Integrability of the Rucklidge system

Lima, Maurício Firmino Silva; Llibre, Jaume; Valls, Clàudìa

doi:10.1007/s11071-014-1389-y

Cited by 13 publications

(9 citation statements)

References 8 publications

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“…We recall some definitions given in [5,7]. Some well-known results on the Darboux theory of integrability and analytic first integrals may be found in [1,2,5,6].…”

Section: Elementary Resultsmentioning

confidence: 99%

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Integrability Analysis of the Smallest 3D Biochemical Reaction Model

Amen¹

2023

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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.

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“…We recall some definitions given in [5,7]. Some well-known results on the Darboux theory of integrability and analytic first integrals may be found in [1,2,5,6].…”

Section: Elementary Resultsmentioning

confidence: 99%

“…We call L the cofactor of e f g . For more details see [1,7]. We recall that a Darboux first integral is a product of complex powers of invariant algebraic surfaces and exponential factors.…”

Section: Elementary Resultsmentioning

confidence: 99%

Integrability Analysis of the Smallest 3D Biochemical Reaction Model

Amen¹

2023

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“…where h is an arbitrary constant. According to Liouville theorem for the equivalent Hamiltonian system, system ( 11) is completely integrable if it has another first integral independent on the Jacobi integral (12). Executing the time transformation (see Appendix A for more details about time transformation),…”

Section: Basic Equationsmentioning

confidence: 99%

“…e first one is finding the sufficient conditions for the integrability, and this requires the construction of a sufficient number of first integrals of motion. Numerous methods can be utilized to construct the first integrals of motion such as the direct method, Darboux method, and Yehia method (see, e.g., [3][4][5][6][7][8][9][10][11][12][13]). e second one deals with obtaining the necessary conditions of the integrability (see, e.g., [14][15][16][17][18][19][20]), but we must introduce the required number of the integrals to confirm the integrability.…”

Section: Introductionmentioning

confidence: 99%

Quartic Integral in Rigid Body-Gyrostat Dynamics

Mnasri

Elmandouh

2020

Advances in Astronomy

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In this work, we investigate the problem of constructing new integrable problems in the dynamics of the rigid body rotating about its fixed point as results of the effect of a combination of potential and gyroscopic forces possessing a common symmetry axis. We introduce two new integrable problems in a rigid body dynamics that generalize some integrable problems in this field, known by names of Chaplygin and Yehia–Elmandouh.

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“…Control of the chaotic behavior of the Rucklidge system was analyzed through various control techniques in [20][21][22]. In [23], the analytic integrability of the Rucklidge system was explored. Recently, a novel method to generate double-scroll chaotic attractors from the Rucklidge system via the arctangent function series was proposed in [24].…”

Section: Introductionmentioning

confidence: 99%

Symbolic Encoding of Periodic Orbits and Chaos in the Rucklidge System

Dong

Jia

et al. 2021

Complexity

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To describe and analyze the unstable periodic orbits of the Rucklidge system, a so-called symbolic encoding method is introduced, which has been proven to be an efficient tool to explore the topological properties concealed in these periodic orbits. In this work, the unstable periodic orbits up to a certain topological length in the Rucklidge system are systematically investigated via a proposed variational method. The dynamics in the Rucklidge system are explored by using phase portrait analysis, Lyapunov exponents, and Poincaré first return maps. Symbolic encodings of the periodic orbits with two and four letters based on the trajectory topology in the phase space are implemented under two sets of parameter values. Meanwhile, the bifurcations of the periodic orbits are explored, significantly improving the understanding of the dynamics of the Rucklidge system. The multiple-letter symbolic encoding method could also be applicable to other nonlinear dynamical systems.

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Integrability of the Rucklidge system

Abstract: Abstract. We study the Darboux and the analytic integrability of the Rucklidge system.

Cited by 13 publications

References 8 publications

Integrability Analysis of the Smallest 3D Biochemical Reaction Model

Integrability Analysis of the Smallest 3D Biochemical Reaction Model

Quartic Integral in Rigid Body-Gyrostat Dynamics

Symbolic Encoding of Periodic Orbits and Chaos in the Rucklidge System

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