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13 pages, 517 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

First and Second Integrals of Hopf–Langford-Type Systems

by Vassil M. Vassilev and Svetoslav G. Nikolov

Axioms 2025, 14(1), 8; https://doi.org/10.3390/axioms14010008 - 27 Dec 2024

Viewed by 798

The work examines a seven-parameter, three-dimensional, autonomous, cubic nonlinear differential system. This system extends and generalizes the previously studied quadratic nonlinear Hopf–Langford-type systems. First, by introducing cylindrical coordinates in its phase space, we show that the regarded system can be reduced to a two-dimensional Liénard system, which corresponds to a second-order Liénard equation. Then, we present (in explicit form) polynomial first and second integrals of Liénard systems of the considered type identifying those values of their parameters for which these integrals exist. It is also proved that a generic Liénard equation is factorizable if and only if the corresponding Liénard system admits a second integral of a special form. It is established that each Liénard system corresponding to a Hopf–Langford system of the considered type admits such a second integral, and hence, the respective Liénard equation is factorizable. Full article

(This article belongs to the Special Issue Complex Networks and Dynamical Systems)

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19 pages, 1541 KiB

Open AccessArticle

The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron

by Akanksha Verma, Wojciech Sumelka and Pramod Kumar Yadav

Symmetry 2023, 15(9), 1753; https://doi.org/10.3390/sym15091753 - 13 Sep 2023

Cited by 5 | Viewed by 1419

Abstract

This paper proposes an approximation algorithm based on the Legendre and Chebyshev artificial neural network to explore the approximate solution of fractional Lienard and Duffing equations with a Caputo fractional derivative. These equations show the oscillating circuit and generalize the spring–mass device equation. The proposed approach transforms the given nonlinear fractional differential equation (FDE) into an unconstrained minimization problem. The simulated annealing (SA) algorithm minimizes the mean square error. The proposed techniques examine various non-integer order problems to verify the theoretical results. The numerical results show that the proposed approach yields better results than existing methods. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

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9 pages, 233 KiB

Open AccessArticle

Existence of Periodic Solutions for a Class of the Generalized Liénard Equations

by María Teresa de Bustos, Zouhair Diab, Juan Luis G. Guirao, Miguel A. López and Raquel Martínez

Symmetry 2022, 14(5), 944; https://doi.org/10.3390/sym14050944 - 6 May 2022

Cited by 3 | Viewed by 2251

Abstract

We study analytically the existence of periodic solutions of the generalized Liénard differential equations of the form [...] Read more.

We study analytically the existence of periodic solutions of the generalized Liénard differential equations of the form

\ddot{x} + f (x, \dot{x}) \dot{x} + n^{2} x + g (x) = ε^{2} p_{1} (t) + ε^{3} p_{2} (t),

where n

\in N^{*}

, the functions

f, g

are of class

C^{3}, C^{4}

in a neighborhood of the origin, respectively, the functions

p_{i}

are of class

C^{0}

2 π -

periodic in the variable t, with

i =

1, 2

, and

ε

is a small parameter as usual. The mathematical tool that we have used is the averaging theory of dynamical systems of second order. Full article

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