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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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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).

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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

[41]

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

[42]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[50]

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

[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.

[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.

[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.

[54]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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).

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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

[41]

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

[42]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[50]

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

[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.

[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.

[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.

[54]

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

[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.

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Paul L. Salceanu. Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents. Mathematical Biosciences & Engineering, 2011, 8 (3) : 807-825. doi: 10.3934/mbe.2011.8.807

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