Study of a simple 3D quadratic system with homoclinic flip bifurcations of inward twist case C

Algaba, Antonio; Domínguez-Moreno, M. Cinta; Merino, Manuel; Rodríguez-Luis, Alejandro J.

doi:10.1016/j.cnsns.2019.05.005

Cited by 6 publications

(14 citation statements)

References 56 publications

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“…[4][5][6][7][8][9] For the three-dimensional case whose Jacobian matrix at the equilibrium has a pair of purely imaginary eigenvalues and a negative eigenvalue, there are few methods; see Edneral et al, Sang et al, and Wang et al [10][11][12] For the four-dimensional case whose Jacobian matrix at the equilibrium has a pair of purely imaginary eigenvalues and a pair of complex conjugate eigenvalues with negative real parts, see Sang et al 13 For the n-dimensional case whose Jacobian matrix at the equilibrium has a pair of purely imaginary eigenvalues and n − 2 eigenvalues with nonzero real parts, see previous studies. 5,9,[14][15][16][17][18] For a n-dimensional system whose Jacobian matrix at the equilibrium has a pair of purely imaginary eigenvalues and n − 2 eigenvalues with negative real parts, let us denote the kth focus quantity by V k . If V k = 0 for all k ≥ 1, then the equilibrium is a center for the flow on the center manifold.…”

Section: Introductionmentioning

confidence: 99%

“…For the two‐dimensional case whose Jacobian matrix at the equilibrium has a pair of purely imaginary eigenvalues, there are few methods; see Giné and Santallusia, Kuznetsov, Liu et al, Romanovski and Shafer, Sang, and Yu and Chen 4‐9 . For the three‐dimensional case whose Jacobian matrix at the equilibrium has a pair of purely imaginary eigenvalues and a negative eigenvalue, there are few methods; see Edneral et al, Sang et al, and Wang et al 10‐12 For the four‐dimensional case whose Jacobian matrix at the equilibrium has a pair of purely imaginary eigenvalues and a pair of complex conjugate eigenvalues with negative real parts, see Sang et al 13 For the n ‐dimensional case whose Jacobian matrix at the equilibrium has a pair of purely imaginary eigenvalues and n −2 eigenvalues with nonzero real parts, see previous studies 5,9,14‐18 …”

Section: Introductionmentioning

confidence: 99%

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Focus quantities with applications to some finite‐dimensional systems

Sang

2020

Math Methods in App Sciences

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In studying small limit cycles of finite-dimensional systems, one of the central problem is the computation of focus quantities. In practice, the computation is a challenging problem even for some simple low-dimensional systems. This paper is devoted to the computation of focus quantities of all orders and to the study of Hopf bifurcations in some chaotic systems. A recursive formula for computing focus quantities is presented for a K + 2-dimensional system. The formula is a generalization of previous results on low-dimensional systems with K = 0 and K = 1. For a four-dimensional hyper-chaotic system, according to the sign of the first focus quantity, we prove that the simple Hopf bifurcation of the system is supercritical. For a five-dimensional chaotic system with four equilibria of Hopf type, according to the signs of the first focus quantities, we prove that the simple Hopf bifurcations of the system are subcritical.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Focus quantities with applications to some finite‐dimensional systems

Sang

2020

Math Methods in App Sciences

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“…This has changed very recently, when Algaba, Domínguez-Moreno, Merino, and Rodríguez-Luis [3] found an example of a three-dimensional quadratic system with an inward-twisted homoclinic flip bifurcation. More precisely, they presented the system…”

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confidence: 99%

“…Note that the homoclinic orbit is not to the origin but to the equilibrium p = (x 1 , x 2 , x 3 ) = (b/4, b 2 /16, −16 a/b 2 ), which exists provided b = 0. Algaba et al [3] studied the local bifurcation structure near C in in quite some detail. We are interested here in how the unfolding of C in is embedded more globally in an overall bifurcation diagram.…”

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confidence: 99%

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Computing connecting orbits to infinity associated with a homoclinic flip bifurcation

Giraldo¹,

Krauskopf²,

Osinga³

2020

Journal of Computational Dynamics

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We consider the bifurcation diagram in a suitable parameter plane of a quadratic vector field in R 3 that features a homoclinic flip bifurcation of the most complicated type. This codimension-two bifurcation is characterized by a change of orientability of associated two-dimensional manifolds and generates infinite families of secondary bifurcations. We show that curves of secondary n-homoclinic bifurcations accumulate on a curve of a heteroclinic bifurcation involving infinity.We present an adaptation of the technique known as Lin's method that enables us to compute such connecting orbits to infinity. We first perform a weighted directional compactification of R 3 with a subsequent blow-up of a non-hyperbolic saddle at infinity. We then set up boundary-value problems for two orbit segments from and to a common two-dimensional section: the first is to a finite saddle in the regular coordinates, and the second is from the vicinity of the saddle at infinity in the blown-up chart. The so-called Lin gap along a fixed one-dimensional direction in the section is then brought to zero by continuation. Once a connecting orbit has been found in this way, its locus can be traced out as a curve in a parameter plane.

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New insights into a chaotic system with only a Lyapunov stable equilibrium

Chen

Liu

Wei

et al. 2020

Math Methods in App Sciences

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This paper gives some new insights into a chaotic system. The considered system has only one Lyapunov stable equilibrium and positive Lyapunov exponent in some certain parameter range, which means there coexists chaotic attractors and Lyapunov stable equibirium in system. First, from the perspective of analyzing the global structure of the system, based on the Poincaré compactification technique, the complete description of its dynamical behavior on the sphere at infinity is presented. The obtaining results show that its global dynamical behavior is very complex. Second, from a geometric perspective, the Jacobi stability of the system is investigated, including the unique equilibrium and a periodic orbit. Interestingly, we find that the unique Lyapunov stable equilibrium is always Jacobi unstable, a Lyapunov stable periodic orbit falls into both Jacobi stable and Jacobi unstable regions. It is shown that we might witness chaotic behavior of the system before the trajectories enter a neighborhood of the equilibrium point. It is hoped that the investigation of this paper can contribute to better understand and study the complex chaotic system with stable equilibria.

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Study of a simple 3D quadratic system with homoclinic flip bifurcations of inward twist case C

Cited by 6 publications

References 56 publications

Focus quantities with applications to some finite‐dimensional systems

Focus quantities with applications to some finite‐dimensional systems

Computing connecting orbits to infinity associated with a homoclinic flip bifurcation

New insights into a chaotic system with only a Lyapunov stable equilibrium

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