Darboux and Analytic First Integrals of the Generalized Michelson System

Comments on ‘Global dynamics of the generalized Lorenz systems having invariant algebraic surfaces’

In this paper the complex dynamics of a smallest biochemical system model in three-dimensional systems with the reaction scheme. This model is described by a system of three nonlinear ordinary differential equations with five positive real parameters, are analyzed and studied. We present a thorough analysis of their invariant algebraic surfaces and exponential factors and investigate the integrability and nonintegrabilty of this model. Particularly, we show the non-existence of polynomial, rational, Darboux and local analytic first integrals in a neighborhood of the equilibrium. Moreover, we prove that, the model is not integrable in the sense of Bogoyavlensky in the class of rational functions.

In this paper, we study the Darboux integrability of the Rikitake system ẋ=−μx+yz, ẏ=−μy+x(z−a), ż=1−xy. More precisely, we characterize all the invariant algebraic surfaces, the exponential factors, and the polynomial, rational, and Darboux first integrals of this system according to the values of its parameters a and μ.

Invariant algebraic surfaces are commonly observed in differential systems arising in mathematical modeling of natural phenomena. In this paper, we study the integrability and dynamics of quadratic polynomial differential systems defined in [Formula: see text] having an elliptic paraboloid as an invariant algebraic surface. We obtain the normal form for these kind of systems and, by using the invariant paraboloid, we prove the existence of first integrals, exponential factors, Darboux invariants and inverse Jacobi multipliers, for suitable choices of parameter values. We characterize all the possible configurations of invariant parallels and invariant meridians on the invariant paraboloid and give necessary conditions for the invariant parallel to be a limit cycle and for the invariant meridian to have two orbits heteroclinic to a point at infinity. We also study the dynamics of a particular class of the quadratic polynomial differential systems having an invariant paraboloid, giving information about localization and local stability of finite singular points and, by using the Poincaré compactification, we study their dynamics on the Poincaré sphere (at infinity). Finally, we study the well-known Rabinovich system in the case of invariant paraboloids, performing a detailed study of its dynamics restricted to these invariant algebraic surfaces.

The famous and well-studied Lorenz system is considered a paradigm for chaotic behavior in three-dimensional continuous differential systems. After the appearance of such a system in 1963, several Lorenz-like chaotic systems have been proposed and studied in the related literature, as Rössler system, Chen-Ueta system, Rabinovich system, Rikitake system, among others. However, these systems are parameter dependent and are chaotic only for suitable combinations of parameter values. This raises the question of when such systems are not chaotic, which can be seen as a dual problem regarding chaotic systems. In this paper, we give sufficient algebraic conditions for a generalized class of Lorenz-like systems to be nonchaotic. Using the general results obtained, we give some examples of nonchaotic behavior of some classical ``chaotic'' Lorenz-like systems, including the Lorenz system itself. The nonchaotic differential systems presented here have invariant algebraic surfaces, which contain the stable (or unstable) invariant manifolds of their equilibrium points. We show that, in some cases, the deformation of these invariant manifolds through the destruction of the invariant algebraic surfaces, by perturbing the parameter values, can reorganize the global structure of the phase space, leading to a transition from nonchaotic to chaotic behavior of such differential systems.

We go beyond in the study of the integrability of the classical model of competition between three species studied by May and Leonard [19], by considering a more realistic asymmetric model. Our results show that there are no global analytic first integrals and we provide all proper rational first integrals of this extended model by classifying its invariant algebraic surfaces.

In 2011 Pehlivan proposed a three—dimensional forced—damped autonomous differential system which can display simultaneously unbounded and chaotic solutions. This untypical phenomenon has been studied recently by several authors. In this paper we study the opposite to its chaotic motion, i.e. its integrability, mainly the existence of polynomial, rational and Darboux first integrals through the analysis of its invariant algebraic surfaces and its exponential factors.

In this paper we give the normal form of all polynomial differential systems in $$\mathbb {R}^3$$ having a weighted homogeneous surface $$f=0$$ as an invariant algebraic surface and characterize among these systems those having a Darboux invariant constructed uniquely using this invariant surface. Using the obtained results we give some examples of stratified vector fields, when $$f=0$$ is a singular surface. We also apply the obtained results to study the Vallis system, which is related to the so-called El Nino atmospheric phenomenon, when it has a cone as an invariant algebraic surface, performing a dynamical analysis of the flow of this system restricted to the invariant cone and providing a stratification for this singular surface.

In this paper we use the method of characteristic curves for solving linear partial differential equations to study the invariant algebraic surfaces of the Rikitake system = -µ x + y(z + β) = -µ y + x(z-β) z = α-xy. Our main results are the following. First, we show that the cofactor of any invariant algebraic surface is of the form rz + c, where r is an integer. Second, we characterize all invariant algebraic surfaces. Moreover, as a corollary we characterize all values of the parameters for which the Rikitake system has a rational or algebraic first integral.

In a very recent paper by Deng (Z Angew Math Phys 64:1443–1449, 2013), the author claims to have successfully found all the invariant algebraic surfaces of the generalized Lorenz system, \({\dot{x} = a(y - x), \ \dot{y} = bx + cy - xz, \ \dot{z} = dz + xy}\). He provides six invariant algebraic surfaces, found according to the idea of the weight of a polynomial introduced by Swinnerton-Dyer (Math Proc Camb Philos Soc 132:385–393, 2002). Unfortunately, his result is incorrect because a seventh invariant algebraic surface is missed. Moreover, those six invariant algebraic surfaces can be obtained in a much simpler manner: Since the Lorenz system and the generalized Lorenz system are equivalent through a homothetic scaling in time and state variables (for c ≠ 0), it is trivial to obtain the corresponding results for the generalized Lorenz system from the well-known results on invariant algebraic surfaces of the Lorenz system.

On first integrals of a family of generalized Lorenz-like systems

In this paper, we study the dynamics of the FitzHugh–Nagumo system $$\dot{x}=z,\;\dot{y}=b\left( x-dy\right) ,\;\dot{z}=x\left( x-1\right) \left( x-a\right) +y+cz$$ having invariant algebraic surfaces. This system has four different types of invariant algebraic surfaces. The dynamics of the FitzHugh–Nagumo system having two of these classes of invariant algebraic surfaces have been characterized in Valls (J Nonlinear Math Phys 26:569–578, 2019). Using the quasi-homogeneous directional blow-up and the Poincaré compactification, we describe the dynamics of the FitzHugh–Nagumo system having the two remaining classes of invariant algebraic surfaces. Moreover, for these FitzHugh–Nagumo systems we prove that they do not have limit cycles.

For a three-dimensional chaotic system, little seems to be known about the perturbation of invariant algebraic surface and the center on this surface. This question is very interesting and worth investigating. This paper is devoted to analyzing the limit cycles from perturbed center (trivial and nontrivial equilibria) on the invariant algebraic surface of the unified Lorenz-type system (ULTS), which contains some common chaotic systems as its particular cases. First, based on the parameter-dependent center manifold, we obtain the approximate two-dimensional center manifold from the perturbation of invariant algebraic surface, as well as the two-dimensional system on this center manifold. Second, by applying the averaging method of third order to the above two-dimensional system, we show that under suitable perturbation of parameters of the ULTS, there is one limit cycle bifurcating from the perturbed center on the invariant algebraic surface of the ULTS, and the stability of this limit cycle is determined as well. By using the averaging method of fourth order, we show the same results with the averaging method of third order. Finally, numerical simulation is used to verify the theoretical analyses.

Global dynamics of the generalized Lorenz systems having invariant algebraic surfaces

We present a global dynamical analysis of the following quadratic differential system $$\dot{x}{=}a(y{-}x), \dot{y}\!=\!dy-xz, \dot{z}\!=\!-bz+fx^2+gxy$$ , where $$(x,y,z)\in {\mathbb {R}}^3$$ are the state variables and a, b, d, f, g are real parameters. This system has been proposed as a new type of chaotic system, having additional complex dynamical properties to the well-known chaotic systems defined in $${\mathbb {R}}^3$$ , alike Lorenz, Rossler, Chen and other. By using the Poincare compactification for a polynomial vector field in $${\mathbb {R}}^3$$ , we study the dynamics of this system on the Poincare ball, showing that it undergoes interesting types of bifurcations at infinity. We also investigate the existence of first integrals and study the dynamical behavior of the system on the invariant algebraic surfaces defined by these first integrals, showing the existence of families of homoclinic and heteroclinic orbits and centers contained on these invariant surfaces.