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March  2015, 10(1): 209-221. doi: 10.3934/nhm.2015.10.209

Effect of anisotropies on the magnetization dynamics

1. 

Departamento de Física y Matemáticas Aplicadas, Universidad de Navarra, Pamplona, 31080, Spain, Spain, Spain

2. 

Departamento de Física, Universidade Federal da Paraba, 58051-970 João Pessoa, Brazil

3. 

Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile, Chile, Chile

Received  July 2014 Revised  December 2014 Published  February 2015

We report a systematic investigation of the magnetic anisotropy effects observed in the deterministic spin dynamics of a magnetic particle in the presence of a time-dependent magnetic field. The system is modeled by the Landau-Lifshitz-Gilbert equation and the magnetic field consists of two terms, a constant term and a term involving a harmonic time modulation. We consider a general quadratic anisotropic energy with three different preferential axes. The dynamical behavior of the system is represented in Lyapunov phase diagrams, and by calculating bifurcation diagrams, Poincaré sections and Fourier spectra. We find an intricate distribution of shrimp-shaped regular island embedded in wide chaotic phases. Anisotropy effects are found to play a key role in defining the symmetries of regular and chaotic stability phases.
Citation: Laura M. Pérez, Jean Bragard, Hector Mancini, Jason A. C. Gallas, Ana M. Cabanas, Omar J. Suarez, David Laroze. Effect of anisotropies on the magnetization dynamics. Networks & Heterogeneous Media, 2015, 10 (1) : 209-221. doi: 10.3934/nhm.2015.10.209
References:
[1]

F. M. de Aguiar, A. Azevedo and S. M. Rezende, Characterization of strange attractors in spin-wave chaos,, Phys. Rev. B, 39 (1989), 9448. Google Scholar

[2]

H. A. Albuquerque and P. C. Rech, Spiral periodic structure inside chaotic region in parameter-space of a Chua circuit,, Int. J. Circuit Theory. App., 40 (2012), 189. doi: 10.1002/cta.713. Google Scholar

[3]

L. F. Alvarez, O. Pla and O. Chubykalo, Quasiperiodicity, bistability, and chaos in the Landau-Lifshitz equation,, Phys. Rev. B, 61 (2000), 11613. doi: 10.1103/PhysRevB.61.11613. Google Scholar

[4]

I. V. Barashenkov, M. M. Bogdan and V. I. Korobov, Stability diagram for the phase-locked soliton of the parametrically driven, damped nonlinear Schrödinger equation,, Europhys. Lett., 15 (1991), 113. doi: 10.1209/0295-5075/15/2/001. Google Scholar

[5]

R. Barrio, A. Shilnikov and L. P. Shilnikov, Kneadings, symbolic dynamics and painting Lorenz chaos,, Int. J. Bif. Chaos, 22 (2012). doi: 10.1142/S0218127412300169. Google Scholar

[6]

R. Barrio, F. Blesa and S. Serrano, Topological changes in periodicity hubs of dissipative systems,, Phys. Rev. Lett., 108 (2012). doi: 10.1103/PhysRevLett.108.214102. Google Scholar

[7]

X. Batlle and A. Labarta, Finite-size effects in fine particles: Magnetic and transport properties,, J. Phys. D, 35 (2002). Google Scholar

[8]

J. Becker, F. Rodelsperger, Th. Weyrauch, H. Benner, W. Just and A. Cenys, Intermittency in spin-wave instabilities,, Phys. Rev. E, 59 (1999), 1622. doi: 10.1103/PhysRevE.59.1622. Google Scholar

[9]

M. Beleggia, S. Tandon, Y. Zhu and M. De Graef, On the magnetostatic interactions between nanoparticles of arbitrary shape,, J. Magn.Magn. Mater, 278 (2004), 270. doi: 10.1016/j.jmmm.2003.12.1314. Google Scholar

[10]

M. Beleggia and M. De Graef, General magnetostatic shape-shape interactions,, J. Magn.Magn. Mater, 285 (2005). doi: 10.1016/j.jmmm.2004.09.004. Google Scholar

[11]

C. Bonatto, J. Garreau and J. A. C. Gallas, Self-similarities in the frequency-amplitude space of a loss-modulated CO2 laser,, Phys. Rev. Lett., 95 (2005). doi: 10.1103/PhysRevLett.95.143905. Google Scholar

[12]

C. Bonatto and J. A. C. Gallas, Periodicity hub and nested spirals in the phase diagram of a simple resistive circuit,, Phys. Rev. Lett., 101 (2008). doi: 10.1103/PhysRevLett.101.054101. Google Scholar

[13]

C. Bonatto and J. A. C. Gallas, Accumulation boundaries: Codimension-two accumulation of accumulations in phase diagrams of semiconductor lasers, electric circuits, atmospheric and chemical oscillators,, Phil. Trans. R. Soc. A, 366 (2008), 505. doi: 10.1098/rsta.2007.2107. Google Scholar

[14]

C. Bonatto, J. A. C. Gallas and Y. Ueda, Chaotic phase similarities and recurrences in a damped-driven Duffing oscillator,, Phys Rev. E, 77 (2008). doi: 10.1103/PhysRevE.77.026217. Google Scholar

[15]

J. Bragard, H. Pleiner, O. J. Suarez, P. Vargas, J. A. C. Gallas and D. Laroze, Chaotic dynamics of a magnetic nanoparticle,, Phys. Rev. E, 84 (2011). doi: 10.1103/PhysRevE.84.037202. Google Scholar

[16]

J. Cai, Y. Kato, A. Ogawa, Y. Harada, M. Chiba and T. Hirata, Chaotic dynamics during slow relaxation process in magnon systems,, J. Phys. Soc. Jap., 71 (2002), 3087. doi: 10.1143/JPSJ.71.3087. Google Scholar

[17]

M. G. Clerc, S. Coulibaly and D. Laroze, Interaction law of 2D localized precession states,, Europhys. Lett., 90 (2010). Google Scholar

[18]

M. G. Clerc, S. Coulibaly and D. Laroze, Localized waves in a parametrically driven magnetic nanowire,, Europhys. Lett., 97 (2012). Google Scholar

[19]

B. D. Cullity and C. D. Graham, Introduction to Magnetic Materials,, $2^{nd}$ edition, (2009). doi: 10.1002/9780470386323. Google Scholar

[20]

W. L. Ditto, M. L. Spano, H. T. Savage, S. N. Rauseo, J. Heagy and E. Ott, Experimental observation of a strange nonchaotic attractor,, Phys. Rev. Lett., 65 (1990), 533. doi: 10.1103/PhysRevLett.65.533. Google Scholar

[21]

W. Façanha, B. Oldeman and L. Glass, Bifurcation structures in two-dimensional maps: The endoskeletons of shrimps,, Phys. Lett. A, 377 (2013), 1264. doi: 10.1016/j.physleta.2013.03.025. Google Scholar

[22]

R. E. Francke, T. Pöschel and J. A. C. Gallas, Zig-zag networks of self-excited periodic oscillations in a tunnel diode and a fiber-ring laser,, Phys. Rev. E, 87 (2013). doi: 10.1103/PhysRevE.87.042907. Google Scholar

[23]

J. G. Freire and J. A. C. Gallas, Non-Shilnikov cascades of spikes and hubs in a semiconductor laser with optoelectronic feedback,, Phys. Rev. E, 82 (2010). doi: 10.1103/PhysRevE.82.037202. Google Scholar

[24]

J. G. Freire and J. A. C. Gallas, Stern-Brocot trees in the periodicity of mixed-mode oscillations,, Phys. Chem. Chem. Phys., 13 (2011), 12191. doi: 10.1039/c0cp02776f. Google Scholar

[25]

J. G. Freire and J. A. C. Gallas, Stern-Brocot trees in cascades of mixed-mode oscillations and canards in the extended Bonhoeffer-van der Pol and the FitzHugh-Nagumo models of excitable systems,, Phys. Lett. A, 375 (2011), 1097. doi: 10.1016/j.physleta.2011.01.017. Google Scholar

[26]

J. G. Freire, C. Cabeza, A. Marti, T. Pöschel and J. A. C. Gallas, Antiperiodic oscillations,, Nature Sci. Rep., 3 (2013). doi: 10.1038/srep01958. Google Scholar

[27]

J. A. C. Gallas, Structure of the parameter space of the Hénon map,, Phys. Rev. Lett., 70 (1993), 2714. doi: 10.1103/PhysRevLett.70.2714. Google Scholar

[28]

J. A. C. Gallas, The structure of infinite periodic and chaotic hub cascades in phase diagrams of simple autonomous flows,, Int. J. Bifur. Chaos, 20 (2010), 197. doi: 10.1142/S0218127410025636. Google Scholar

[29]

M. R. Gallas, M. R. Gallas and J. A. C. Gallas, Distribution of chaos and periodic spikes in a three-cell population model of cancer,, Eur. Phys. J. Special Topics, 223 (2014), 2131. Google Scholar

[30]

G. Gibson and C. Jeffries, Observation of period doubling and chaos in spin-wave instabilities in yttrium iron garnet,, Phys. Rev. A, 29 (1984), 811. doi: 10.1103/PhysRevA.29.811. Google Scholar

[31]

A. Hoff, D. T. da Silva, C. Manchein and H. A. Albuquerque, Bifurcation structures and transient chaos in a four-dimensional Chua model,, Phys. Lett. A, 378 (2014), 171. doi: 10.1016/j.physleta.2013.11.003. Google Scholar

[32]

M. Lakshmanan, The fascinating world of the Landau-Lifshitz-Gilbert equation: An overview, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 369 (2011), 1280. doi: 10.1098/rsta.2010.0319. Google Scholar

[33]

P. Landeros, J. Escrig, D. Altbir, D. Laroze, J. d'Albuquerque e Castro and P. Vargas, Scaling relations for magnetic nanoparticles,, Phys. Rev. B, 65 (2005). Google Scholar

[34]

D. Laroze and P. Vargas, Dynamical behavior of two interacting magnetic nanoparticles,, Phys. B, 372 (2006), 332. doi: 10.1016/j.physb.2005.10.079. Google Scholar

[35]

D. Laroze and L. M. Perez, Classical spin dynamics of four interacting magnetic particles on a ring,, Phys. B, 403 (2008), 473. doi: 10.1016/j.physb.2007.08.078. Google Scholar

[36]

D. Laroze, P. Vargas, C. Cortes and G. Gutierrez, Dynamics of two interacting dipoles,, J. Magn. Magn. Mater., 320 (2008), 1440. doi: 10.1016/j.jmmm.2007.12.010. Google Scholar

[37]

D. Laroze, O. J. Suarez, J. Bragard and H. Pleiner, Characterization of the chaotic magnetic particle dynamics,, IEEE Trans. On Magnetics, 47 (2011), 3032. doi: 10.1109/TMAG.2011.2158072. Google Scholar

[38]

D. Laroze, D. Becerra-Alonso, J. A. C. Gallas and H. Pleiner, Magnetization dynamics under a quasiperiodic magnetic field,, IEEE Trans. On Magnetics, 48 (2012), 3567. doi: 10.1109/TMAG.2012.2207378. Google Scholar

[39]

D. Laroze, P. G. Siddheshwar and H. Pleiner, Chaotic convection in a ferrofluid,, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 2436. doi: 10.1016/j.cnsns.2013.01.016. Google Scholar

[40]

E. N. Lorenz, Compound windows of the Hénon map,, Physica D, 237 (2008), 1689. doi: 10.1016/j.physd.2007.11.014. Google Scholar

[41]

D. Mayergoyz, G. Bertotti and C. Serpico, Nonlinear Magnetization Dynamics in Nanosystems,, Elsevier, (2009). Google Scholar

[42]

R. C. O'Handley, Modern Magnetic Materials: Principles and Applications,, Wiley Interscience, (1999). Google Scholar

[43]

D. F. M. Oliveira, M. Robnik and E. D. Leonel, Shrimp-shape domains in a dissipative kicked rotator,, Chaos, 21 (2011). doi: 10.1063/1.3657917. Google Scholar

[44]

L. M. Pérez, O. J. Suarez, D. Laroze and H. L. Mancini, Classical spin dynamics of anisotropic Heisenberg dimers,, Cent. Eur. J. Phys., 11 (2013), 1629. Google Scholar

[45]

A. Sack, J. G. Freire, E. Lindberg, T. Pöschel and J. A. C. Gallas, Discontinuous spirals of stable periodic oscillations,, Nature Sci. Rep., 3 (2013). doi: 10.1038/srep03350. Google Scholar

[46]

R. K. Smith, M. Grabowski and R. E. Camley, Period doubling toward chaos in a driven magnetic macrospin,, J. Magn. Magn. Mater., 322 (2010), 2127. doi: 10.1016/j.jmmm.2010.01.045. Google Scholar

[47]

S. L. T. Souza, A. A. Lima, I. R. Caldas, R. O. Medrano-T and Z. O. Guimaã es-Filho, Self-similarities of periodic structures for a discrete model of a two-gene system,, Phys. Lett. A, 376 (2012), 1290. Google Scholar

[48]

S. Tandon, M. Beleggia, Y. Zhu and M. De Graef, On the computation of the demagnetization tensor for uniformly magnetized particles of arbitrary shape. Part I: Analytical approach,, J. Magn.Magn. Mater, 271 (2004), 9. Google Scholar

[49]

S. Tandon, M. Beleggia, Y. Zhu and M. De Graef, On the computation of the demagnetization tensor for uniformly magnetized particles of arbitrary shape. Part II: Numerical approach,, J. Magn.Magn. Mater, 271 (2004), 27. Google Scholar

[50]

D. Urzagasti, D. Laroze, M. G. Clerc and H. Pleiner, Breather soliton solutions in a parametrically driven magnetic wire,, Europhys. Lett., 104 (2013). Google Scholar

[51]

D. Urzagasti, A. Aramayo and D. Laroze, Soliton-antisoliton interaction in a parametrically driven easy-plane magnetic wire,, Phys. Lett. A, 378 (2014), 2614. doi: 10.1016/j.physleta.2014.07.013. Google Scholar

[52]

D. V. Vagin and P. Polyakov, Control of chaotic and deterministic magnetization dynamics regimes by means of sample shape varying,, J. App. Phys, 105 (2009). doi: 10.1063/1.3075838. Google Scholar

[53]

R. Vitolo, P. Glendinning and J. A. C. Gallas, Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows,, Phys. Rev. E, 84 (2011). doi: 10.1103/PhysRevE.84.016216. Google Scholar

[54]

P. E. Wigen (Ed.), Nonlinear Phenomena and Chaos in Magnetic Materials,, World Scientific, (1994). Google Scholar

[55]

A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, Determining Lyapunov exponents from a time series,, Physica D, 16 (1985), 285. doi: 10.1016/0167-2789(85)90011-9. Google Scholar

show all references

References:
[1]

F. M. de Aguiar, A. Azevedo and S. M. Rezende, Characterization of strange attractors in spin-wave chaos,, Phys. Rev. B, 39 (1989), 9448. Google Scholar

[2]

H. A. Albuquerque and P. C. Rech, Spiral periodic structure inside chaotic region in parameter-space of a Chua circuit,, Int. J. Circuit Theory. App., 40 (2012), 189. doi: 10.1002/cta.713. Google Scholar

[3]

L. F. Alvarez, O. Pla and O. Chubykalo, Quasiperiodicity, bistability, and chaos in the Landau-Lifshitz equation,, Phys. Rev. B, 61 (2000), 11613. doi: 10.1103/PhysRevB.61.11613. Google Scholar

[4]

I. V. Barashenkov, M. M. Bogdan and V. I. Korobov, Stability diagram for the phase-locked soliton of the parametrically driven, damped nonlinear Schrödinger equation,, Europhys. Lett., 15 (1991), 113. doi: 10.1209/0295-5075/15/2/001. Google Scholar

[5]

R. Barrio, A. Shilnikov and L. P. Shilnikov, Kneadings, symbolic dynamics and painting Lorenz chaos,, Int. J. Bif. Chaos, 22 (2012). doi: 10.1142/S0218127412300169. Google Scholar

[6]

R. Barrio, F. Blesa and S. Serrano, Topological changes in periodicity hubs of dissipative systems,, Phys. Rev. Lett., 108 (2012). doi: 10.1103/PhysRevLett.108.214102. Google Scholar

[7]

X. Batlle and A. Labarta, Finite-size effects in fine particles: Magnetic and transport properties,, J. Phys. D, 35 (2002). Google Scholar

[8]

J. Becker, F. Rodelsperger, Th. Weyrauch, H. Benner, W. Just and A. Cenys, Intermittency in spin-wave instabilities,, Phys. Rev. E, 59 (1999), 1622. doi: 10.1103/PhysRevE.59.1622. Google Scholar

[9]

M. Beleggia, S. Tandon, Y. Zhu and M. De Graef, On the magnetostatic interactions between nanoparticles of arbitrary shape,, J. Magn.Magn. Mater, 278 (2004), 270. doi: 10.1016/j.jmmm.2003.12.1314. Google Scholar

[10]

M. Beleggia and M. De Graef, General magnetostatic shape-shape interactions,, J. Magn.Magn. Mater, 285 (2005). doi: 10.1016/j.jmmm.2004.09.004. Google Scholar

[11]

C. Bonatto, J. Garreau and J. A. C. Gallas, Self-similarities in the frequency-amplitude space of a loss-modulated CO2 laser,, Phys. Rev. Lett., 95 (2005). doi: 10.1103/PhysRevLett.95.143905. Google Scholar

[12]

C. Bonatto and J. A. C. Gallas, Periodicity hub and nested spirals in the phase diagram of a simple resistive circuit,, Phys. Rev. Lett., 101 (2008). doi: 10.1103/PhysRevLett.101.054101. Google Scholar

[13]

C. Bonatto and J. A. C. Gallas, Accumulation boundaries: Codimension-two accumulation of accumulations in phase diagrams of semiconductor lasers, electric circuits, atmospheric and chemical oscillators,, Phil. Trans. R. Soc. A, 366 (2008), 505. doi: 10.1098/rsta.2007.2107. Google Scholar

[14]

C. Bonatto, J. A. C. Gallas and Y. Ueda, Chaotic phase similarities and recurrences in a damped-driven Duffing oscillator,, Phys Rev. E, 77 (2008). doi: 10.1103/PhysRevE.77.026217. Google Scholar

[15]

J. Bragard, H. Pleiner, O. J. Suarez, P. Vargas, J. A. C. Gallas and D. Laroze, Chaotic dynamics of a magnetic nanoparticle,, Phys. Rev. E, 84 (2011). doi: 10.1103/PhysRevE.84.037202. Google Scholar

[16]

J. Cai, Y. Kato, A. Ogawa, Y. Harada, M. Chiba and T. Hirata, Chaotic dynamics during slow relaxation process in magnon systems,, J. Phys. Soc. Jap., 71 (2002), 3087. doi: 10.1143/JPSJ.71.3087. Google Scholar

[17]

M. G. Clerc, S. Coulibaly and D. Laroze, Interaction law of 2D localized precession states,, Europhys. Lett., 90 (2010). Google Scholar

[18]

M. G. Clerc, S. Coulibaly and D. Laroze, Localized waves in a parametrically driven magnetic nanowire,, Europhys. Lett., 97 (2012). Google Scholar

[19]

B. D. Cullity and C. D. Graham, Introduction to Magnetic Materials,, $2^{nd}$ edition, (2009). doi: 10.1002/9780470386323. Google Scholar

[20]

W. L. Ditto, M. L. Spano, H. T. Savage, S. N. Rauseo, J. Heagy and E. Ott, Experimental observation of a strange nonchaotic attractor,, Phys. Rev. Lett., 65 (1990), 533. doi: 10.1103/PhysRevLett.65.533. Google Scholar

[21]

W. Façanha, B. Oldeman and L. Glass, Bifurcation structures in two-dimensional maps: The endoskeletons of shrimps,, Phys. Lett. A, 377 (2013), 1264. doi: 10.1016/j.physleta.2013.03.025. Google Scholar

[22]

R. E. Francke, T. Pöschel and J. A. C. Gallas, Zig-zag networks of self-excited periodic oscillations in a tunnel diode and a fiber-ring laser,, Phys. Rev. E, 87 (2013). doi: 10.1103/PhysRevE.87.042907. Google Scholar

[23]

J. G. Freire and J. A. C. Gallas, Non-Shilnikov cascades of spikes and hubs in a semiconductor laser with optoelectronic feedback,, Phys. Rev. E, 82 (2010). doi: 10.1103/PhysRevE.82.037202. Google Scholar

[24]

J. G. Freire and J. A. C. Gallas, Stern-Brocot trees in the periodicity of mixed-mode oscillations,, Phys. Chem. Chem. Phys., 13 (2011), 12191. doi: 10.1039/c0cp02776f. Google Scholar

[25]

J. G. Freire and J. A. C. Gallas, Stern-Brocot trees in cascades of mixed-mode oscillations and canards in the extended Bonhoeffer-van der Pol and the FitzHugh-Nagumo models of excitable systems,, Phys. Lett. A, 375 (2011), 1097. doi: 10.1016/j.physleta.2011.01.017. Google Scholar

[26]

J. G. Freire, C. Cabeza, A. Marti, T. Pöschel and J. A. C. Gallas, Antiperiodic oscillations,, Nature Sci. Rep., 3 (2013). doi: 10.1038/srep01958. Google Scholar

[27]

J. A. C. Gallas, Structure of the parameter space of the Hénon map,, Phys. Rev. Lett., 70 (1993), 2714. doi: 10.1103/PhysRevLett.70.2714. Google Scholar

[28]

J. A. C. Gallas, The structure of infinite periodic and chaotic hub cascades in phase diagrams of simple autonomous flows,, Int. J. Bifur. Chaos, 20 (2010), 197. doi: 10.1142/S0218127410025636. Google Scholar

[29]

M. R. Gallas, M. R. Gallas and J. A. C. Gallas, Distribution of chaos and periodic spikes in a three-cell population model of cancer,, Eur. Phys. J. Special Topics, 223 (2014), 2131. Google Scholar

[30]

G. Gibson and C. Jeffries, Observation of period doubling and chaos in spin-wave instabilities in yttrium iron garnet,, Phys. Rev. A, 29 (1984), 811. doi: 10.1103/PhysRevA.29.811. Google Scholar

[31]

A. Hoff, D. T. da Silva, C. Manchein and H. A. Albuquerque, Bifurcation structures and transient chaos in a four-dimensional Chua model,, Phys. Lett. A, 378 (2014), 171. doi: 10.1016/j.physleta.2013.11.003. Google Scholar

[32]

M. Lakshmanan, The fascinating world of the Landau-Lifshitz-Gilbert equation: An overview, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 369 (2011), 1280. doi: 10.1098/rsta.2010.0319. Google Scholar

[33]

P. Landeros, J. Escrig, D. Altbir, D. Laroze, J. d'Albuquerque e Castro and P. Vargas, Scaling relations for magnetic nanoparticles,, Phys. Rev. B, 65 (2005). Google Scholar

[34]

D. Laroze and P. Vargas, Dynamical behavior of two interacting magnetic nanoparticles,, Phys. B, 372 (2006), 332. doi: 10.1016/j.physb.2005.10.079. Google Scholar

[35]

D. Laroze and L. M. Perez, Classical spin dynamics of four interacting magnetic particles on a ring,, Phys. B, 403 (2008), 473. doi: 10.1016/j.physb.2007.08.078. Google Scholar

[36]

D. Laroze, P. Vargas, C. Cortes and G. Gutierrez, Dynamics of two interacting dipoles,, J. Magn. Magn. Mater., 320 (2008), 1440. doi: 10.1016/j.jmmm.2007.12.010. Google Scholar

[37]

D. Laroze, O. J. Suarez, J. Bragard and H. Pleiner, Characterization of the chaotic magnetic particle dynamics,, IEEE Trans. On Magnetics, 47 (2011), 3032. doi: 10.1109/TMAG.2011.2158072. Google Scholar

[38]

D. Laroze, D. Becerra-Alonso, J. A. C. Gallas and H. Pleiner, Magnetization dynamics under a quasiperiodic magnetic field,, IEEE Trans. On Magnetics, 48 (2012), 3567. doi: 10.1109/TMAG.2012.2207378. Google Scholar

[39]

D. Laroze, P. G. Siddheshwar and H. Pleiner, Chaotic convection in a ferrofluid,, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 2436. doi: 10.1016/j.cnsns.2013.01.016. Google Scholar

[40]

E. N. Lorenz, Compound windows of the Hénon map,, Physica D, 237 (2008), 1689. doi: 10.1016/j.physd.2007.11.014. Google Scholar

[41]

D. Mayergoyz, G. Bertotti and C. Serpico, Nonlinear Magnetization Dynamics in Nanosystems,, Elsevier, (2009). Google Scholar

[42]

R. C. O'Handley, Modern Magnetic Materials: Principles and Applications,, Wiley Interscience, (1999). Google Scholar

[43]

D. F. M. Oliveira, M. Robnik and E. D. Leonel, Shrimp-shape domains in a dissipative kicked rotator,, Chaos, 21 (2011). doi: 10.1063/1.3657917. Google Scholar

[44]

L. M. Pérez, O. J. Suarez, D. Laroze and H. L. Mancini, Classical spin dynamics of anisotropic Heisenberg dimers,, Cent. Eur. J. Phys., 11 (2013), 1629. Google Scholar

[45]

A. Sack, J. G. Freire, E. Lindberg, T. Pöschel and J. A. C. Gallas, Discontinuous spirals of stable periodic oscillations,, Nature Sci. Rep., 3 (2013). doi: 10.1038/srep03350. Google Scholar

[46]

R. K. Smith, M. Grabowski and R. E. Camley, Period doubling toward chaos in a driven magnetic macrospin,, J. Magn. Magn. Mater., 322 (2010), 2127. doi: 10.1016/j.jmmm.2010.01.045. Google Scholar

[47]

S. L. T. Souza, A. A. Lima, I. R. Caldas, R. O. Medrano-T and Z. O. Guimaã es-Filho, Self-similarities of periodic structures for a discrete model of a two-gene system,, Phys. Lett. A, 376 (2012), 1290. Google Scholar

[48]

S. Tandon, M. Beleggia, Y. Zhu and M. De Graef, On the computation of the demagnetization tensor for uniformly magnetized particles of arbitrary shape. Part I: Analytical approach,, J. Magn.Magn. Mater, 271 (2004), 9. Google Scholar

[49]

S. Tandon, M. Beleggia, Y. Zhu and M. De Graef, On the computation of the demagnetization tensor for uniformly magnetized particles of arbitrary shape. Part II: Numerical approach,, J. Magn.Magn. Mater, 271 (2004), 27. Google Scholar

[50]

D. Urzagasti, D. Laroze, M. G. Clerc and H. Pleiner, Breather soliton solutions in a parametrically driven magnetic wire,, Europhys. Lett., 104 (2013). Google Scholar

[51]

D. Urzagasti, A. Aramayo and D. Laroze, Soliton-antisoliton interaction in a parametrically driven easy-plane magnetic wire,, Phys. Lett. A, 378 (2014), 2614. doi: 10.1016/j.physleta.2014.07.013. Google Scholar

[52]

D. V. Vagin and P. Polyakov, Control of chaotic and deterministic magnetization dynamics regimes by means of sample shape varying,, J. App. Phys, 105 (2009). doi: 10.1063/1.3075838. Google Scholar

[53]

R. Vitolo, P. Glendinning and J. A. C. Gallas, Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows,, Phys. Rev. E, 84 (2011). doi: 10.1103/PhysRevE.84.016216. Google Scholar

[54]

P. E. Wigen (Ed.), Nonlinear Phenomena and Chaos in Magnetic Materials,, World Scientific, (1994). Google Scholar

[55]

A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, Determining Lyapunov exponents from a time series,, Physica D, 16 (1985), 285. doi: 10.1016/0167-2789(85)90011-9. Google Scholar

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[5]

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[6]

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[7]

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[8]

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[9]

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[10]

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[11]

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[12]

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[13]

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[14]

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[15]

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[16]

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[17]

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[18]

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[19]

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[20]

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