Closed form integration of a hyperelliptic, odd powers, undamped oscillator

Scarpello, Giovanni Mingari; Ritelli, Daniele

doi:10.1007/s11012-011-9455-8

Cited by 5 publications

(4 citation statements)

References 10 publications

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“…Numerous solutions of the non-linear system (2) using a variety of techniques have been proposed including trigonometric series [9], mathematical transformations [10], Taylor series expansions [11], perturbation techniques [12,13], numeric-analytic techniques [14] and Lambert W-functions [15,16]. Also, an exact solution has previously been derived in the special case when the prey growth rate and predator decay rate are identical in magnitude, but with opposite signs, i.e.…”

Section: Normalized Equations and Single Coupling Parametermentioning

confidence: 99%

“…From Eqs. ( 10) and ( 7) together with the definition (11) of the G-function, the LV system's Hamiltonian is ( )…”

Section: Appendixmentioning

confidence: 99%

“…The positive function ( ) G η λ is a generalized hyperbolic cosine function that reaches its minimum 1 G λ = at 0 η = for any value of λ : hence its inverse function 1 G λ − exists, and, for any value of λ , Eq. (12) admits two respective positive and negthe η-function is associated with the coupling ratio λ only, invariance λ − of the G-function(11) implies that, for a given λ , any positive solution solution associated with the ratio 1 / λ , and reciprocally…”

mentioning

confidence: 99%

“…latter as an analytical function of ξ only through(12), an ap-proximate yet accurate ODE for ( ) t ξ is proposed below.Case1 λ = .The LV problem is solved exactly in the particular case 1 λ = ,[2]. The G-function(11) (omitting the index for simplicity) reduces to the hyperbolic cosine function and the energy conservation equation (12) becomes…”

mentioning

confidence: 99%

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"Hybrid Species and Closed-Form Solutions of the Lotka-Volterra Equations"

Boulnois

2023

AJBSR

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The classical Lotka-Volterra system of two coupled non-linear ordinary differential equations is expressed in terms of a single positive coupling parameter λ , ratio of the respective natural growth and decay rates of the prey and predator populations. "Hybrid-species" are introduced resulting in a novel invariant λ −Hamiltonian of two coupled first-order ODE albeit with one being linear ; a new exact, closed-form, single quadrature solution valid for any value of λ and the system's energy is derived. In the particular case 1 λ =the ODE system partially uncouples and new, exact, closed-form time-dependent solutions are derived for each individual species. In the case 1 λ ≠ an accurate practical approximation uncoupling the non-linear system is proposed; solutions are provided in terms of explicit quadratures together with analytical high energy asymptotic solutions. A novel, exact, closed-form expression of the system's oscillation period valid for any value of λ and orbital energy is derived; two fundamental properties of the period are established; for 1 λ = the period is expressed in terms of a universal energy function and shown to be the shortest.

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Section: Normalized Equations and Single Coupling Parametermentioning

confidence: 99%

“…From Eqs. ( 10) and ( 7) together with the definition (11) of the G-function, the LV system's Hamiltonian is ( )…”

Section: Appendixmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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"Hybrid Species and Closed-Form Solutions of the Lotka-Volterra Equations"

Boulnois

2023

AJBSR

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Nonlinear oscillators with real valued powers: an analytic treatment

Citterio

Talamo

2016

Meccanica

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Hypergeometric solutions to a three dimensional dissipative oscillator driven by aperiodic forces

Bocci

Scarpello²,

Ritelli

2018

Applied Mathematical Modelling

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We model the dynamical behavior of a three dimensional (3-D) dissipative oscillator consisting of a m-block whose vertical fall occurs against a spring and which can also slide horizontally on a rigid truss rotating at an assigned angular speed ω(t). The bead's z-vertical time law is obvious, whilst its x-motion along the horizontal arm is ruled by a linear differential equation we solve through the Hermite functions and the Kummer [1] confluent Hypergeometric Function (CHF) 1F1 . After the rotation θ(t) has been computed, we know completely the m-motion in a cylindrical frame reference so that some transients have then been analyzed. Finally, further effects as an inclined slide and a contact dry friction have been added to the problem, so that the motion differential equation becomes inhomogeneous: we resort to Lagrange method of variation of constants, helped by a Fourier-Bessel expansion, in order to manage the relevant intractable integrations.

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Closed form integration of a hyperelliptic, odd powers, undamped oscillator

Cited by 5 publications

References 10 publications

"Hybrid Species and Closed-Form Solutions of the Lotka-Volterra Equations"

"Hybrid Species and Closed-Form Solutions of the Lotka-Volterra Equations"

Nonlinear oscillators with real valued powers: an analytic treatment

Hypergeometric solutions to a three dimensional dissipative oscillator driven by aperiodic forces

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