a linear equation one cannot observe the (nonlinear) effects which accompany the transition from the nonlinear resonance to the parametric one. So, let us consider a wider neighborhood U<sup>2</sup> ð Þ n ¼ 2 of the origin. In Eq. (25) we discard the terms O ε<sup>2</sup> � � and, for the resulting system, consider the resonance cases when ω ¼ 1 ¼ qβ=p (p and q being relatively prime integers). We then study the bifurcations pertaining to the transition from the parametric resonance to the ordinary one. We once again introduce the detuning 1 � qβ=p ¼ γ1ε. As a result, the system in question will be rewritten as

$$\begin{aligned} \dot{\xi} &= \left( q\beta/p \right) \eta + \gamma\_1 \varepsilon \\ \dot{\eta} &= - \left( q\beta/p \right) \xi + \varepsilon \left[ \mathbb{C}\_1 \xi \cos \beta t + (\mathbb{C}\_2 + \mathbb{C}\_3) \eta - \gamma\_1 \xi + \xi^3/\mathfrak{G} \right]. \end{aligned} \tag{27}$$

Now, we introduce the action (I) – angle (ϑ) variables. Since the unperturbed system is linear, the substitution has the simple form <sup>ξ</sup> <sup>¼</sup> ffiffiffiffi <sup>2</sup><sup>I</sup> <sup>p</sup> sin<sup>ϑ</sup> and <sup>η</sup> <sup>¼</sup> ffiffiffiffi <sup>2</sup><sup>I</sup> <sup>p</sup> cosϑ. In terms of this variables, system (27) will be written as

$$\dot{I} = \varepsilon F(I, \mathfrak{P}, \mathfrak{q}), \quad \dot{\mathfrak{F}} = \mathfrak{q}\mathfrak{P}/p - \varepsilon R(I, \mathfrak{P}, \mathfrak{q}), \quad \dot{\mathfrak{q}} = \mathfrak{p}, \tag{28}$$

where F ¼ 2IGcosϑ � γ<sup>1</sup> ffiffiffiffi <sup>2</sup><sup>I</sup> <sup>p</sup> sinϑ, <sup>R</sup> <sup>¼</sup> <sup>G</sup>sin<sup>ϑ</sup> <sup>þ</sup> <sup>γ</sup>1cosϑ<sup>=</sup> ffiffiffiffi <sup>2</sup><sup>I</sup> <sup>p</sup>

$$G = \mathbb{C}\_1 \sin \theta \cos \varphi + (\mathbb{C}\_2 + \mathbb{C}\_3) \cos \theta - \gamma\_1 \sin \theta + (I/3) \sin^3 \theta \dots$$

Let us introduce in Eq. (28) the "resonance phase" ψ ¼ ϑ � qφ=p and average the resulting system over the "fast" variable φ. As a result, we arrive at the two-dimensional autonomous system

$$\begin{aligned} \dot{u} &= \varepsilon \left[ \mathbb{C}\_1 / 2 \right] \iota \sin 2v + (\mathbb{C}\_2 + \mathbb{C}\_3) u \big] \\ \dot{v} &= \varepsilon \left[ (\mathbb{C}\_1 / 4) \cos 2v - u / 8 - \gamma\_1 / 2 \right] \end{aligned} \tag{29}$$

when p ¼ 2 and q ¼ 1 and to the system

The structure of individual resonance zones U<sup>μ</sup> is described (up to the terms O ε<sup>3</sup>=<sup>2</sup> ) by the pendulum-type Eq. (6). Since functions A<sup>0</sup> and σ have different forms in the oscillatory and

Figure 7. Invariant curves (separatrices) of Poincaré map for Eq. (22) with p<sup>1</sup> ¼ �0, 1, p<sup>3</sup> ¼ 0, 1, β ¼ 1:6, and p<sup>2</sup> ¼ �0, 07

In our case the divergence of the vector field of Eq. (23) contains no terms explicitly depending

A computer-generated picture of invariant curves of the Poincaré map for Eq. (22), with β ¼ 1:6, is shown in Figure 7. In Eq. 21(a) a case of synchronization of oscillations in the subharmonic with p ¼ 2, q ¼ 1 (p<sup>1</sup> ¼ 0, 1, p<sup>2</sup> ¼ 0, 07, p<sup>3</sup> ¼ �0, 1Þ is shown, and in Figure 7(b), a partly passable resonance with <sup>p</sup> <sup>¼</sup> <sup>2</sup>, q <sup>¼</sup> <sup>1</sup> <sup>p</sup><sup>1</sup> <sup>¼</sup> <sup>0</sup>, <sup>1</sup> , <sup>p</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>=30, <sup>p</sup><sup>3</sup> ¼ �0, <sup>1</sup><sup>Þ</sup> is shown. In the domain <sup>G</sup><sup>2</sup> the synchronization of oscillations on the main resonance (<sup>p</sup> <sup>¼</sup> <sup>q</sup> <sup>¼</sup> 1) takes place.

<sup>x</sup> <sup>¼</sup> <sup>ε</sup><sup>1</sup>=<sup>n</sup>ξ, y <sup>¼</sup> <sup>x</sup>\_ <sup>¼</sup> <sup>ε</sup><sup>1</sup>=<sup>n</sup><sup>η</sup>

<sup>ξ</sup>\_ <sup>¼</sup> <sup>η</sup>, <sup>η</sup>\_ ¼ �<sup>ξ</sup> <sup>þ</sup> <sup>ε</sup> <sup>C</sup>1ξcos <sup>β</sup><sup>t</sup> <sup>þ</sup> ð Þ <sup>C</sup><sup>2</sup> <sup>þ</sup> <sup>C</sup><sup>3</sup> <sup>η</sup> <sup>þ</sup> <sup>ε</sup><sup>2</sup>=<sup>n</sup>ξ<sup>3</sup>

cos <sup>β</sup><sup>t</sup> <sup>=</sup><sup>6</sup> <sup>þ</sup> <sup>ξ</sup><sup>2</sup>

width of the resonance zone decreases rapidly with the increase of p when q ¼ 1.

<sup>0</sup> v; hpq

<sup>0</sup> and σð Þ<sup>s</sup> in an explicit form were obtained in [13]. It is also found that the

and σð Þ<sup>s</sup> v; hpq

, where <sup>s</sup> <sup>¼</sup> 1 corre-

=6�

<sup>η</sup> <sup>þ</sup> … (25)

rotational domains, we introduce the notations Að Þ<sup>s</sup>

94 Perturbation Methods with Applications in Science and Engineering

on t; hence, σ does not depend on v, i.e., σ ¼ const.

The functions Að Þ<sup>s</sup>

(a) with p<sup>2</sup> ≃ � 1=30 (b).

4.2. Neighborhood of the origin

As a result, we arrive at the system

�ε<sup>1</sup>þ2=<sup>n</sup> <sup>C</sup>1ξ<sup>3</sup>

sponds to the oscillatory domain and s ¼ 2 to the rotational one.

Denote Un <sup>¼</sup> ð Þ <sup>x</sup>; <sup>y</sup> : <sup>0</sup> <sup>≤</sup> H xð Þ ; <sup>y</sup> <sup>≤</sup>Cε<sup>2</sup>=<sup>n</sup> and substitute in Eq. (23):

$$\begin{aligned} \dot{u} &= \varepsilon (\mathbb{C}\_2 + \mathbb{C}\_3) \\ \dot{v} &= \varepsilon \left( -u/8 - \gamma\_1/2 \right) \end{aligned} \tag{30}$$

when <sup>p</sup> 6¼ 2 and/or <sup>q</sup> <sup>&</sup>gt; 1. As we know, <sup>u</sup> <sup>¼</sup> <sup>I</sup> <sup>þ</sup> <sup>O</sup>ð Þ<sup>ε</sup> , v <sup>¼</sup> <sup>ψ</sup> <sup>þ</sup> <sup>O</sup> <sup>ε</sup><sup>2</sup> � �. From Eq. (29) and (30), it follows that (in our approximation) only one resonance with p ¼ 2, q ¼ 1 appears in the neighborhood U2.

resonance regime with p ¼ 2, q ¼ 1 in the oscillatory domain and the one with p ¼ 1, q ¼ 1 in

Periodic Perturbations: Parametric Systems http://dx.doi.org/10.5772/intechopen.79513 97

1. The transition from Figure 8(a)–(c) corresponds to two period-doubling bifurcations, while the passage from the parametric resonance (Figure 8(b)) to the ordinary nonlinear resonance (Figure 8(c)) corresponds to the birth of two periodic (of period 2) saddle points and

2. The bifurcation which involves the birth of a quasi-attractor (Figure 7(b)) in the neighborhood of the unperturbed separatrix is the most interesting one. It may take place at any magnitude of the external force (parameter <sup>C</sup>1). It suffices to have <sup>B</sup>ð Þ<sup>s</sup> ð Þ¼ <sup>1</sup> <sup>0</sup>, C<sup>ð</sup> <sup>2</sup> <sup>¼</sup>

3. In the quasi-integrable nonconservative case, there appear no resonances with q > 1 and odd p in the oscillatory domain and no resonances with q > 1 and even p in the rotational

This work was supported in part by the Russian Foundation for Basic Research under grant

[1] Morozov AD, Shil'nikov LP. On nonconservative periodic systems similar to twodimensional Hamiltonian ones. Pricl. Mat. i Mekh. (Russian). 1983;47(3):385-394

[2] Morozov AD. Quasi-conservative systems: Cycles, resonances and chaos. World Scientific Series on Nonlinear Science Series A. 1998;30. http://www.worldscientific.com/worldsci

[3] Morozov AD. Resonances and chaos in parametric systems. Journal of Applied Mathe-

no. 18-01-00306, by the Russian Science Foundation under grant no. 14-41-00044.

In conclusion we make the following remarks on Eqs. (22) and (23).

�C3=3Þ, εð Þ C<sup>2</sup> � C<sup>3</sup> < 0, for example, C<sup>2</sup> ¼ �1=30, C<sup>3</sup> ¼ 0:1, ε > 0:.

a node (focus) from a multiple saddle fixed point.

Address all correspondence to: morozov@mm.unn.ru

matics and Mechanics. 1994;58(3):413-423

Lobachevsky State University of Nizhny Novgorod, Russia

the rotational domain.

domain.

Author details

Albert Morozov

References

books/10.1142/3238

Acknowledgements

Figure 8. Phase portraits of system (29) with C<sup>2</sup> <sup>2</sup> <sup>þ</sup> <sup>C</sup><sup>2</sup> <sup>3</sup> 6¼ 0.

The investigation of system (29) when C<sup>2</sup> <sup>2</sup> <sup>þ</sup> <sup>C</sup><sup>2</sup> <sup>3</sup> 6¼ 0 for different values of detuning γ<sup>1</sup> presents no difficulty, because, according to the Bendixson criterion, there are no limit cycles. The most typical rough phase portraits are presented in Figure 8 where, parallel with the phase portraits in the ð Þ u; v plane, the corresponding phase portraits in Cartesian coordinates ð Þ x; y ¼ x\_ are shown. Figure 8(a) corresponds to the case when we have <sup>γ</sup><sup>1</sup> <sup>&</sup>gt; <sup>γ</sup><sup>∗</sup> <sup>&</sup>gt; 0, <sup>γ</sup><sup>∗</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C2 <sup>1</sup> � 4ð Þ C<sup>2</sup> þ C<sup>3</sup> 2 q =2, Figure 8(b) when ∣γ1∣ ≤ γ∗, and Figure 8(c) when ∣γ1∣ > γ<sup>∗</sup> and γ<sup>1</sup> < 0. In addition, in all three cases, we assume C<sup>2</sup> þ C<sup>3</sup> < 0.

#### 4.3. Conclusion

The number of splittable resonances is bounded, when C<sup>2</sup> <sup>2</sup> <sup>þ</sup> <sup>C</sup><sup>2</sup> <sup>3</sup> 6¼ 0. For the actual pendulum (Eq. (22)), when the small nonconservative forces are present, we, most likely, have one resonance regime with p ¼ 2, q ¼ 1 in the oscillatory domain and the one with p ¼ 1, q ¼ 1 in the rotational domain.

In conclusion we make the following remarks on Eqs. (22) and (23).

