Modified Van der Pol-Helmholtz oscillator equation with exact harmonic solutions

AKPLOGAN, Abdou Raimi Olagnika; Adjaï, K. K. D.; Akande, J.; Avossevou, Gabriel Y. H.; Monsia, M. D.

doi:10.21203/rs.3.rs-1229125/v1

Cited by 2 publications

(4 citation statements)

References 7 publications

(17 reference statements)

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“…The previous analysis also holds for Equation (3.4). The above shows that the classical theorems for the existence of isochronous centres clearly exclude a number of Lienard equations, as seen in several previous papers [7,8].…”

Section: 3supporting

confidence: 75%

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Unusual Nonpolynomial Van der Pol Oscillator Equations With Exact Harmonic and Isochronous Solutions

Adjaï¹,

Nonti²,

Akande³

et al. 2022

Int. J. Anal. Appl.

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We do not know Van der Pol-type equations with nonlinear restoring force having explicitly an exact periodic solution. We present, for the first time, nonpolynomial Van der Pol oscillator equations that do not satisfy the classical existence theorems. We exhibit their exact harmonic and isochronous solutions and prove the existence of limit cycles by using averaging theory. We also present first integrals and exact solutions of polynomial Van der Pol-Duffing equations to show that they do not have any limit cycle. Additionally, we prove that the damped Duffing-type equations are equivalent to the conservative Duffing equations exhibiting nonoscillatory solutions.

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Section: 3supporting

confidence: 75%

“…studied by Akplogan et al [8], where the parameters q i , i = 1, ...., 7 are arbitrary constants. In this situation, let us consider the following equation with Van der Pol damping called Van der Pol-type equation:…”

Section: Introductionmentioning

confidence: 99%

Unusual Nonpolynomial Van der Pol Oscillator Equations With Exact Harmonic and Isochronous Solutions

Adjaï¹,

Nonti²,

Akande³

et al. 2022

Int. J. Anal. Appl.

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“…( 20) are the n þ 1 ðÞ Hamiltonians of the n þ 1 ðÞ equations given by the class of Eq. (21). Theorem 1.1 shows that H 1 x, _ x ðÞ is a time-independent constant.…”

Section: Remarkmentioning

confidence: 96%

“…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.…”

Section: Introductionmentioning

confidence: 99%

Isochronous Oscillations of Nonlinear Systems

Akande¹,

Adjaï²,

Nonti³

et al. 2023

Nonlinear Systems - Recent Developments and Advances

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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.

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Modified Van der Pol-Helmholtz oscillator equation with exact harmonic solutions

Cited by 2 publications

References 7 publications

Unusual Nonpolynomial Van der Pol Oscillator Equations With Exact Harmonic and Isochronous Solutions

Unusual Nonpolynomial Van der Pol Oscillator Equations With Exact Harmonic and Isochronous Solutions

Isochronous Oscillations of Nonlinear Systems

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