Isochronous Oscillations of Nonlinear Systems

Abstract

Real-world systems, such as physical and living systems, are generally subject to vibrations that can affect their long-term integrity and safety. Thus, the determination of the law that governs the evolution of the oscillatory quantity has become a major topic in modern engineering design. The process often leads to solving nonlinear differential equations. However, one can admit that the main objective of the theory of differential equations to obtain explicit solutions is far from being carried out. If we know how to solve linear systems, the case of systems of nonlinear differential equations is not in general solved. Isochronous nonlinear systems have therefore received particular attention. This chapter is devoted to presenting some recent developments and advances in the theory of isochronous oscillations of nonlinear systems. The harmonic oscillator as a prototype of isochronous systems is investigated to state some useful definitions (section 2), and the existence of second-order isochronous nonlinear systems having explicit elementary first integrals with an exact sinusoidal solution and higher-order autonomous nonlinear systems that reproduce the dynamics of the harmonic oscillator is proven (section 3). Finally, higher-order nonautonomous nonlinear systems that can exhibit isochronous oscillations are shown (section 4), and a conclusion for the chapter is presented.

1. Introduction

The enormous literature generated by the qualitative theory of dynamic systems suggests that all questions about nonlinear systems are well studied and answered. Far from it, we can see that there are many interesting questions that are not fully resolved. We must note that a dynamic system is a time-dependent system that can undergo regular or chaotic processes. The dynamics of such systems are often described by finite-difference equations (discrete dynamic systems) or differential equations (continuous dynamic systems). Since many problems in engineering physics, biology, and applied mathematics are formulated in terms of differential equations, continuous dynamic systems have been the subject of an intensive investigation in the literature. In particular, planar polynomial dynamic systems given by [1, 2, 3, 4]

$$\dot{x}=P\left(x,y\right),\dot{y}=Q\left(x,y\right),$$E1

where overdot indicates differentiation with respect to time, and $P$ and $Q$ are polynomials in $x$ and $y$, are widely investigated from the perspective of the existence of isochronous centers, limit cycles, and elementary first integrals. One question that has been the object of special attention is the determination of the maximum number of limit cycles of the polynomial system (1), motivated by the second part of the Hilbert ${16}^{\mathit{th}}$ problem [1]. Another important question is the determination of polynomial and rational first integrals that ensure the complete integrability of polynomial dynamic systems of type (1) inspired by the work of Darboux [2, 3, 4]. When the first integrals do not explicitly depend on the time, the system is said, in the case of autonomous systems, conservative system and can exhibit periodic solution. A prototype of dynamic systems that can experience conservative oscillations is the harmonic oscillator described by [5, 6, 7]

$$\ddot{x}+x=0,$$E2

such that

$$x\left(t\right)=cost.$$E3

The harmonic oscillator (2) is characterized by a fixed constant period $T=2\pi$. Therefore, the system (2) is said to be an isochronous dynamic system in contrast to dynamic systems exhibiting amplitude-dependent frequency oscillations known as nonlinear systems. Nonlinear systems differ from linear systems that exhibit amplitude-independent frequency. A typical example of a nonlinear dynamical system is given by the well-known cubic Duffing equation [5, 6, 7, 8]

$$\ddot{x}+a_1\dot{x}+a_{2}x+a_3{x}^3=0,$$E4

which can exhibit $a_1=0$, an amplitude-dependent period, and experience softening and hardening phenomena under a periodic forcing function, where $a_1>0$, $a_2>0$, and $a_3$ are constants. Linear systems, such as Eq. (2), cannot exhibit softening and hardening, leading in general to fatigue and failure of material systems [9, 10]. Consequently, the problem of finding dynamic systems, more precisely nonlinear dynamic systems since real-world systems are nonlinear systems, preserving the feature of amplitude-independent frequency, has become a vital question for modern engineering design. Thus, the design and identification of nonlinear isochronous systems have generated a major and attractive research field in the theory of dynamic systems such that the well-established qualitative theory of dynamic systems has been widely applied to identify isochronous centers or systems that can exhibit amplitude-independent periods. In this way, theorems for the existence of a center and an isochronous center are established, particularly for the system (1), where $P\left(x,y\right)$ and $Q\left(x,y\right)$ are not necessarily polynomials in $x$ and $y$ [5, 6, 11, 12, 13, 14, 15, 16, 17]. Additionally, a multitude of approximation methods for periodic solutions has been developed in the literature on the basis of the predictions of the qualitative theory of differential equations and numerical results, while no exact explicit solutions are known. However, many of these studies are not mathematically consistent, as shown by the recent developments and advances in the theory of differential equations due to the considerable contribution of Monsia and coworkers. Consider as an example of illustration the unusual Lienard equation

$$\ddot{x}-\frac{\mathit{ax}}{\sqrt{{\mu }^2-x^2}}\dot{x}=0,$$E5

investigated by Akande et al. [18], where $a$ and $\mu$ are constants. The authors [18] showed that Eq. (5) has the exact isochronous harmonic solution

$$x\left(t\right)=\mu sin\left[-a\left(t+K\right)\right],$$E6

where $a<0$, $\mu >0$, $a\ne -\mu$ and $K$ is an arbitrary constant, while Eq. (5) obviously does not satisfy the classical existence theorems for a center for the Lienard equation of the form

$$\ddot{x}+f\left(x\right)\dot{x}+g\left(x\right)=0,$$E7

where $f\left(x\right)$ and $g\left(x\right)$ are functions of $x$ [5, 11, 14, 16, 17]. The inadequacy of the mentioned theorems can also be shown by considering the following exceptional quadratic Lienard-type [19]

$$\ddot{x}+\frac{u^{\prime }\left(x\right)}{u\left(x\right)}{\dot{x}}^2=0,$$E8

where $u\left(x\right)$ is a function of $x$ and prime means differentiation with respect to $x$. The authors [19] proved that Eq. (8) can exhibit, for example, that when $u\left(x\right)={\left({\mu }^2-x^2\right)}^{-\frac{1}{2}}$, the exact isochronous harmonic solution

$$x\left(t\right)=\mu sin\left[b\left(t+K_1\right)\right],$$E9

where $\mu >0$, $b>0$, and $K_1$ are arbitrary parameters such that $b\ne \mu$. Eq. (8) belongs to the general class of Lienard-type equation

$$\ddot{x}+\vartheta \left(x\right){\dot{x}}^2+g\left(x\right)=0,$$E10

where $\vartheta \left(x\right)$ is a function of $x$. Eq. (10) can be generalized in the form

$$\ddot{x}+h\left(x,\dot{x}\right)\dot{x}+g\left(x\right)=0,$$E11

where $h\left(x,\dot{x}\right)$ is a function of $x$ and $\dot{x}$. Obviously, Eq. (8), where

$$u\left(x\right)={\left({\mu }^2-x^2\right)}^{-\frac{1}{2}},$$E12

does not satisfy the classical theorems for the existence of at least one periodic solution [5, 6] or for the existence of an isochronous center, as stated in Refs. [12, 16, 17]. Other counterexamples of classical existence theorems can be seen in Refs. [20, 21, 22, 23, 24, 25, 26, 27]. If some progress has been made with the work of Calogero and coworkers [28], it will be very difficult to say the same thing concerning the dynamic systems represented by nonlinear differential equations having an exact elementary function solution, more precisely an exact explicit isochronous sinusoidal solution before the contribution of Monsia and his group (see Refs. [29, 30, 31] and References therein). The work of Monsia and his group revealed not only the inadequacy of the qualitative theory of dynamic systems to predict the effective behavior of nonlinear systems but also showed the existence of many autonomous and nonautonomous nonlinear dynamic systems with an exact explicit isochronous sinusoidal solution of a second and high order. The present chapter aims to contribute to these recent developments and advances in identifying and generating second-order and higher-order autonomous and nonautonomous nonlinear dynamic systems with an exact isochronous sinusoidal solution. To do so, we study the harmonic oscillator considered as the prototype of isochronous systems (section 2), the isochronous oscillations of higher-order autonomous nonlinear systems (section 3), and the isochronous oscillations of higher-order nonautonomous nonlinear systems (section 4). Finally, we present a conclusion for the chapter.

2. Harmonic oscillator

The equation of the harmonic oscillator (2) can be rewritten as a dynamical system in the form

$$\dot{x}=y,\dot{y}=-x,$$E13

such that the integral curves are given by

$$\frac{\mathit{dy}}{\mathit{dx}}=-\frac{x}{y}.$$E14

By separation of variables and integration, we have

$$H\left(x,y\right)=\frac{1}{2}{y}^2+\frac{1}{2}{x}^2,$$E15

where $H$ is a constant of integration known as the Hamiltonian or

$$H\left(x,\dot{x}\right)=\frac{1}{2}{\dot{x}}^2+\frac{1}{2}{x}^2,$$E16

so that Eq. (2) is said to be a Hamiltonian system. When $H\left(x,\dot{x}\right)=\frac{1}{2}$, the formula

$$x\left(t\right)=cos\left(t+\phi \right),$$E17

such that

$$\dot{x}\left(t\right)=-sin\left(t+\phi \right),$$E18

where $\phi$ is an arbitrary constant, verifies the first integral (16). Thus, Eq. (17) is the general solution of the harmonic oscillator (2), which exhibits periodic oscillations of period $T=2\pi$, independent of the oscillation amplitude, as shown in Figure 1.

Figure 1.

Typical behavior of solution (17) when $\phi =0$.

Such oscillations are said to be isochronous. Since all solutions given by Eq (17) are periodic with a fixed constant period $T$, the harmonic oscillator is called an isochronous system. Therefore we can state the following definitions.

Definition 1: A system exhibits isochronous oscillations if the period $T$ is independent of amplitude.

Definition 2: If the periodic general solution with a fixed constant period $T$ of a system (S) verifies $H\left(x,\dot{x}\right)=c$ where $c$ is a constant, then such a system (S) of differential equations corresponding to the Hamiltonian $H$ is an isochronous system.

On the basis of these definitions, we can investigate the isochronicity of nonlinear systems below.

3. Autonomous nonlinear systems

Recently, Monsia and coworkers introduced a new class of first integrals in the literature [29, 30, 32]. This type of class of first integrals contains $\left(n+1\right)$ first integrals $H\left(x,\dot{x}\right)$ such that $H\left(x,\dot{x}\right)=c$ when $x\left(t\right)=cos\left(t+\phi \right)$, where $n\ge 0$ is an integer. The corresponding $\left(n+1\right)$ second-order autonomous nonlinear differential equations admit the exact sinusoidal general solution $cos\left(t+\phi \right)$. In this part, we consider such classes of first integrals to secure isochronous oscillations of autonomous nonlinear systems.

3.1 Isochronous nonlinear systems

Consider a second-order autonomous equation

$$E\left(x,\dot{x},\ddot{x}\right)=0.$$E19

Thus, we have the following results.

Theorem 1.1: Assume that

$$H_1\left(x,\dot{x}\right)=b={\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3+\dot{x}\left(x^2-1\right)+x^{2n+2},$$E20

is a class of $\left(n+1\right)$ first integrals of Eq. (19), where $b$ is a constant and $n\ge 0$ is an integer. Then, Eq. (19) takes the form

$$\frac{d}{\mathit{dt}}{H}_1\left(x,\dot{x}\right)=\ddot{x}\left(2\dot{x}\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+3{\dot{x}}^2+x^2-1\right)+2\dot{x}\left[{\dot{x}}^2\sum _{\mathrm{\ell }=0}^n\mathrm{\ell }{x}^{2\mathrm{\ell }-1}+x\dot{x}+\left(n+1\right)x^{2n+1}\right]=0,$$E21

with the exact sinusoidal general solution

$$x\left(t\right)=\mathit{cos}\left(t+\phi \right)$$E22

where $\phi$ is an arbitrary constant.

Proof. Differentiating with respect to time, the first integral (20) immediately yields Eq. (21). To prove that formula (22) is a solution of Eq. (21), it suffices to prove that Eq. (22) verifies Eq. (20). However, it is also possible to give direct proof by substituting Eq. (22) into Eq. (21). From Eq. (22),

$$\dot{x}\left(t\right)=-sin\left(t+\phi \right),$$E23

and

$$\ddot{x}\left(t\right)=-cos\left(t+\phi \right),$$E24

Inserting Eqs. (22)–(24) and the trigonometric equation

$${cos}^2\left(t+\phi \right)+{sin}^2\left(t+\phi \right)=1,$$E25

into Eq. (21), leads to

$$\begin{array}{l}\ddot{x}\left(2\dot{x}\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+3{\dot{x}}^2+x^2-1\right)+2\dot{x}\left[{\dot{x}}^2\sum _{\mathrm{\ell }=0}^n\mathrm{\ell }{x}^{2\mathrm{\ell }-1}+x\dot{x}+\left(n+1\right)x^{2n+1}\right]\\ =\begin{array}{l}-cos\left(t+\phi \right)\left[-2sin\left(t+\phi \right)\sum _{\mathrm{\ell }=0}^n{cos}^{2\mathrm{\ell }}\left(t+\phi \right)+3{sin}^2\left(t+\phi \right)+{cos}^2\left(t+\phi \right)-1\right]\\ -2sin\left(t+\phi \right)[{sin}^2\left(t+\phi \right)\sum _{\mathrm{\ell }=0}^n\mathrm{\ell }{cos}^{2\mathrm{\ell }-1}\left(t+\phi \right)-cos\left(t+\phi \right)sin\left(t+\phi \right)\\ +\left(n+1\right){cos}^{2n+1}\left(t+\phi \right)]\end{array}\\ =\begin{array}{l}2sin\left(t+\phi \right)\sum _{\mathrm{\ell }=0}^n{cos}^{2\mathrm{\ell }+1}\left(t+\phi \right)-3cos\left(t+\phi \right){sin}^2\left(t+\phi \right)-{cos}^3\left(t+\phi \right)+cos\left(t+\phi \right)\\ -2{sin}^3\left(t+\phi \right)\sum _{\mathrm{\ell }=0}^n\mathrm{\ell }{cos}^{2\mathrm{\ell }-1}\left(t+\phi \right)+2cos\left(t+\phi \right){sin}^2\left(t+\phi \right)\\ -2\left(n+1\right)sin\left(t+\phi \right){cos}^{2n+1}\left(t+\phi \right)\end{array}\\ =\begin{array}{l}2sin\left(t+\phi \right)\sum _{\mathrm{\ell }=0}^n{cos}^{2\mathrm{\ell }+1}\left(t+\phi \right)-cos\left(t+\phi \right){sin}^2\left(t+\phi \right)-{cos}^3\left(t+\phi \right)+cos\left(t+\phi \right)\\ -2sin\left(t+\phi \right)\left[1-{cos}^2\left(t+\phi \right)\right]\sum _{\mathrm{\ell }=0}^n\mathrm{\ell }{cos}^{2\mathrm{\ell }-1}\left(t+\phi \right)-2\left(n+1\right)sin\left(t+\phi \right){cos}^{2n+1}\left(t+\phi \right)\end{array}\\ =\begin{array}{l}2sin\left(t+\phi \right)\sum _{\mathrm{\ell }=0}^n{cos}^{2\mathrm{\ell }+1}\left(t+\phi \right)-cos\left(t+\phi \right)\left[{sin}^2\left(t+\phi \right)+{cos}^2\left(t+\phi \right)-1\right]\\ -2sin\left(t+\phi \right)\sum _{\mathrm{\ell }=0}^n\mathrm{\ell }{cos}^{2\mathrm{\ell }-1}\left(t+\phi \right)+2sin\left(t+\phi \right)\sum _{\mathrm{\ell }=0}^n\mathrm{\ell }{cos}^{2\mathrm{\ell }+1}\left(t+\phi \right)\\ -2\left(n+1\right)sin\left(t+\phi \right){cos}^{2n+1}\left(t+\phi \right)\end{array}\\ =\begin{array}{l}sin\left(t+\phi \right)\sum _{\mathrm{\ell }=0}^n\left(2\mathrm{\ell }+2\right){cos}^{2\mathrm{\ell }+1}\left(t+\phi \right)-sin\left(t+\phi \right)\sum _{\mathrm{\ell }=0}^{n}2\mathrm{\ell }{cos}^{2\mathrm{\ell }-1}\left(t+\phi \right)\\ -2\left(n+1\right)sin\left(t+\phi \right){cos}^{2n+1}\left(t+\phi \right)\end{array}\\ =sin\left(t+\phi \right)\left[\sum _{\mathrm{\ell }=0}^{n-1}\left(2\mathrm{\ell }+2\right){cos}^{2\mathrm{\ell }+1}\left(t+\phi \right)-\sum _{\mathrm{\ell }=0}^{n}2\mathrm{\ell }{cos}^{2\mathrm{\ell }-1}\left(t+\phi \right)\right]\\ =\begin{array}{l}sin\left(t+\phi \right)[2cos\left(t+\phi \right)+4{cos}^3\left(t+\phi \right)+6{cos}^5\left(t+\phi \right)+\dots +2n{cos}^{2n-1}\left(t+\phi \right)\\ -\sum _{\mathrm{\ell }=0}^{n}2\mathrm{\ell }{cos}^{2\mathrm{\ell }-1}\left(t+\phi \right)]\end{array}\\ =\mathit{sin}\left(t+\phi \right)\left[\sum _{\mathrm{\ell }=0}^{n}2\mathrm{\ell }{cos}^{2\mathrm{\ell }-1}\left(t+\phi \right)-\sum _{\mathrm{\ell }=0}^{n}2\mathrm{\ell }{cos}^{2\mathrm{\ell }-1}\left(t+\phi \right)\right]\\ =0,\end{array}$$E26

such that Theorem 1.1 is proved.

Remark 1. The $\left(n+1\right)$ first integrals given by Eq. (20) are the $\left(n+1\right)$ Hamiltonians of the $\left(n+1\right)$ equations given by the class of Eq. (21). Theorem 1.1 shows that $H_1\left(x,\dot{x}\right)$ is a time-independent constant. One can check that $H_1\left(x,\dot{x}\right)=1$ under $x\left(t\right)=cos\left(t+\phi \right)$. Therefore, the $\left(n+1\right)$ systems of differential Eq. (21) are isochronous and exactly reproduce the dynamics of the harmonic oscillator. These results are impossible to predict by the qualitative theory of dynamic systems, mainly by the classical existence theorems [5, 6]. Indeed, the class of Eq. (21) can be rewritten as

$$\ddot{x}+\frac{2\left[{\dot{x}}^2{\sum }_{\mathrm{\ell }=0}^n\mathrm{\ell }{x}^{2\mathrm{\ell }-1}+x\dot{x}+\left(n+1\right)x^{2n+1}\right]}{3{\dot{x}}^2+x^2-1+2\dot{x}{\sum }_{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}}\dot{x}=0$$E27

Eq. (27) has the form of the mixed Lienard-type differential Eq. (11), where

$$h\left(x,\dot{x}\right)=\frac{2\left[{\dot{x}}^2{\sum }_{\mathrm{\ell }=0}^n\mathrm{\ell }{x}^{2\mathrm{\ell }-1}+x\dot{x}+\left(n+1\right)x^{2n+1}\right]}{3{\dot{x}}^2+x^2-1+2\dot{x}{\sum }_{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}},$$E28

and

$$g\left(x\right)=0.$$E29

Since $h\left(0,0\right)=0$ is not negative, $g\left(x\right)=0$ for $x\ne 0$ and $g\left(x\right)$ is not odd, then Eq. (21) does not satisfy the classical theorems for the existence of at least one periodic solution (see Theorem $11.2$ of ([5], p. 387) and the Lienard-Levinson-Smith theorem of [6]) in contrast to Theorem 1.1. As an example of illustration, let $n=0$. Thus, Eq. (27) becomes

$$\ddot{x}+\frac{2x\left(1+\dot{x}\right)}{3{\dot{x}}^2+2\dot{x}+x^2-1}\dot{x}=0$$E30

The phase portrait and vector field of Eq. (27) are shown in Figures 2 and 3 for $n=1$ and $n=2$. Consider now the class of the first-order differential equation

Figure 2.

Phase portrait and vector field of Eq. (27) for $n=1$.

Figure 3.

Phase portrait and vector field of Eq. (27) for $n=2$.

$$H_2\left(x,\dot{x}\right)=b={\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3\left(1+\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }+2}\right)+\dot{x}\left(x^{2n+4}-1\right)+x^{2n+2},$$E31

where $n\ge 0$ is an integer and $b$ is a constant. Therefore, we have the following theorem.

Theorem 1.2: If Eq. (31) is a first integral or Hamiltonian of Eq. (19), then Eq. (19) can be written as

$$\begin{array}{ll}& \ddot{x}\left[3{\dot{x}}^2\left(1+\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }+2}\right)+2\dot{x}\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+x^{2n+4}-1\right]\\ & +\left\{{\dot{x}}^3\sum _{\mathrm{\ell }=0}^n\left(2\mathrm{\ell }+2\right)x^{2\mathrm{\ell }+1}+{\dot{x}}^2\sum _{\mathrm{\ell }=0}^{n}2\mathrm{\ell }{x}^{2\mathrm{\ell }-1}+\left[\left(2n+4\right)x^{2n+3}\right]\dot{x}+\left(2n+2\right)x^{2n+1}\right\}\dot{x}=0,\end{array}$$E32

with the exact general solution

$$x\left(t\right)=cos\left(t+\phi \right).$$E33

Proof. By differentiation with respect to time, Eq. (31) immediately leads to Eq. (32). It suffices to show that Eq. (33) verifies Eq. (31) to prove that formula (33) is a solution of Eq. (32). Substituting Eqs. (22)–(25) into Eq. (31) yields

$$\begin{array}{rcl}{H}_2\left(x,\dot{x}\right)& =& {sin}^2\left(t+\phi \right)\sum _{\mathrm{\ell }=0}^n{cos}^{2\mathrm{\ell }}\left(t+\phi \right)-{sin}^3\left(t+\phi \right)\left[1+\sum _{\mathrm{\ell }=0}^n{cos}^{2\mathrm{\ell }+2}\left(t+\phi \right)\right]\\ & & -sin\left(t+\phi \right)\left[{cos}^{2n+4}\left(t+\phi \right)-1\right]+{cos}^{2n+2}\left(t+\phi \right)\\ & =& \left[1-{cos}^2\left(t+\phi \right)\right]\sum _{\mathrm{\ell }=0}^n{cos}^{2\mathrm{\ell }}\left(t+\phi \right)-sin\left(t+\phi \right)\left[1-{cos}^2\left(t+\phi \right)\right]\left[1+\sum _{\mathrm{\ell }=0}^n{cos}^{2\mathrm{\ell }+2}\left(t+\phi \right)\right]\\ & & -sin\left(t+\phi \right)\left[{cos}^{2n+4}\left(t+\phi \right)-1\right]+{cos}^{2n+2}\left(t+\phi \right)\\ & =& \sum _{\mathrm{\ell }=0}^n{cos}^{2\mathrm{\ell }}\left(t+\phi \right)-\sum _{\mathrm{\ell }=0}^n{cos}^{2\mathrm{\ell }+2}\left(t+\phi \right)-sin\left(t+\phi \right)\\ & & -sin\left(t+\phi \right)\sum _{\mathrm{\ell }=0}^n{cos}^{2\mathrm{\ell }+2}\left(t+\phi \right)+sin\left(t+\phi \right){cos}^2\left(t+\phi \right)\\ & & +sin\left(t+\phi \right)\sum _{\mathrm{\ell }=0}^n{cos}^{2\mathrm{\ell }+4}\left(t+\phi \right)-sin\left(t+\phi \right){cos}^{2n+4}\left(t+\phi \right)+sin\left(t+\phi \right)\\ & & +{cos}^{2n+2}\left(t+\phi \right)\\ & =& sin\left(t+\phi \right)[{cos}^2\left(t+\phi \right)+\sum _{\mathrm{\ell }=0}^n{cos}^{2\mathrm{\ell }+4}\left(t+\phi \right)-\sum _{\mathrm{\ell }=0}^n{cos}^{2\mathrm{\ell }+2}\left(t+\phi \right)\\ & & -{cos}^{2n+4}\left(t+\phi \right)]+\sum _{\mathrm{\ell }=0}^n{cos}^{2\mathrm{\ell }}\left(t+\phi \right)-\sum _{\mathrm{\ell }=0}^n{cos}^{2\mathrm{\ell }+2}\left(t+\phi \right)+{cos}^{2n+2}\left(t+\phi \right)\\ & =& sin\left(t+\phi \right)[{cos}^2\left(t+\phi \right)+{cos}^{2n+4}\left(t+\phi \right)+\sum _{\mathrm{\ell }=0}^{n-1}{cos}^{2\mathrm{\ell }+4}\left(t+\phi \right)-{cos}^2\left(t+\phi \right)\\ & & -\sum _{\mathrm{\ell }=1}^n{cos}^{2\mathrm{\ell }+2}\left(t+\phi \right)-{cos}^{2n+4}\left(t+\phi \right)]+\sum _{\mathrm{\ell }=0}^n{cos}^{2\mathrm{\ell }}\left(t+\phi \right)\\ & & -\sum _{\mathrm{\ell }=0}^{n-1}{cos}^{2\mathrm{\ell }+2}\left(t+\phi \right)-{cos}^{2n+2}\left(t+\phi \right)+{cos}^{2n+2}\left(t+\phi \right)\\ & =& sin\left(t+\phi \right)\left[\sum _{\mathrm{\ell }=0}^{n-1}{cos}^{2\mathrm{\ell }+4}\left(t+\phi \right)-\sum _{\mathrm{\ell }=1}^n{cos}^{2\mathrm{\ell }+2}\left(t+\phi \right)\right]\\ & & +\left[1+\sum _{\mathrm{\ell }=1}^n{cos}^{2\mathrm{\ell }}\left(t+\phi \right)-\sum _{\mathrm{\ell }=0}^{n-1}{cos}^{2\mathrm{\ell }+2}\left(t+\phi \right)\right]\\ & =& sin\left(t+\phi \right)[{cos}^4\left(t+\phi \right)+{cos}^6\left(t+\phi \right)+\dots +{cos}^{2n+2}\left(t+\phi \right)\\ & & -\sum _{\mathrm{\ell }=1}^n{cos}^{2\mathrm{\ell }+2}\left(t+\phi \right)]+1+[\sum _{\mathrm{\ell }=1}^n{cos}^{2\mathrm{\ell }}\left(t+\phi \right)\\ & & -\left({cos}^2\left(t+\phi \right)+{cos}^4\left(t+\phi \right)+\dots +{cos}^{2n}\left(t+\phi \right)\right)]\\ & =& 1+sin\left(t+\phi \right)\left[\sum _{\mathrm{\ell }=1}^n{cos}^{2\mathrm{\ell }+2}\left(t+\phi \right)-\sum _{\mathrm{\ell }=1}^n{cos}^{2\mathrm{\ell }+2}\left(t+\phi \right)\right]\\ & & +\left[\sum _{\mathrm{\ell }=1}^n{cos}^{2\mathrm{\ell }}\left(t+\phi \right)-\sum _{\mathrm{\ell }=1}^n{cos}^{2\mathrm{\ell }}\left(t+\phi \right)\right]\\ & =& 1,\end{array}$$E34

proving Theorem 1.2.

Remark 2. Eq. (32) takes the form

$$\ddot{x}+\frac{\left\{{\dot{x}}^3{\sum }_{\mathrm{\ell }=0}^n\left(2\mathrm{\ell }+2\right)x^{2\mathrm{\ell }+1}+{\dot{x}}^2{\sum }_{\mathrm{\ell }=0}^{n}2\mathrm{\ell }{x}^{2\mathrm{\ell }-1}+\left[\left(2n+4\right)x^{2n+3}\right]\dot{x}+\left(2n+2\right)x^{2n+1}\right\}}{\left[3{\dot{x}}^2\left(1+{\sum }_{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }+2}\right)+2\dot{x}{\sum }_{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+x^{2n+4}-1\right]}\dot{x}=0,$$E35

Obviously, Eq. (35) has the form of Eq. (27), so the classical existence theorems cannot predict its general solution (33). As previously stated, Eq. (35) contains $\left(n+1\right)$ nonlinear isochronous Hamiltonian oscillators. One can verify that $H_2\left(x,\dot{x}\right)=1$, when $x\left(t\right)=cos\left(t+\phi \right)$. As an example of Eq. (35), put $n=0$. Then, Eq. (35) becomes

$$\ddot{x}+\frac{2x{\dot{x}}^3+\left(4x^3+2x\right)\dot{x}}{3{\dot{x}}^2\left(1+x^2\right)+2\dot{x}+x^4+x^2-1}\dot{x}=0.$$E36

The phase diagram and vector field of Eq. (36) are shown in Figure 4. Figures 5 and 6 show the phase portrait and vector field of Eq. (35) for $n=1$ and $n=2$.

Figure 4.

Phase portrait and vector field of Eq. (36).

Figure 5.

Phase portrait and vector field of Eq. (35) for $n=1$.

Figure 6.

Phase portrait and vector field of Eq. (35) for $n=2$.

3.2 Higher-order nonlinear equation

Nonlinear systems have been extensively investigated from the perspective of chaotic behavior. Chaos in higher-order systems has been widely studied in the literature since they are subject in general to a dramatic change in their qualitative behavior under a small change in initial conditions. Consequently, the determination of exact explicit solutions has been less explored in the literature. It follows the high importance of finding higher-order systems that admit a general solution with a regular predictable behavior when the initial conditions change. In this regard, higher-order systems having a sinusoidal general solution such as the harmonic oscillator cannot, in an analytic way, exhibit chaotic behavior. We, therefore, focus on these systems in this part. It is obvious that [30, 32], if

$$H\left(x,\dot{x}\right)=b,$$E37

where $b$ is a constant, and

$$x\left(t\right)=cos\left(t+\phi \right)$$E38

then

$$\frac{d^m}{{\mathit{dt}}^m}\left[H\left(x,\dot{x}\right)-b\right]=0,$$E39

where $m\ge 0$ is an integer, with the exact solution (38). Indeed

$$\frac{d^m}{{\mathit{dt}}^m}\left[H\left(x,\dot{x}\right)-b\right]=\frac{d^{m-1}}{{\mathit{dt}}^{m-1}}\left\{\frac{d}{\mathit{dt}}\left[H\left(x,\dot{x}\right)-b\right]\right\}=0.$$E40

Therefore, the following theorems have been proven.

Theorem 1.3: Consider the Hamiltonian (20). Then, the equation

$$\frac{d^m}{{\mathit{dt}}^m}\left[{\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3+\dot{x}\left(x^2-1\right)+x^{2n+2}-1\right]=0,$$E41

where $b=+1$, has the general solution

$$x\left(t\right)=cos\left(t+\phi \right).$$E42

Remark 3. Eq. (41) is a class of $\left(n+1\right)$ nonlinear $\left(m+1\right)\mathit{th}$ order autonomous systems that can exhibit isochronous oscillations.

Theorem 1.4: Consider the Hamiltonian or first integral (31), where $b=1$. Then, equation

$$\frac{d^m}{{\mathit{dt}}^m}\left[{\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3\left(1+\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }+2}\right)+\dot{x}\left(x^{2n+4}-1\right)+x^{2n+2}\right]=0,$$E43

possesses the general and harmonic isochronous solution

$$x\left(t\right)=cos\left(t+\phi \right).$$E44

Remark 4. Eq. (43) is a class of $\left(n+1\right)$ nonlinear $\left(m+1\right)\mathit{th}$ order autonomous systems that can exhibit isochronous oscillations. It is interesting to note that the constant $\phi$ can be determined by using two initial conditions

$$x\left(t=0\right)=x_0,\dot{x}\left(t=0\right)=v_0$$E45

whereas the Cauchy initial value problem requires $q$ initial conditions for $\mathit{qth}$ order systems of differential equations. Additionally, we can prove the following results.

Theorem 1.5: Let

$$\frac{d^m}{{\mathit{dt}}^m}\left\{\left[{\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3+\dot{x}\left(x^2-1\right)+x^{2n+2}-1\right]x\right\}=0.$$E46

Then, Eq. (46) has the general and exact isochronous harmonic solution

$$x\left(t\right)=cos\left(t+\phi \right).$$E47

Proof. Eq. (46) can be rewritten in the form

$$\begin{array}{l}\left[{\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3+\dot{x}\left(x^2-1\right)+x^{2n+2}-1\right]\frac{d^{m}x}{{\mathit{dt}}^m}\\ +x\frac{d^m}{{\mathit{dt}}^m}\left[{\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3+\dot{x}\left(x^2-1\right)+x^{2n+2}-1\right]=0.\end{array}$$E48

Since ${\dot{x}}^2{\sum }_{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3+\dot{x}\left(x^2-1\right)+x^{2n+2}-1=0$, when $x\left(t\right)=cos\left(t+\phi \right)$, the first term of Eq. (48) is zero. The second term is also zero under Theorem 1.3. Thus, Theorem 1.5 is proved.

Theorem 1.6: Let

$$\frac{d^m}{{\mathit{dt}}^m}\left\{\left[{\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3+\dot{x}\left(x^2-1\right)+x^{2n+2}-1\right]e^x\right\}=0.$$E49

Then, Eq. (49) admits the general and exact isochronous sinusoidal solution

$$x\left(t\right)=cos\left(t+\phi \right)$$E50

Proof. Writing Eq. (49) yields

$$\begin{array}{l}\left[{\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3+\dot{x}\left(x^2-1\right)+x^{2n+2}-1\right]e^x\\ +e^x\frac{d^m}{{\mathit{dt}}^m}\left[{\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3+\dot{x}\left(x^2-1\right)+x^{2n+2}-1\right]=0.\end{array}$$E51

allows us to note that the second term is zero under Theorem 1.3. Since ${\dot{x}}^2{\sum }_{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3+\dot{x}\left(x^2-1\right)+x^{2n+2}-1=0$ for $x\left(t\right)=cos\left(t+\phi \right)$, the first term of Eq. (51) is equal to zero, so Theorem 1.6 is proved.

Theorem 1.7: Let

$$\frac{d^m}{{\mathit{dt}}^m}\left\{\left[{\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3\left(1+\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }+2}\right)+\dot{x}\left(x^{2n+4}-1\right)+x^{2n+2}-1\right]x\right\}=0$$E52

Then, Eq. (52) exhibits the general and exact isochronous harmonic solution

$$x\left(t\right)=cos\left(t+\phi \right).$$E53

Proof. Eq. (52) can take the form

$$\begin{array}{l}\left[{\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3\left(1+\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }+2}\right)+\dot{x}\left(x^{2n+4}-1\right)+x^{2n+2}-1\right]\frac{d^{m}x}{{\mathit{dt}}^m}\\ +x\frac{d^m}{{\mathit{dt}}^m}\left[{\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3\left(1+\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }+2}\right)+\dot{x}\left(x^{2n+4}-1\right)+x^{2n+2}-1\right]=0\end{array}$$E54

Since ${\dot{x}}^2{\sum }_{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3\left(1+{\sum }_{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }+2}\right)+\dot{x}\left(x^{2n+4}-1\right)+x^{2n+2}-1=0$ when $x\left(t\right)=cos\left(t+\phi \right)$, the first term of Eq. (54) is equal to zero. The second term is equal to zero under Theorem 1.4. Therefore, Theorem 1.7 is proved.

Theorem 1.8: Let

$$\frac{d^m}{{\mathit{dt}}^m}\left\{\left[{\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3\left(1+\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }+2}\right)+\dot{x}\left(x^{2n+4}-1\right)+x^{2n+2}-1\right]e^x\right\}=0.$$E55

Then, Eq. (55) admits the general and exact isochronous solution

$$x\left(t\right)=cos\left(t+\phi \right).$$E56

Proof. Applying the rule of differentiation of a product of two functions, Eq. (55) can be written as

$$\begin{array}{ll}& \left[{\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3\left(1+\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }+2}\right)+\dot{x}\left(x^{2n+4}-1\right)+x^{2n+2}-1\right]e^x+\\ & e^x\frac{d^m}{{\mathit{dt}}^m}\left[{\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3\left(1+\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }+2}\right)+\dot{x}\left(x^{2n+4}-1\right)+x^{2n+2}-1\right]=0.\end{array}$$E57

Under Theorem 1.4, the second term of Eq. (57) is equal to zero when $x\left(t\right)=cos\left(t+\phi \right)$. The first term is also zero when $x\left(t\right)=cos\left(t+\phi \right)$ since

$${\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3\left(1+\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }+2}\right)+\dot{x}\left(x^{2n+4}-1\right)+x^{2n+2}-1=0.$$

This completes the proof of Theorem 1.8.

Remark 5. If $H\left(x,\dot{x}\right)=b$, when $x=cos\left(t+\phi \right)$, then

$$\frac{d^m}{{\mathit{dt}}^m}\left\{\left[H\left(x,\dot{x}\right)-b\right]Q\left(x,\dot{x}\right)\right\}=0,$$E58

has the exact solution $cost$, where $Q\left(x,\dot{x}\right)\ne 0$ is a function of its arguments. Now, we can investigate higher-order nonautonomous nonlinear systems.

4. Nonautonomous nonlinear systems

In recent decades, nonautonomous systems have been the subject of intensive investigation in the literature, given their applications in physics and applied mathematics [33, 34, 35]. In particular, these systems have been used to describe time-varying parameter processes in many areas of physical and life sciences [33, 35]. Nonautonomous systems are generally investigated within the framework of the qualitative theory of differential equations. The Lyapunov method is often used to study the stability, boundedness, and conditions for the existence of periodic solutions of these systems. However, the recent literature shows that the classical existence theorems are not sufficient to predict the behavior of nonlinear dynamic systems. Additionally, qualitative results are not sufficient for engineering and industrial applications [23]. By definition [34, 35], a nonautonomous dynamic system is distinguished from an autonomous system by the fact that the solution of the associated initial value problem depends not only on the elapsed time $t-t_0$ but also on the initial time $t_0$. In this part, we prove the existence of nonautonomous dynamic systems whose solution to the initial value problem does not depend on the initial time $t_0$. To that end, we have the following result.

Theorem 1.9: Consider the Hamiltonian $H\left(x,\dot{x}\right)$ such that

$$H\left(x,\dot{x}\right)=b,$$E59

when $x=cos\left(t+\phi \right)$, where $b$ and $\phi$ are constants. Then, the nonautonomous equation

$$\frac{d^m}{{\mathit{dt}}^m}\left[H\left(x,\dot{x}\right)-b\right]Q\left(t\right)=0,$$E60

has the general and exact isochronous sinusoidal solution

$$x\left(t\right)=cos\left(t+\phi \right),$$E61

where $Q\left(t\right)\ne 0$ is a function of $t$.

Proof. Using the rule of differentiation of a product of two functions, we can rewrite Eq. (60) in the form

$$\left(H-b\right)\frac{d^{m}Q\left(t\right)}{{\mathit{dt}}^m}+Q\left(t\right)\frac{d^m}{{\mathit{dt}}^m}\left(H-b\right)=0.$$E62

From Eq. (59), the first term of Eq. (62) is equal to zero for $x\left(t\right)=cos\left(t+\phi \right)$. Now $\frac{d^m}{{\mathit{dt}}^m}\left(H-b\right)=\frac{d^{m-1}}{{\mathit{dt}}^{m-1}}\left[\frac{d}{\mathit{dt}}\left(H-b\right)\right]$ so that $\left[\frac{d}{\mathit{dt}}\left(H-b\right)\right]=\frac{\mathit{dH}}{\mathit{dt}}=0$, using Eq. (59) when $x\left(t\right)=cos\left(t+\phi \right)$. This completes the proof of Theorem 1.9.

4.1 Examples of illustration

4.1.1 Example 1

Let us consider Eq. (20), where $b=1$. Then, the nonlinear $\left(m+1\right)\mathit{th}$ order nonautonomous equation

$$\frac{d^m}{{\mathit{dt}}^m}\left\{\left[{\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3+\dot{x}\left(x^2-1\right)+x^{2n+2}-1\right]cost\right\}=0,$$E63

exhibits isochronous oscillations corresponding to the general and exact sinusoidal solution

$$x\left(t\right)=cos\left(t+\phi \right).$$E64

Solution (64) is also the general and exact isochronous sinusoidal solution of the harmonic oscillator

$$\ddot{x}+x=0.$$E65

Thus, the constant $\phi$ can be determined using two initial conditions

$$x\left(t=0\right)=x_0,\dot{x}\left(t=0\right)=v_0$$E66

such that, as is well-known, solution (64) does not depend on the initial time $t_0$.

4.1.2 Example 2

Consider Eq. (31) where $b=1$. Then, we have the nonlinear $\left(m+1\right)\mathit{th}$ order nonautonomous equation

$$\frac{d^m}{{\mathit{dt}}^m}\left\{\left[{\dot{x}}^2\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }}+{\dot{x}}^3\left(1+\sum _{\mathrm{\ell }=0}^n{x}^{2\mathrm{\ell }+2}\right)+\dot{x}\left(x^{4n+2}-1\right)+x^{2n+2}-1\right]cost\right\}=0,$$E67

that can exhibit isochronous sinusoidal oscillations with the general and exact solution $cos\left(t+\phi \right)$. It is interesting to note that the classes of Eqs. (63) and (67) contain $\left(n+1\right)$ nonlinear $\left(m+1\right)\mathit{th}$ order nonautonomous systems that reproduce in an exact way the isochronous harmonic oscillations of the harmonic oscillator. In this context, we can present a conclusion for the chapter.

5. Conclusion

In this chapter, we explicitly proved some results concerning isochronous sinusoidal oscillations of nonlinear systems. These results contribute to recent developments and major advances in the field of second-order and higher-order autonomous and nonautonomous nonlinear dynamic system theory.