Complete factorisation and analytic solutions of generalized Lotka-Volterra equations

Брениг, Леон

doi:10.1016/0375-9601(88)90920-6

Cited by 88 publications

(62 citation statements)

References 8 publications

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“…It is easy to see from (4) and (15) that the explicitness of x 1/β i is generally not preserved under QMTs. Assuming that the index of the explicit variable is i = n + 1 in (4), it is clear that a QMT with the following special C matrix preserves x n+1 as the explicit variable:…”

Section: The Effect Of Quasi-monomial Transformationsmentioning

confidence: 98%

“…The class of quasi-polynomial (QP) systems has gained a significant interest in the modeling of nonlinear dynamical systems since the majority of smooth nonlinear systems occurring in practice can be algorithmically transformed to QP form [4,12,13]. Recent research indicates that QP system-representation can be used as a very useful tool in nonlinear systems and control theory [17,20,21], too.…”

Section: Introductionmentioning

confidence: 99%

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An algorithm for determining a class of invariants in quasi-polynomial systems

Pongrácz

Szederkényi

Hangos

2006

Computer Physics Communications

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Section: The Effect Of Quasi-monomial Transformationsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

An algorithm for determining a class of invariants in quasi-polynomial systems

Pongrácz

Szederkényi

Hangos

2006

Computer Physics Communications

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“…However, in recent years this model has appeared widely in applied mathematics and in a large variety of physics topics such as laser physics, plasma physics, convective instabilities, neural networks, etc. (Minorsky 1962;Brenig 1988;Murray 1989). The integrability properties of the system (4.1) alone have been analysed by many authors; see for example Cairo & Llibre (2000) and references therein.…”

Section: Two Coupled Odes: Application (A ) Example: Two-dimensional mentioning

confidence: 99%

On the complete integrability and linearization of nonlinear ordinary differential equations. III. Coupled first-order equations

2008

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Continuing our study on the complete integrability of nonlinear ordinary differential equations (ODEs), in this paper we consider the integrability of a system of coupled first-order nonlinear ODEs of both autonomous and non-autonomous types. For this purpose, we modify the original Prelle-Singer (PS) procedure so as to apply it to both autonomous and non-autonomous systems of coupled first-order ODEs. We briefly explain the method of finding integrals of motion (time-independent as well as timedependent integrals) for two and three coupled first-order ODEs by extending the PS method. From this we try to answer some of the open questions in the original PS method. We also identify integrable cases for the two-dimensional Lotka-Volterra system and three-dimensional Rössler system as well as other examples including nonautonomous systems in a straightforward way using this procedure. Finally, we develop a linearization procedure for coupled first-order ODEs.

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“…This problem can be solved in many cases either symbolically with DSolve or numerically with NDSolve. Symbolic solutions can also be based on recent results on the explicit Taylor representation of the solutions of polynomial differential equations (Barton et al, 1971;Brenig, 1988).…”

Section: Temporal Evolution Of Relevant Concentrationsmentioning

confidence: 99%

Dynamic modeling of biochemical reactions with application to signal transduction: principles and tools using Mathematica

Tóth

Rospars

2005

Biosystems

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Modeling of biochemical phenomena is based on formal reaction kinetics. This requires the translation of the original reaction systems into sets of differential equations expressing the effects of the various reaction steps. The temporal behavior of the system is obtained by solving the differential equations. We present the main concepts on which the formal approach of these two problems is based and we show how the amount of work needed to treat them can be significantly reduced by using a mathematical program package (Mathematica). Symbolic and numerical calculations can be performed with the programs presented and graphic presentations of the behavior of the system be obtained. The basic ideas are illustrated with three examples taken from the area of signal transduction and ion signaling.

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Complete factorisation and analytic solutions of generalized Lotka-Volterra equations

Cited by 88 publications

References 8 publications

An algorithm for determining a class of invariants in quasi-polynomial systems

An algorithm for determining a class of invariants in quasi-polynomial systems

On the complete integrability and linearization of nonlinear ordinary differential equations. III. Coupled first-order equations

Dynamic modeling of biochemical reactions with application to signal transduction: principles and tools using Mathematica

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