Integrable Deformations and Dynamical Properties of Systems with Constant Population

Cristian Lăzureanu

doi:10.3390/math9121378

Integrable Deformations and Dynamical Properties of Systems with Constant Population

Mathematics ◽

10.3390/math9121378 ◽

2021 ◽

Vol 9 (12) ◽

pp. 1378

Author(s):

Cristian Lăzureanu

Keyword(s):

Differential Equations ◽

Three Dimensional ◽

Poisson Geometry ◽

Volterra System ◽

First Order ◽

Integrable Deformations ◽

Dynamical Properties ◽

Points Of View ◽

First Order Differential Equations

In this paper we consider systems of three autonomous first-order differential equations x˙=f(x),x=(x,y,z),f=(f1,f2,f3) such that x(t)+y(t)+z(t) is constant for all t. We present some Hamilton–Poisson formulations and integrable deformations. We also analyze the case of Kolmogorov systems. We study from some standard and nonstandard Poisson geometry points of view the three-dimensional Lotka–Volterra system with constant population.

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The Hopf bifurcation theorem in three dimensions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410005413x ◽

1977 ◽

Vol 82 (3) ◽

pp. 469-483 ◽

Cited By ~ 4

Author(s):

Peter Swinnerton-Dyer

Keyword(s):

Hopf Bifurcation ◽

Singular Point ◽

Differential Equations ◽

Limit Cycle ◽

Three Dimensional ◽

Three Dimensions ◽

Bifurcation Theorem ◽

First Order ◽

Range Of Values ◽

First Order Differential Equations

AbstractThe Hopf bifurcation theorem describes the creation of a limit cycle from an isolated singular point of a system of first-order differential equations depending on a parameter. This paper describes a method for determining explicitly a range of values of the parameter throughout which the Hopf configuration continues to exist; only the three-dimensional case is described in this paper, but the method can be generalized.

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Asymptotics for third-order nonlinear differential equations: Non-oscillatory and oscillatory cases

Asymptotic Analysis ◽

10.3233/asy-211710 ◽

2021 ◽

pp. 1-19

Author(s):

Calogero Vetro ◽

Dariusz Wardowski

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Order Differential Equation ◽

Nonlinear Differential Equations ◽

Oscillatory Behavior ◽

Third Order ◽

First Order ◽

Third Order Differential Equation ◽

Comparison Technique ◽

First Order Differential Equations

We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.

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