Strange Attractor in the Ising Model with Competing Interactions on the Cayley Tree

Yokoi, Carlos S. O.; Oliveira, Mário J. de; Salinas, S. R.

doi:10.1103/physrevlett.54.163

Cited by 80 publications

(53 citation statements)

References 13 publications

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“…Besides the fact that, on such graphs, renormalization procedures can lead to exact results [3], they have been explored as models for systems that are not translational invariant, neither in the positions of the spins nor in the coupling constants mediating the interactions between them. In this respect, the analysis of disordered and aperiodic models on scale invariant graphs, which include hierarchical lattices [4,5,6], Cayley trees [7] or Sierpinski gaskets and carpets [8,9], have provided valuable insight into the behavior of critical phenomena of non homogeneous systems on Euclidean lattices.…”

Section: Introductionmentioning

confidence: 99%

Magnetic models on Apollonian networks

Andrade

Herrmann

2005

Phys. Rev. E

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Section: Introductionmentioning

confidence: 99%

Magnetic models on Apollonian networks

Andrade

Herrmann

2005

Phys. Rev. E

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“…which is equivalent to a physical model of a magnetic system, ANNNI model [10,11], that was intensively investigated in the past. This map is capable of generating stable attractors (nontrivial fixed points, periodic and quasi-periodic orbits) as well as unstable chaotic behavior.…”

Section: Monotonic Functionsmentioning

confidence: 99%

Long-term properties of time series generated by a perceptron with various transfer functions

Priel

Kanter

1999

Phys. Rev. E

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We study the effect of various transfer functions on the properties of a time series generated by a continuous-valued feed-forward network in which the next input vector is determined from past output values. The parameter space for monotonic and non-monotonic transfer functions is analyzed in the unstable regions with the following main finding; non-monotonic functions can produce robust chaos whereas monotonic functions generate fragile chaos only. In the case of nonmonotonic functions, the number of positive Lyapunov exponents increases as a function of one of the free parameters in the model, hence, high dimensional chaotic attractors can be generated. We extend the analysis to a combination of monotonic and non-monotonic functions.

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“…Note that da Silva and Coutinho [16] have generalized the approach used by Thompson [11] for the Ising model on the Cayley tree with only nearest-neighbour interactions and an external field. Yokoi et al [14] extended the calculations of Vannimenus [9] for a tree of arbitrary order and gave strong numerical evidence for the existence of chaotic phases associated with strange attractors.…”

Section: Introductionmentioning

confidence: 99%

Strange Attractors in the Vannimenus Model on an Arbitrary Order Cayley Tree

Ganikhodjaev

Akın

Temir

et al. 2013

J. Phys.: Conf. Ser.

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Abstract. We consider the Vannimenus model on a Cayley tree of arbitrary order k with competing nearest-neighbour interactions J1 and next-nearest-neighbour interactions J2 and J3 in the presence of an external magnetic field h. In this paper we study the phase diagram of the model using an iterative scheme for a renormalized effective nearest-neighbour coupling Kr and effective field per site Xr for spins on the rth level; it recovers, as particular cases, previous works by Vannimenus, Inawashiro et al, Mariz et al and Ganikhodjaev and Uguz. Each phase is characterized by a particular attractor and the phase diagram is obtained by following the evolution and detecting the qualitative changements of these attractors. These changements can be either continuous or abrupt, respectively characterizing second-or first-order phase transitions. We present a few typical attractors and at finite temperatures, several interesting features (evolution of reentrances, separation of the modulated region into few disconnected pieces, etc) are exhibited for typical values of parameters. IntroductionThe anisotropic next-nearest-neighbour Ising (ANNNI) model, which consists of an Ising model with nearest-neighbour interactions augmented by competing next-nearest-neighbour couplings acting parallel to a single axis direction, is one of the simplest model displaying a rich phase diagram with a Lifshitz point and many spatially modulated phases [1]- [8]. There has been a considerable theoretical effort to obtain the structure of the global phase diagram of the ANNNI model in the T − p space, where T is temperature and p = −J 2 /J 1 is the ratio between the competing exchange interactions. On the basis of numerical mean-field calculations, Bak and von Boehm [7] suggested the existence of an infinite succession of commensurate phases, the so-called devil's staircase, at low temperatures. This mean-field picture has been supported by low-temperature series expansions performed by Fisher and Selke [8]. An Ising model with competing interactions on the Cayley tree has recently been studied extensively because of the appearance of nontrivial magnetic orderings (see [9] -[19] and references therein). The Cayley tree is not a realistic lattice; however, its amazing topology makes the exact calculation of various quantities possible. For many problems the solution on a tree is much simpler than on a regular lattice and is equivalent to the standard Bethe-Peierls theory [20].

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Strange Attractor in the Ising Model with Competing Interactions on the Cayley Tree

Cited by 80 publications

References 13 publications

Magnetic models on Apollonian networks

Magnetic models on Apollonian networks

Long-term properties of time series generated by a perceptron with various transfer functions

Strange Attractors in the Vannimenus Model on an Arbitrary Order Cayley Tree

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