Magnetic field induced dynamical chaos Somrita Ray, Alendu Baura, and Bidhan Chandra Bag Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 23, 043121 (2013); doi: 10.1063/1.4832175 View online: http://dx.doi.org/10.1063/1.4832175 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/23/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Low-dimensional dynamical system for Rayleigh-Bénard convection subjected to magnetic field J. Appl. Phys. 113, 124902 (2013); 10.1063/1.4795264 Oscillons: An encounter with dynamical chaos in 1953? Chaos 21, 023123 (2011); 10.1063/1.3562545 The transient dynamics leading to spin turbulence in high-field solution magnetic resonance: A numerical study J. Chem. Phys. 124, 154501 (2006); 10.1063/1.2181568 Experimental observation of Lorenz chaos in the Quincke rotor dynamics Chaos 15, 013102 (2005); 10.1063/1.1827411 Chaos in integrate-and-fire dynamical systems AIP Conf. Proc. 502, 88 (2000); 10.1063/1.1302370
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CHAOS 23, 043121 (2013)
Magnetic field induced dynamical chaos Somrita Ray, Alendu Baura, and Bidhan Chandra Baga) Department of Chemistry, Visva-Bharati, Santiniketan 731 235, India
(Received 10 May 2013; accepted 6 November 2013; published online 20 November 2013) In this article, we have studied the dynamics of a particle having charge in the presence of a magnetic field. The motion of the particle is confined in the x–y plane under a two dimensional nonlinear potential. We have shown that constant magnetic field induced dynamical chaos is possible even for a force which is derived from a simple potential. For a given strength of the magnetic field, initial position, and velocity of the particle, the dynamics may be regular, but it may become chaotic when the field is time dependent. Chaotic dynamics is very often if the field is time dependent. Origin of chaos has been explored using the Hamiltonian function of the dynamics in terms of action and angle variables. Applicability of the present study has been discussed with a C 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4832175] few examples. V
Velocity dependent magnetic force cannot create or destroy fixed point. It implies that usual bifurcation route to chaos is not the origin of magnetic field (MF) induced chaos. We have explored origin of chaotic dynamics of a charged particle (CP) in the presence of a magnetic and potential energy fields. Our investigation shows that overlapping of resonance zones is necessary for nonintegrable motion of the particle. Another observation is that regular motion of the charged particle becomes chaotic if the applied magnetic field becomes time dependent (TD).
I. INTRODUCTION
Study of dynamics of a particle having charge is always an intriguing issue in physics, since electron or ion is the major constituent of most of the natural systems which belong to subjects like physics, chemistry, and biology. Dynamics of a particle in the presence of magnetic field has been studied in different context in recent past.1–10 In these studies, the particle exhibits random motion as a signature of a statistical system, having infinite dimension due to coupling of the particle with the thermal environment. The effective dynamics is described by Langevin equation of motion which is a stochastic differential equation. But in the present paper, we are exploring a dynamics which is stochastic in nature in a very low dimensional (deterministic) system. This is popularly known as chaotic motion in the field of nonlinear dynamics (NLD). Several models11,12 have been studied to investigate nonlinear dynamics of CP in the presence of a magnetic field. Effect of a uniform magnetic field (MF) on the dynamics of a CP in a conducting rod has been explored in Ref. 11. In this study, it has been shown that an extensible current-carrying rod in a uniform magnetic field is nonintegrable. Nonintegrabale dynamics is also observed in a spatially periodic billiard chain, in which electrons move under the influence of a perpendicular magnetic field.12 a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
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Quantum mechanical study shows that spatial periodic variation of magnetic field can induce chaos.13 Nonlinear dynamics of charged particle in the presence of space dependent magnetic field has been studied classically in Ref. 14. In these studies, boundary effect and velocity dependent magnetic force have been only considered. But in conducting solids, electrons experience a periodic force in space from a periodic potential energy function. Dynamics of a charged particle (CP) in x–y plane in the presence of periodic potential and constant magnetic field quantum mechanically in Ref. 15. Here, it has been shown that the dynamics is chaotic at the classical limit. In the present paper, we have studied similar problem based on classical mechanics and explored origin of chaos in a simple way. We have considered double well potential along x-direction, and motion is bounded harmonically in the y-direction. As a result of velocity dependent coupling through magnetic force between two degrees of freedom, there is a possibility of chaotic motion. For some initial conditions (in terms of position and velocity of the particle), the dynamics may be regular in nature even in the presence of the field. But the dynamics is very often chaotic in nature when the external magnetic field is time dependent. The origin of the stochastic nature of the dynamics has been explored by writing the Hamiltonian function in terms of action and angle variables. We now consider the relevance of the present model study. Electrons in solid experience a nonlinear periodic potential. If the solid is a one dimensional nanowire, that periodic potential is along the wire and one may assume that electron is harmonically bound along the direction perpendicular to the axis of the wire. In the present study, two consecutive wells of the periodic potential are accounted by a single double-well potential, which gives the benefit of studying the dynamics of the system in a finite region. Thus, dynamics of electron in nanowire in the presence of magnetic field may mimic the present study. Similarly dynamics of an ion (which is trapped by nonlinear potential in solid electrolytes) in the presence of magnetic field may also mimic the present model. Study of conductivity of ion in solid electrolytes becomes an important issue because of potential applications of these materials in a
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C 2013 AIP Publishing LLC V
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diverse range of all-solid-state devices such as rechargeable lithium batteries, flexible electrochromic displays, and smart windows.16,17 One can study the effect of chaotic dynamics on the conductivity of solid electrolytes in the presence of magnetic field. It may help to tune the conductivity of ion by controlling chaotic diffusion. Tuning of conductivity is very important for specific needs. The properties of the electrolyte are tuned by varying chemical composition to a large extent and hence are adapted to specific needs.17 Physical method like controlling of chaotic diffusion may be easier route in this context. Interest grows rapidly to study dynamics of particles confined to finite size containers of various geometries.18–21 In Ref. 19, it has been shown that a simpler instance of temperature controlled resonance can naturally occur in a confined magnetized electron gas. Extension of present study may be interesting in the case of confined magnetized system. Present analysis may also be extended following Ref. 22 to the case of relativistic two dimensional dynamics of a charged particle in a magnetic field. Before leaving this part, we would like to mention that chaotic dynamics as a result of interplay of constant magnetic field and force derived from potential has also been studied both experimentally23,24 and theoretically25 for hydrogen atom in a uniform magnetic filed. In the present paper, we have shown that constant magnetic field induced dynamical chaos is possible even for a force which is derived from a simple potential. The outline of the paper is as follows: In Sec. II, we have presented the model. Constant magnetic field induced chaos, and its origin have been discussed in Sec. III. Dynamics of a particle in the presence of time dependent magnetic field has been presented in Sec. IV. The paper is concluded in Sec. V.
field is pointing along the z-axis of the Cartesian reference frame, i.e., B ¼ (0,0,B). For this choice of magnetic field, Eq. (1) can be described by means of two independent processes. One is described in the x–y plane (which is perpendicular to the direction of the applied magnetic field) with the potential energy function, Vðx; yÞ ¼ ax4 bx2 þx22 y2 =2 and the other is along the direction of the magnetic field having potential energy function, VðzÞ ¼ x22 z2 =2. In these cases, Eq. (1) can be written as
II. THE MODEL
III. CONSTANT MAGNETIC FIELD INDUCED CHAOS
To start with, we consider the equation of motion of a particle (having charge q and mass m) in three dimension in the presence of a magnetic field B. It can be written for the velocity vector u (Refs. 5 and 10) as
To study the present problem, we have solved Eqs. (4)–(7) numerically using the well known fourth order Runge-Kutta method. Relevant parameters for the numerical study are a ¼ 0.25, b ¼ 0.5, and x2 ¼ 1:0. The results of numerical integration of Eqs. (4)–(7) in the absence of magnetic field ðX ¼ 0:0Þ and for the initial condition of the oscillators: x(0) ¼ 0.005, y(0) ¼ 0.001, ux(0) ¼ 0.0, and uy(0) ¼ 0.0 are shown in Poincare plot (Fig. 1). This is a characteristic of regular motion of uncoupled oscillators. It is in sharp contrast to what we have observed (in the presence of magnetic field) in Fig. 2 plotting the results of the numerical integration of equations of motion with the same initial condition, x(0) ¼ 0.005, y(0) ¼ 0.001, ux(0) ¼ 0.0, and uy(0) ¼ 0.0. Thus, magnetic field can induce chaos through velocity dependent coupling between two oscillators. To become sure about this phenomenon, we have studied dynamics for other initial condition and observed chaos (results are shown in Fig. 3). This is a new observation with respect to known coordinate dependent coupling induced chaos.26 We now calculate an important parameter (a precursor of largest Lyapunov exponent used as a measure of regularity or chaoticity of nonlinear dynamical system) proposed many years ago in Ref. 27. It is the long time average of ln dðtÞ=d0 , where d0 is the initial separation of the two
u_ ¼ F;
(1)
where F¼
1 q rVðrÞ þ u B: m mc
(2)
Here, rV is the gradient of the potential V and c is the speed of light. In the present study, we have considered that the particle is bounded by harmonic potential along y-z directions and experience a double-well potential along x-direction. Thus, the total potential energy can be represented as Vðx; y; zÞ ¼ ax4 bx2 þ x22 ðy2 þ z2 Þ=2;
(3)
where a and b are constants which determine the fixed points corresponding to the potential as well as the barrier height. x2 in the last term of the above equation is the characteristic frequency of the oscillation along y and z directions, respectively. For simplicity, we consider that the applied magnetic
x_ ¼ ux ;
(4)
y_ ¼ uy ;
(5)
u_x ¼ 4ax3 þ 2bx þ Xuy ;
(6)
u_y ¼ x22 y Xux ;
(7)
z_ ¼ uz ;
(8)
u_z ¼ x22 z;
(9)
where X¼
qB : mc
(10)
The above Eqs.(4)–(9) imply that the motion along zcomponent is independent of the applied magnetic field and it has no effect on the motion in x–y plane. In this plane, we have studied non-linear dynamics of a charged particle in the presence of an external magnetic field. In the next section, i.e., in Sec. III, we shall demonstrate that constant magnetic field can induce chaos.
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FIG. 1. Plot of ux vs x on the Poincare surface of section (y ¼ 0) with initial condition x ¼ 0.005, y ¼ 0.001, ux ¼ 0.0, uy ¼ 0.0, and X ¼ 0:0.
initially nearby trajectories and d(t) is the corresponding separation at some time, t. d(t) having dimension of length is expressed as "
Du2 Du2y Þ dðtÞ ¼ Dx2 þ Dy2 þ 2x þ 2 x1 x2
#1=2 ;
(11)
where x1 corresponds to the frequency around the stable fixed points of the nonliner potential along x-direction. Dx; Dy; Dux , and Duy in above equation are related to two nearby trajectories, x, y, ux, uy, and x þ Dx; y þ Dy; ux þ Dux ; uy þ Duy at the same time t in four dimensional phase space. Based on this relationship and Eqs. (4)–(7), one can write time evolution equation for Dx; Dy; Dux , and Duy as _ ¼ Dux ; Dx
(12)
_ ¼ Duy ; Dy
(13)
_ x ¼ ð3ax2 þ 2bxÞDx þ XDuy ; Du
(14)
_ y ¼ x2 Dy XDux : Du 2
(15)
FIG. 2. Plot of ux vs x on the Poincare surface of section (y ¼ 0) with initial condition x ¼ 0.005, y ¼ 0.001, ux ¼ 0.0, uy ¼ 0.0, and X ¼ 0:5.
FIG. 3. Plot of ux vs x on the Poincare surface of section (y ¼ 0) with initial condition x ¼ 2.4, y ¼ 2.0, ux ¼ 0.0, uy ¼ 0.0, and X ¼ 0.5.
Assuming that Dx; Dy; Dux , and Duy are small, we have ignored the terms in the above set of equations which are nonlinear in terms of these quantities. To determine d(t), we have numerically solved Eqs. (4)–(7) and Eqs. (12)–(15), simultaneously. In going from the j-th step to the jþ1-th step of the iteration in course of time evolution, Dx has to be initialized as Dxðjþ1Þ0 ¼ ðDxj0 =dj Þd0 . Similar procedure has been followed for related quantities. This initialization implies that at each step, the iteration starts with same magnitude of d0 but the direction of d0 for step j þ 1 is that of d(t) for j-th step. For a pictorial illustration, we refer to Fig. 1 of Ref. 28. The j-th time of iteration is determined by t ¼ jTðj ¼ 1; 2; ::::1Þ. Here, T is the characteristic time that corresponds to the shortest ensemble averaged period of a nonlinear dynamical system. Thus, following Casartelli et al.,27 a stochastic parameter can be defined by the following time average of lndj =d0 as bn ðt; xð0Þ; yð0Þ; ux ð0Þ; uy ð0Þ; d0 Þ ¼
n dj 1X ln : n j¼1 d0
(16)
In Ref. 27, it has been shown that as n ! 1; bn has a definite value. It is positive for the chaotic systems. For the regular system the stochastic parameter is zero. bn has been calculated for n ¼ 100000000. We have checked that with this large value of n the stochastic parameter becomes n-independent. The value of this parameter for Figs. 1 and 2 are 0.000005058 and 0.010427679, respectively. Thus, magnetic field induced chaos is further confirmed. b1 is 0.017863778 for Fig. 3. Large positive value of the stochastic parameter for Fig. 3 implies that the dynamics corresponding to this figure is chaotic. It is apparent from Figs. 2 and 3 that chaos is strong in Fig. 3 compared to Fig. 2. This inspection is supported by the values of the stochastic parameter corresponding to these figures. We now explore qualitatively the origin of magnetic field induced chaos. At this end, we write the Hamiltonian for the present system in terms of the canonical momentum, scalar and vector potentials as
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H¼
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Chaos 23, 043121 (2013)
ðpx Ax ðx; yÞÞ2 ðpy Ay ðx; yÞÞ2 y2 þ þ ax4 bx2 þ x22 : 2 2 2 (17)
In the above equation, we have used q ¼ 1, m ¼ 1, and c ¼ 1. px and Ax in Eq. (17) are components (along x-direction) of ~ respectively. momentum vector ð~ p Þ and vector potential ðAÞ, Similarly py and Ay are components of these quantities along y-direction. Relation between px and Ax is px ¼ ux þ Ax :
where J1 and h1 are the action and angle variables corresponding to oscillation along x-direction. Similarly, J2 and h2 are the action angle variables related to oscillation along y-direction. Making use of Eqs. (27)–(30) in Eq. (26), one may write that H¼
(18)
Similarly, py and Ay are related by py ¼ uy þ Ay :
p_x ¼ u_x þ A_x
þ ðx1 x2 Þsinðh1 þ h2 ÞÞ:
(19)
From Eqs. (18) and (19), one may write (20)
4aJ12 2bJ1 cos4 h1 cos2 h1 þ x1 J1 sin2 h1 þ x2 J2 2 x1 x1 X2 2J1 2J2 þ cos2 h1 þ cos2 h2 8 x1 x2 1=2 X J1 J 2 ððx1 þ x2 Þsinðh1 h2 Þ 2 x1 x2
Making use of simple trigonometric relations,30 the above Hamiltonian can be transformed as H ¼ 2g cosðh1 þ h2 Þcosðh1 h2 Þ
and p_y ¼ u_y þ A_y :
þ 2g1 cosð2h1 þ h2 Þcosð2h1 h2 Þ ! X 2 J2 g g1 cos 2h2 8x2 X J1 J2 1=2 ððx1 þ x2 Þsinðh1 h2 Þ 2 x1 x2
(21)
Using Hamilton’s equations of motion in Eqs. (20) and (21), we have u_x ¼
@Ay @Ax @Ax uy þ uy 4ax3 þ 2bx; @t @y @x
u_y ¼
@Ay @Ay @Ax ux þ ux x22 y: @t @x @y
(22) (23)
þ ðx1 x2 Þsinðh1 þ h2 ÞÞ þ g2 ;
Xy ; Ax ¼ 2 Ay ¼
g¼
(25)
We now introduce action and angle variables through the following relations:29 x¼
1=2
px ¼ ð2J1 x1 Þ
cos h1 ; 1=2
(33)
aJ12 ; 2x21
(34)
and
Xx : 2
p2x p2y Xðypx xpy Þ X2 ðx2 þ y2 Þ y2 þ þ þ ax4 bx2 þ x22 : þ 2 2 2 8 2 (26)
2J1 x1
2aJ12 bJ1 x1 J1 X2 J1 þ ; x1 2 8x1 x21 g1 ¼
(24)
This choice is nothing but the popularly known symmetric gauge for A. For the above relations (24) and (25) Hamiltonian (17) becomes
(32)
where
Comparing Eqs. (6) and (7) with Eqs. (22) and (23), one can identify that
H¼
(31)
sin h1 ;
(27) (28)
g2 ¼
3aJ12 bJ1 x1 J1 X 2 J1 J2 : þ x þ J þ þ 2 2 x1 2 8 x1 x2 2x21
(35)
It is apparent in the first three terms in Hamiltonian (32) that there are possibilities of 1:1 and 2:1 resonances. Overlapping of these two resonance zones may result in chaos as demonstrated in Figs. 2 and 3. It is supported by the equations of motion. Equations (4)–(7) imply that magnetic force cannot create or destroy fixed point. Hence bifurcation route to chaos is not the origin of chaos for the present case. Before leaving this part, we would like to mention that one can also do above analysis using other possible choices ~ such as for A ~ ¼ ð0; Xx; 0Þ A
(36)
(37)
or
1=2 2J2 cos h2 ; y¼ x2
(29)
~ ¼ ðXy; 0; 0Þ: A
py ¼ ð2J2 x2 Þ1=2 sin h2 ;
(30)
One can easily check that the above gauges satisfy Eqs. (6) and (7) of motion. These gauges only modify some of the
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coefficients in the Hamiltonian (32). Thus, possibilities of resonances are invariant for different gauges as we expect. To explore the origin of chaos we have just used the symmetric gauge which is generally considered in the field of electrodynamics.31 IV. TIME DEPENDENT MAGNETIC FIELD INDUCED CHAOS
In the presence of time dependent magnetic field, B ¼ (0, 0, B(t)), Eqs. (6) and (7) become u_x ¼ 4ax3 þ 2bx þ XðtÞuy þ u_y ¼ x22 y XðtÞux
X0 ðtÞy ; 2
X0 ðtÞx ; 2
(38) (39) FIG. 4. Plot of ux vs x on the Poincare surface of section (y ¼ 0) with initial condition x ¼ 0.005, y ¼ 0.001, ux ¼ 0.0, uy ¼ 0.0, X0 ¼ 0:16, and X1 ¼ 1:0
where XðtÞ ¼
qBðtÞ : mc
(40)
0
X ðtÞ in Eqs. (38) and (39) is the time derivative of XðtÞ. In the present problem, we have considered the following form of B(t): BðtÞ ¼ B0 cosðX1 tÞ:
(41)
Thus, the applied magnetic field oscillates with amplitude B0 and frequency X1 . Using B(t) in Eq. (40), we have XðtÞ ¼ X0 cosðX1 tÞ ¼
qBðtÞ ; mc
(42)
where X0 ¼
qB0 : mc
(43)
To identify the signature of TDMF in the nonlinear dynamics of a charged particle, we have solved numerically Eqs. (4) and (5) and Eqs. (38) and (39), simultaneously. The results of numerical integration of Eqs. (4) and (5) and Eqs. (38) and (39) for the initial condition of the oscillators: xð0Þ ¼ 0:005; yð0Þ ¼ 0:001; ux ð0Þ ¼ 0:0; uy ð0Þ ¼ 0:0; X0 ¼ 0:16, and X1 ¼ 1:0 are shown in Poincare plot (Fig. 4). It is in sharp contrast to what we observe (in the presence of constant magnetic field, X1 ¼ 0:0) in Fig. 5 plotting the results of the numerical integration of equations of motion with the same initial condition, x(0) ¼ 0.005, y(0) ¼ 0.001, ux(0) ¼ 0.0, and uy(0) ¼ 0.0. Thus, time dependent magnetic field can induce chaos. This is further confirmed by calculating the stochastic parameter, bn , for n ¼ 100000000. It is 0.201331304 and 0.000176666 for Figs. 4 and 5, respectively. Thus, identification of chaotic dynamics by inspection of Poincare plot is supported by calculation of the stochastic parameter. There is an interesting distinguishable feature between sets of Figs. 1 and 2 and Figs. 4 and 5. By switching on the constant magnetic field, regular dynamics corresponding to Fig. 1 becomes chaotic as demonstrated in Fig. 2. Energy is same for both Figs. 1 and 2 as the magnetic force does not
work. It is in sharp contrast to coordinate dependent coupling induced chaos, where at a particular energy both regular and chaotic dynamics are not possible. Henon-Heiles model is a common example of coordinate dependent coupling induced chaos.26 We now consider Figs. 4 and 5. In Fig. 5, energy is a constant of motion and trajectory of the dynamics is confined within the right well. But energy is time dependent in case of Fig. 4 as a result of time dependent induced electric field which works on the particle. Thus, if the amplitude of the oscillating magnetic field is appreciably large and frequency is not high then motion may shift from low energy state to the separatrix zone by virtue of the work done by the induced electric field. As a result of that, chaos is observed for any initial position and velocity in the presence of time dependent magnetic filed. But in the presence of constant magnetic field, dynamics may or may not be chaotic. We now explore qualitatively why chaotic dynamics is so frequent in the presence of time dependent magnetic field compared to that in the presence of a time independent field.
FIG. 5. Plot of ux vs x on the Poincare surface of section y ¼ 0 with initial condition x ¼ 0.005, y ¼ 0.001, ux ¼ 0.0, uy ¼ 0.0, X0 ¼ 0:16, and X1 ¼ 0:0
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Following earlier procedure, Hamiltonian in the presence of TDMF can be written as H ¼ h1 cos 2h1 þ g1 cos 4h1 X20 2J1 2J2 2 2 2 2 þ cos ðX1 tÞcos h1 þ cos ðX1 tÞcos h2 8 x1 x2 X0 J1 J2 1=2 cosðX1 tÞððx1 þ x2 Þsinðh1 h2 Þ 2 x1 x2 þ ðx1 x2 Þsinðh1 þ h2 ÞÞ þ h2 ;
the present study general, we have considered that the applied field may be time dependent or not. Our investigation includes the following major points: (i)
(44)
(ii)
where h1 ¼
2aJ12 bJ1 x1 J1 x1 2 x21
(45)
(iii)
and 3aJ12 bJ1 x1 J1 h2 ¼ þ x 2 J2 : þ x1 2 2x21
(iv) (46)
(v)
The above Hamiltonian (44) can be rearranged as H ¼h1 cos 2h1 þ g1 cos 4h1 þ h2 " X20 J1 þ fcosðh1 þ X1 tÞ þ cosðh1 X1 tÞg2 16 x1 J2 þ fcosðh2 þ X1 tÞ þ cosðh2 X1 tÞg2 x2 X0 J1 J2 1=2 ½ðx1 þ x2 Þ½sinðh1 þ h2 þ X1 tÞ 4 x1 x2 þ sinðh1 þ h2 X1 tÞ þ ðx1 x2 Þ ½sinðh1 h2 þ X1 tÞ þ sinðh1 h2 X1 tÞ:
By virtue of coupling between two oscillators through velocity dependent magnetic force, constant magnetic field can induce chaotic motion. As the magnetic force does not work, at a given energy both chaotic and regular dynamics are possible. It is in sharp contrast to coordinate dependent coupling induced chaos. The dynamics may be regular even in the presence of a constant magnetic field. But if the field is an oscillating one then the dynamics becomes chaotic. Thus, time dependent magnetic field can induce chaos. For any initial position and velocity, the dynamics is chaotic for a meaningful amplitude and frequency of the oscillating magnetic field. Chaos in the present system is a result of overlapping of resonance zones. The present study may applicable to understand the dynamics of electrons in nanowire and ions in solid electrolytes in the presence magnetic filed. Extension of present study may be interesting in the case of confined magnetized system. Present analysis may also be extended to the case of relativistic two dimensional dynamics of a charged particle in a magnetic field.
ACKNOWLEDGMENTS
Thanks are due to Council of Scientific and Industrial Research, Govt. of India for partial financial support. (47) 1
The above Hamiltonian implies that h_1 and h_2 are not constant of motion and they are complex functions of cos h1 ; cos h2 ; J1 ; J2 . Therefore, for any given non-zero value of X1 , there is a possibility of resonance by virtue of the terms containing cosðh1 X1 tÞ and cosðh2 X1 tÞ present in the Hamiltonian. There is also a possibility where both the resonances may appear simultaneously and as a result of that chaos may be observed. If the amplitude of the oscillating field is small, the last term of the Hamiltonian as well as resonances would be insignificant and there would be no chaotic dynamics. On the other hand if the amplitude is appreciably large, the force due to induced electric field may be sufficient to drive the particle from the harmonic part of the potential to the anharmonic region. Motion in the nonlinear region easily satisfies the condition of simultaneous resonances. Thus, for any initial position and velocity, chaos is possible in the presence of oscillating magnetic field with meaningful amplitude and frequency. These are consistent with our numerical experiment. V. CONCLUSION
We have studied the dynamics of a particle whose motion is confined in a two-dimensional anharmonic potential in the presence of an external magnetic field. To make
K. Amemiya, J. Phys. Soc. Jpn. 72, 135 (2003). A. S. Moskalenkoa, S. D. Ganicheva, V. I. Perel, and I. N. Yassievicha, Physica B 273–274, 1007 (1999); V. I. Perel and I. N. Yassievich, JETP Lett. 68, 804 (1998); A. S. Moskalenko, V. I. Perel, and I. N. Yassievich, JETP 90, 217 (2000). 3 N. Telang and S. Bandyopadhyay, Appl. Phys. Letts. 66, 1623 (1995). 4 E. E. Vdovin et al., Science 290, 122 (2000). 5 A. Baura, M. K. Sen, and B. C. Bag, Phys. Chem. Chem. Phys. 13, 9445 (2011); A. Baura, M. K. Sen, and B. C. Bag, Chem. Phys. 417, 30 (2013); A. Baura, S. Ray, and B. C. Bag, J. Chem. Phys. 138, 244110 (2013). 6 J. I. Jimenez-Aquino and M. Romero-Bastida, Rev. Mex. Fıs. E 52, 182 (2006). 7 A. M. Jayannavar and M. Sahoo, Phys. Rev. E 75, 032102 (2007). 8 J. I. Jimenez-Aquino, R. M. Velasco, and F. J. Uribe, Phys. Rev. E 77, 051105 (2008). 9 A. Baura, M. K. Sen, and B. C. Bag, Eur. Phys. J. B 75, 267 (2010). 10 A. Baura, M. K. Sen, and B. C. Bag, Phys. Rev. E 82, 041102 (2010); A. Baura, S. Ray, M. K. Sen, and B. C. Bag, J. Appl. Phys. 113, 124905 (2013). 11 D. Sinden and G. H. M. van der Heijden, J. Phys. A: Math. Theor. 42, 375207 (2009). 12 H. Schanz and M. Prusty, J. Phys. A: Math. Theor. 38, 10085 (2005). 13 A. Y. Rom, S. Fishman, R. Kosloff, and T. Maniv, Phys. Rev. B 54, 9819 (1996). 14 J. Aguirre, A. Luque, and D. Peralta-Salas, Physica D 239, 654 (2010). 15 G. Petschel and T. Geisel, Phys. Rev. Lett. 71, 239 (1993). 16 Applications of Electroactive Polymers, edited by B. Scrosati (Chapman & Hall, London, 1993); P. G. Bruce, Solid State Electrochemistry (Cambridge University Press, Cambridge, 1995); F. M. Gray, Polymer Electrolytes (RSC Materials Monographs, The Royal Society of Chemistry, Cambridge, 1997); J.-M. Tarascon and M. Armand, Nature 414, 359 (2001); A. Barnes et al., Smart Mater. Struct. 7, 752 (1998). 2
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C. A. Angell, C. Liu, and E. Sanchez, Nature 362, 137 (1993); A. M. Christie, S. J. Lilley, E. Staunton, Y. G. Andreev and P. G. Bruce, ibid. 433, 50 (2005). 18 P. K. Ghosh, F. Marchesoni, S. E. Savel’ev, and F. Nori, Phys. Rev. Lett. 104, 020601 (2010). 19 A. Pototsky, P. H€anggi, F. Marchesoni, and S. E. Savelev, Phys. Rev. E 84, 011107 (2011). 20 M. Das, D. Mondal, and D. S. Ray, J. Chem. Phys. 136, 114104 (2012). 21 P. K. Ghosh, V. R. Misko, F. Marchesoni, and F. Nori, Phys. Rev. Lett. 110, 268301 (2013). 22 A. Pototsky, F. Marchesoni, F. V. Kusmartsev, P. H€anggi, and S. E. Savelev, Eur. Phys. J. B 85, 356 (2012). 23 A. Holle, G. Wiebusch, J. Main, B. Hager, H. Rottke, and K. H. Welge, Phys. Rev. Lett. 56, 2594 (1986). 24 J. Main, G. Wiebusch, A. Holle, and K. H. Welge, Phys. Rev. Lett. 57, 2789 (1986).
Chaos 23, 043121 (2013) 25
H. Friedrich and D. Wintgen, Phys. Rep. 183, 37 (1989). B. C. Bag, S. Chaudhuri, J. Ray Chaudhuri, and D. S. Ray, Physica D 125, 47 (1999); B. C. Bag and D. S. Ray, J. Stat. Phys. 96, 271 (1999); Phys. Rev. E 61, 3223 (2000); B. C. Bag, S. Chaudhuri, J. Ray Chaudhuri and D. S. Ray, J. Phys. A: Math. Gen. 33, 8331 (2000); B. C. Bag and D. S. Ray, Phys. Rev. E 62, 1927 (2000); 62, 4409 (2000). 27 M. Casartelli, E. Diana, L. Galgani, and A. Scotti, Phys. Rev. A 13, 1921 (1976). 28 G. Benettin, L. Galgani, and J. Strelcyn, Phys. Rev. A 14, 2338 (1976). 29 A. J. Lichtenberg and M. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983). 30 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, 2007). 31 A. Saha and A. M. Jayannavar, Phys. Rev. E 77, 022105 (2008). 26
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CHAOS 23, 043121 (2013)
Magnetic field induced dynamical chaos Somrita Ray, Alendu Baura, and Bidhan Chandra Baga) Department of Chemistry, Visva-Bharati, Santiniketan 731 235, India
(Received 10 May 2013; accepted 6 November 2013; published online 20 November 2013) In this article, we have studied the dynamics of a particle having charge in the presence of a magnetic field. The motion of the particle is confined in the x–y plane under a two dimensional nonlinear potential. We have shown that constant magnetic field induced dynamical chaos is possible even for a force which is derived from a simple potential. For a given strength of the magnetic field, initial position, and velocity of the particle, the dynamics may be regular, but it may become chaotic when the field is time dependent. Chaotic dynamics is very often if the field is time dependent. Origin of chaos has been explored using the Hamiltonian function of the dynamics in terms of action and angle variables. Applicability of the present study has been discussed with a C 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4832175] few examples. V
Velocity dependent magnetic force cannot create or destroy fixed point. It implies that usual bifurcation route to chaos is not the origin of magnetic field (MF) induced chaos. We have explored origin of chaotic dynamics of a charged particle (CP) in the presence of a magnetic and potential energy fields. Our investigation shows that overlapping of resonance zones is necessary for nonintegrable motion of the particle. Another observation is that regular motion of the charged particle becomes chaotic if the applied magnetic field becomes time dependent (TD).
I. INTRODUCTION
Study of dynamics of a particle having charge is always an intriguing issue in physics, since electron or ion is the major constituent of most of the natural systems which belong to subjects like physics, chemistry, and biology. Dynamics of a particle in the presence of magnetic field has been studied in different context in recent past.1–10 In these studies, the particle exhibits random motion as a signature of a statistical system, having infinite dimension due to coupling of the particle with the thermal environment. The effective dynamics is described by Langevin equation of motion which is a stochastic differential equation. But in the present paper, we are exploring a dynamics which is stochastic in nature in a very low dimensional (deterministic) system. This is popularly known as chaotic motion in the field of nonlinear dynamics (NLD). Several models11,12 have been studied to investigate nonlinear dynamics of CP in the presence of a magnetic field. Effect of a uniform magnetic field (MF) on the dynamics of a CP in a conducting rod has been explored in Ref. 11. In this study, it has been shown that an extensible current-carrying rod in a uniform magnetic field is nonintegrable. Nonintegrabale dynamics is also observed in a spatially periodic billiard chain, in which electrons move under the influence of a perpendicular magnetic field.12 a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
1054-1500/2013/23(4)/043121/7/$30.00
Quantum mechanical study shows that spatial periodic variation of magnetic field can induce chaos.13 Nonlinear dynamics of charged particle in the presence of space dependent magnetic field has been studied classically in Ref. 14. In these studies, boundary effect and velocity dependent magnetic force have been only considered. But in conducting solids, electrons experience a periodic force in space from a periodic potential energy function. Dynamics of a charged particle (CP) in x–y plane in the presence of periodic potential and constant magnetic field quantum mechanically in Ref. 15. Here, it has been shown that the dynamics is chaotic at the classical limit. In the present paper, we have studied similar problem based on classical mechanics and explored origin of chaos in a simple way. We have considered double well potential along x-direction, and motion is bounded harmonically in the y-direction. As a result of velocity dependent coupling through magnetic force between two degrees of freedom, there is a possibility of chaotic motion. For some initial conditions (in terms of position and velocity of the particle), the dynamics may be regular in nature even in the presence of the field. But the dynamics is very often chaotic in nature when the external magnetic field is time dependent. The origin of the stochastic nature of the dynamics has been explored by writing the Hamiltonian function in terms of action and angle variables. We now consider the relevance of the present model study. Electrons in solid experience a nonlinear periodic potential. If the solid is a one dimensional nanowire, that periodic potential is along the wire and one may assume that electron is harmonically bound along the direction perpendicular to the axis of the wire. In the present study, two consecutive wells of the periodic potential are accounted by a single double-well potential, which gives the benefit of studying the dynamics of the system in a finite region. Thus, dynamics of electron in nanowire in the presence of magnetic field may mimic the present study. Similarly dynamics of an ion (which is trapped by nonlinear potential in solid electrolytes) in the presence of magnetic field may also mimic the present model. Study of conductivity of ion in solid electrolytes becomes an important issue because of potential applications of these materials in a
23, 043121-1
C 2013 AIP Publishing LLC V
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diverse range of all-solid-state devices such as rechargeable lithium batteries, flexible electrochromic displays, and smart windows.16,17 One can study the effect of chaotic dynamics on the conductivity of solid electrolytes in the presence of magnetic field. It may help to tune the conductivity of ion by controlling chaotic diffusion. Tuning of conductivity is very important for specific needs. The properties of the electrolyte are tuned by varying chemical composition to a large extent and hence are adapted to specific needs.17 Physical method like controlling of chaotic diffusion may be easier route in this context. Interest grows rapidly to study dynamics of particles confined to finite size containers of various geometries.18–21 In Ref. 19, it has been shown that a simpler instance of temperature controlled resonance can naturally occur in a confined magnetized electron gas. Extension of present study may be interesting in the case of confined magnetized system. Present analysis may also be extended following Ref. 22 to the case of relativistic two dimensional dynamics of a charged particle in a magnetic field. Before leaving this part, we would like to mention that chaotic dynamics as a result of interplay of constant magnetic field and force derived from potential has also been studied both experimentally23,24 and theoretically25 for hydrogen atom in a uniform magnetic filed. In the present paper, we have shown that constant magnetic field induced dynamical chaos is possible even for a force which is derived from a simple potential. The outline of the paper is as follows: In Sec. II, we have presented the model. Constant magnetic field induced chaos, and its origin have been discussed in Sec. III. Dynamics of a particle in the presence of time dependent magnetic field has been presented in Sec. IV. The paper is concluded in Sec. V.
field is pointing along the z-axis of the Cartesian reference frame, i.e., B ¼ (0,0,B). For this choice of magnetic field, Eq. (1) can be described by means of two independent processes. One is described in the x–y plane (which is perpendicular to the direction of the applied magnetic field) with the potential energy function, Vðx; yÞ ¼ ax4 bx2 þx22 y2 =2 and the other is along the direction of the magnetic field having potential energy function, VðzÞ ¼ x22 z2 =2. In these cases, Eq. (1) can be written as
II. THE MODEL
III. CONSTANT MAGNETIC FIELD INDUCED CHAOS
To start with, we consider the equation of motion of a particle (having charge q and mass m) in three dimension in the presence of a magnetic field B. It can be written for the velocity vector u (Refs. 5 and 10) as
To study the present problem, we have solved Eqs. (4)–(7) numerically using the well known fourth order Runge-Kutta method. Relevant parameters for the numerical study are a ¼ 0.25, b ¼ 0.5, and x2 ¼ 1:0. The results of numerical integration of Eqs. (4)–(7) in the absence of magnetic field ðX ¼ 0:0Þ and for the initial condition of the oscillators: x(0) ¼ 0.005, y(0) ¼ 0.001, ux(0) ¼ 0.0, and uy(0) ¼ 0.0 are shown in Poincare plot (Fig. 1). This is a characteristic of regular motion of uncoupled oscillators. It is in sharp contrast to what we have observed (in the presence of magnetic field) in Fig. 2 plotting the results of the numerical integration of equations of motion with the same initial condition, x(0) ¼ 0.005, y(0) ¼ 0.001, ux(0) ¼ 0.0, and uy(0) ¼ 0.0. Thus, magnetic field can induce chaos through velocity dependent coupling between two oscillators. To become sure about this phenomenon, we have studied dynamics for other initial condition and observed chaos (results are shown in Fig. 3). This is a new observation with respect to known coordinate dependent coupling induced chaos.26 We now calculate an important parameter (a precursor of largest Lyapunov exponent used as a measure of regularity or chaoticity of nonlinear dynamical system) proposed many years ago in Ref. 27. It is the long time average of ln dðtÞ=d0 , where d0 is the initial separation of the two
u_ ¼ F;
(1)
where F¼
1 q rVðrÞ þ u B: m mc
(2)
Here, rV is the gradient of the potential V and c is the speed of light. In the present study, we have considered that the particle is bounded by harmonic potential along y-z directions and experience a double-well potential along x-direction. Thus, the total potential energy can be represented as Vðx; y; zÞ ¼ ax4 bx2 þ x22 ðy2 þ z2 Þ=2;
(3)
where a and b are constants which determine the fixed points corresponding to the potential as well as the barrier height. x2 in the last term of the above equation is the characteristic frequency of the oscillation along y and z directions, respectively. For simplicity, we consider that the applied magnetic
x_ ¼ ux ;
(4)
y_ ¼ uy ;
(5)
u_x ¼ 4ax3 þ 2bx þ Xuy ;
(6)
u_y ¼ x22 y Xux ;
(7)
z_ ¼ uz ;
(8)
u_z ¼ x22 z;
(9)
where X¼
qB : mc
(10)
The above Eqs.(4)–(9) imply that the motion along zcomponent is independent of the applied magnetic field and it has no effect on the motion in x–y plane. In this plane, we have studied non-linear dynamics of a charged particle in the presence of an external magnetic field. In the next section, i.e., in Sec. III, we shall demonstrate that constant magnetic field can induce chaos.
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FIG. 1. Plot of ux vs x on the Poincare surface of section (y ¼ 0) with initial condition x ¼ 0.005, y ¼ 0.001, ux ¼ 0.0, uy ¼ 0.0, and X ¼ 0:0.
initially nearby trajectories and d(t) is the corresponding separation at some time, t. d(t) having dimension of length is expressed as "
Du2 Du2y Þ dðtÞ ¼ Dx2 þ Dy2 þ 2x þ 2 x1 x2
#1=2 ;
(11)
where x1 corresponds to the frequency around the stable fixed points of the nonliner potential along x-direction. Dx; Dy; Dux , and Duy in above equation are related to two nearby trajectories, x, y, ux, uy, and x þ Dx; y þ Dy; ux þ Dux ; uy þ Duy at the same time t in four dimensional phase space. Based on this relationship and Eqs. (4)–(7), one can write time evolution equation for Dx; Dy; Dux , and Duy as _ ¼ Dux ; Dx
(12)
_ ¼ Duy ; Dy
(13)
_ x ¼ ð3ax2 þ 2bxÞDx þ XDuy ; Du
(14)
_ y ¼ x2 Dy XDux : Du 2
(15)
FIG. 2. Plot of ux vs x on the Poincare surface of section (y ¼ 0) with initial condition x ¼ 0.005, y ¼ 0.001, ux ¼ 0.0, uy ¼ 0.0, and X ¼ 0:5.
FIG. 3. Plot of ux vs x on the Poincare surface of section (y ¼ 0) with initial condition x ¼ 2.4, y ¼ 2.0, ux ¼ 0.0, uy ¼ 0.0, and X ¼ 0.5.
Assuming that Dx; Dy; Dux , and Duy are small, we have ignored the terms in the above set of equations which are nonlinear in terms of these quantities. To determine d(t), we have numerically solved Eqs. (4)–(7) and Eqs. (12)–(15), simultaneously. In going from the j-th step to the jþ1-th step of the iteration in course of time evolution, Dx has to be initialized as Dxðjþ1Þ0 ¼ ðDxj0 =dj Þd0 . Similar procedure has been followed for related quantities. This initialization implies that at each step, the iteration starts with same magnitude of d0 but the direction of d0 for step j þ 1 is that of d(t) for j-th step. For a pictorial illustration, we refer to Fig. 1 of Ref. 28. The j-th time of iteration is determined by t ¼ jTðj ¼ 1; 2; ::::1Þ. Here, T is the characteristic time that corresponds to the shortest ensemble averaged period of a nonlinear dynamical system. Thus, following Casartelli et al.,27 a stochastic parameter can be defined by the following time average of lndj =d0 as bn ðt; xð0Þ; yð0Þ; ux ð0Þ; uy ð0Þ; d0 Þ ¼
n dj 1X ln : n j¼1 d0
(16)
In Ref. 27, it has been shown that as n ! 1; bn has a definite value. It is positive for the chaotic systems. For the regular system the stochastic parameter is zero. bn has been calculated for n ¼ 100000000. We have checked that with this large value of n the stochastic parameter becomes n-independent. The value of this parameter for Figs. 1 and 2 are 0.000005058 and 0.010427679, respectively. Thus, magnetic field induced chaos is further confirmed. b1 is 0.017863778 for Fig. 3. Large positive value of the stochastic parameter for Fig. 3 implies that the dynamics corresponding to this figure is chaotic. It is apparent from Figs. 2 and 3 that chaos is strong in Fig. 3 compared to Fig. 2. This inspection is supported by the values of the stochastic parameter corresponding to these figures. We now explore qualitatively the origin of magnetic field induced chaos. At this end, we write the Hamiltonian for the present system in terms of the canonical momentum, scalar and vector potentials as
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H¼
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Chaos 23, 043121 (2013)
ðpx Ax ðx; yÞÞ2 ðpy Ay ðx; yÞÞ2 y2 þ þ ax4 bx2 þ x22 : 2 2 2 (17)
In the above equation, we have used q ¼ 1, m ¼ 1, and c ¼ 1. px and Ax in Eq. (17) are components (along x-direction) of ~ respectively. momentum vector ð~ p Þ and vector potential ðAÞ, Similarly py and Ay are components of these quantities along y-direction. Relation between px and Ax is px ¼ ux þ Ax :
where J1 and h1 are the action and angle variables corresponding to oscillation along x-direction. Similarly, J2 and h2 are the action angle variables related to oscillation along y-direction. Making use of Eqs. (27)–(30) in Eq. (26), one may write that H¼
(18)
Similarly, py and Ay are related by py ¼ uy þ Ay :
p_x ¼ u_x þ A_x
þ ðx1 x2 Þsinðh1 þ h2 ÞÞ:
(19)
From Eqs. (18) and (19), one may write (20)
4aJ12 2bJ1 cos4 h1 cos2 h1 þ x1 J1 sin2 h1 þ x2 J2 2 x1 x1 X2 2J1 2J2 þ cos2 h1 þ cos2 h2 8 x1 x2 1=2 X J1 J 2 ððx1 þ x2 Þsinðh1 h2 Þ 2 x1 x2
Making use of simple trigonometric relations,30 the above Hamiltonian can be transformed as H ¼ 2g cosðh1 þ h2 Þcosðh1 h2 Þ
and p_y ¼ u_y þ A_y :
þ 2g1 cosð2h1 þ h2 Þcosð2h1 h2 Þ ! X 2 J2 g g1 cos 2h2 8x2 X J1 J2 1=2 ððx1 þ x2 Þsinðh1 h2 Þ 2 x1 x2
(21)
Using Hamilton’s equations of motion in Eqs. (20) and (21), we have u_x ¼
@Ay @Ax @Ax uy þ uy 4ax3 þ 2bx; @t @y @x
u_y ¼
@Ay @Ay @Ax ux þ ux x22 y: @t @x @y
(22) (23)
þ ðx1 x2 Þsinðh1 þ h2 ÞÞ þ g2 ;
Xy ; Ax ¼ 2 Ay ¼
g¼
(25)
We now introduce action and angle variables through the following relations:29 x¼
1=2
px ¼ ð2J1 x1 Þ
cos h1 ; 1=2
(33)
aJ12 ; 2x21
(34)
and
Xx : 2
p2x p2y Xðypx xpy Þ X2 ðx2 þ y2 Þ y2 þ þ þ ax4 bx2 þ x22 : þ 2 2 2 8 2 (26)
2J1 x1
2aJ12 bJ1 x1 J1 X2 J1 þ ; x1 2 8x1 x21 g1 ¼
(24)
This choice is nothing but the popularly known symmetric gauge for A. For the above relations (24) and (25) Hamiltonian (17) becomes
(32)
where
Comparing Eqs. (6) and (7) with Eqs. (22) and (23), one can identify that
H¼
(31)
sin h1 ;
(27) (28)
g2 ¼
3aJ12 bJ1 x1 J1 X 2 J1 J2 : þ x þ J þ þ 2 2 x1 2 8 x1 x2 2x21
(35)
It is apparent in the first three terms in Hamiltonian (32) that there are possibilities of 1:1 and 2:1 resonances. Overlapping of these two resonance zones may result in chaos as demonstrated in Figs. 2 and 3. It is supported by the equations of motion. Equations (4)–(7) imply that magnetic force cannot create or destroy fixed point. Hence bifurcation route to chaos is not the origin of chaos for the present case. Before leaving this part, we would like to mention that one can also do above analysis using other possible choices ~ such as for A ~ ¼ ð0; Xx; 0Þ A
(36)
(37)
or
1=2 2J2 cos h2 ; y¼ x2
(29)
~ ¼ ðXy; 0; 0Þ: A
py ¼ ð2J2 x2 Þ1=2 sin h2 ;
(30)
One can easily check that the above gauges satisfy Eqs. (6) and (7) of motion. These gauges only modify some of the
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coefficients in the Hamiltonian (32). Thus, possibilities of resonances are invariant for different gauges as we expect. To explore the origin of chaos we have just used the symmetric gauge which is generally considered in the field of electrodynamics.31 IV. TIME DEPENDENT MAGNETIC FIELD INDUCED CHAOS
In the presence of time dependent magnetic field, B ¼ (0, 0, B(t)), Eqs. (6) and (7) become u_x ¼ 4ax3 þ 2bx þ XðtÞuy þ u_y ¼ x22 y XðtÞux
X0 ðtÞy ; 2
X0 ðtÞx ; 2
(38) (39) FIG. 4. Plot of ux vs x on the Poincare surface of section (y ¼ 0) with initial condition x ¼ 0.005, y ¼ 0.001, ux ¼ 0.0, uy ¼ 0.0, X0 ¼ 0:16, and X1 ¼ 1:0
where XðtÞ ¼
qBðtÞ : mc
(40)
0
X ðtÞ in Eqs. (38) and (39) is the time derivative of XðtÞ. In the present problem, we have considered the following form of B(t): BðtÞ ¼ B0 cosðX1 tÞ:
(41)
Thus, the applied magnetic field oscillates with amplitude B0 and frequency X1 . Using B(t) in Eq. (40), we have XðtÞ ¼ X0 cosðX1 tÞ ¼
qBðtÞ ; mc
(42)
where X0 ¼
qB0 : mc
(43)
To identify the signature of TDMF in the nonlinear dynamics of a charged particle, we have solved numerically Eqs. (4) and (5) and Eqs. (38) and (39), simultaneously. The results of numerical integration of Eqs. (4) and (5) and Eqs. (38) and (39) for the initial condition of the oscillators: xð0Þ ¼ 0:005; yð0Þ ¼ 0:001; ux ð0Þ ¼ 0:0; uy ð0Þ ¼ 0:0; X0 ¼ 0:16, and X1 ¼ 1:0 are shown in Poincare plot (Fig. 4). It is in sharp contrast to what we observe (in the presence of constant magnetic field, X1 ¼ 0:0) in Fig. 5 plotting the results of the numerical integration of equations of motion with the same initial condition, x(0) ¼ 0.005, y(0) ¼ 0.001, ux(0) ¼ 0.0, and uy(0) ¼ 0.0. Thus, time dependent magnetic field can induce chaos. This is further confirmed by calculating the stochastic parameter, bn , for n ¼ 100000000. It is 0.201331304 and 0.000176666 for Figs. 4 and 5, respectively. Thus, identification of chaotic dynamics by inspection of Poincare plot is supported by calculation of the stochastic parameter. There is an interesting distinguishable feature between sets of Figs. 1 and 2 and Figs. 4 and 5. By switching on the constant magnetic field, regular dynamics corresponding to Fig. 1 becomes chaotic as demonstrated in Fig. 2. Energy is same for both Figs. 1 and 2 as the magnetic force does not
work. It is in sharp contrast to coordinate dependent coupling induced chaos, where at a particular energy both regular and chaotic dynamics are not possible. Henon-Heiles model is a common example of coordinate dependent coupling induced chaos.26 We now consider Figs. 4 and 5. In Fig. 5, energy is a constant of motion and trajectory of the dynamics is confined within the right well. But energy is time dependent in case of Fig. 4 as a result of time dependent induced electric field which works on the particle. Thus, if the amplitude of the oscillating magnetic field is appreciably large and frequency is not high then motion may shift from low energy state to the separatrix zone by virtue of the work done by the induced electric field. As a result of that, chaos is observed for any initial position and velocity in the presence of time dependent magnetic filed. But in the presence of constant magnetic field, dynamics may or may not be chaotic. We now explore qualitatively why chaotic dynamics is so frequent in the presence of time dependent magnetic field compared to that in the presence of a time independent field.
FIG. 5. Plot of ux vs x on the Poincare surface of section y ¼ 0 with initial condition x ¼ 0.005, y ¼ 0.001, ux ¼ 0.0, uy ¼ 0.0, X0 ¼ 0:16, and X1 ¼ 0:0
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Chaos 23, 043121 (2013)
Following earlier procedure, Hamiltonian in the presence of TDMF can be written as H ¼ h1 cos 2h1 þ g1 cos 4h1 X20 2J1 2J2 2 2 2 2 þ cos ðX1 tÞcos h1 þ cos ðX1 tÞcos h2 8 x1 x2 X0 J1 J2 1=2 cosðX1 tÞððx1 þ x2 Þsinðh1 h2 Þ 2 x1 x2 þ ðx1 x2 Þsinðh1 þ h2 ÞÞ þ h2 ;
the present study general, we have considered that the applied field may be time dependent or not. Our investigation includes the following major points: (i)
(44)
(ii)
where h1 ¼
2aJ12 bJ1 x1 J1 x1 2 x21
(45)
(iii)
and 3aJ12 bJ1 x1 J1 h2 ¼ þ x 2 J2 : þ x1 2 2x21
(iv) (46)
(v)
The above Hamiltonian (44) can be rearranged as H ¼h1 cos 2h1 þ g1 cos 4h1 þ h2 " X20 J1 þ fcosðh1 þ X1 tÞ þ cosðh1 X1 tÞg2 16 x1 J2 þ fcosðh2 þ X1 tÞ þ cosðh2 X1 tÞg2 x2 X0 J1 J2 1=2 ½ðx1 þ x2 Þ½sinðh1 þ h2 þ X1 tÞ 4 x1 x2 þ sinðh1 þ h2 X1 tÞ þ ðx1 x2 Þ ½sinðh1 h2 þ X1 tÞ þ sinðh1 h2 X1 tÞ:
By virtue of coupling between two oscillators through velocity dependent magnetic force, constant magnetic field can induce chaotic motion. As the magnetic force does not work, at a given energy both chaotic and regular dynamics are possible. It is in sharp contrast to coordinate dependent coupling induced chaos. The dynamics may be regular even in the presence of a constant magnetic field. But if the field is an oscillating one then the dynamics becomes chaotic. Thus, time dependent magnetic field can induce chaos. For any initial position and velocity, the dynamics is chaotic for a meaningful amplitude and frequency of the oscillating magnetic field. Chaos in the present system is a result of overlapping of resonance zones. The present study may applicable to understand the dynamics of electrons in nanowire and ions in solid electrolytes in the presence magnetic filed. Extension of present study may be interesting in the case of confined magnetized system. Present analysis may also be extended to the case of relativistic two dimensional dynamics of a charged particle in a magnetic field.
ACKNOWLEDGMENTS
Thanks are due to Council of Scientific and Industrial Research, Govt. of India for partial financial support. (47) 1
The above Hamiltonian implies that h_1 and h_2 are not constant of motion and they are complex functions of cos h1 ; cos h2 ; J1 ; J2 . Therefore, for any given non-zero value of X1 , there is a possibility of resonance by virtue of the terms containing cosðh1 X1 tÞ and cosðh2 X1 tÞ present in the Hamiltonian. There is also a possibility where both the resonances may appear simultaneously and as a result of that chaos may be observed. If the amplitude of the oscillating field is small, the last term of the Hamiltonian as well as resonances would be insignificant and there would be no chaotic dynamics. On the other hand if the amplitude is appreciably large, the force due to induced electric field may be sufficient to drive the particle from the harmonic part of the potential to the anharmonic region. Motion in the nonlinear region easily satisfies the condition of simultaneous resonances. Thus, for any initial position and velocity, chaos is possible in the presence of oscillating magnetic field with meaningful amplitude and frequency. These are consistent with our numerical experiment. V. CONCLUSION
We have studied the dynamics of a particle whose motion is confined in a two-dimensional anharmonic potential in the presence of an external magnetic field. To make
K. Amemiya, J. Phys. Soc. Jpn. 72, 135 (2003). A. S. Moskalenkoa, S. D. Ganicheva, V. I. Perel, and I. N. Yassievicha, Physica B 273–274, 1007 (1999); V. I. Perel and I. N. Yassievich, JETP Lett. 68, 804 (1998); A. S. Moskalenko, V. I. Perel, and I. N. Yassievich, JETP 90, 217 (2000). 3 N. Telang and S. Bandyopadhyay, Appl. Phys. Letts. 66, 1623 (1995). 4 E. E. Vdovin et al., Science 290, 122 (2000). 5 A. Baura, M. K. Sen, and B. C. Bag, Phys. Chem. Chem. Phys. 13, 9445 (2011); A. Baura, M. K. Sen, and B. C. Bag, Chem. Phys. 417, 30 (2013); A. Baura, S. Ray, and B. C. Bag, J. Chem. Phys. 138, 244110 (2013). 6 J. I. Jimenez-Aquino and M. Romero-Bastida, Rev. Mex. Fıs. E 52, 182 (2006). 7 A. M. Jayannavar and M. Sahoo, Phys. Rev. E 75, 032102 (2007). 8 J. I. Jimenez-Aquino, R. M. Velasco, and F. J. Uribe, Phys. Rev. E 77, 051105 (2008). 9 A. Baura, M. K. Sen, and B. C. Bag, Eur. Phys. J. B 75, 267 (2010). 10 A. Baura, M. K. Sen, and B. C. Bag, Phys. Rev. E 82, 041102 (2010); A. Baura, S. Ray, M. K. Sen, and B. C. Bag, J. Appl. Phys. 113, 124905 (2013). 11 D. Sinden and G. H. M. van der Heijden, J. Phys. A: Math. Theor. 42, 375207 (2009). 12 H. Schanz and M. Prusty, J. Phys. A: Math. Theor. 38, 10085 (2005). 13 A. Y. Rom, S. Fishman, R. Kosloff, and T. Maniv, Phys. Rev. B 54, 9819 (1996). 14 J. Aguirre, A. Luque, and D. Peralta-Salas, Physica D 239, 654 (2010). 15 G. Petschel and T. Geisel, Phys. Rev. Lett. 71, 239 (1993). 16 Applications of Electroactive Polymers, edited by B. Scrosati (Chapman & Hall, London, 1993); P. G. Bruce, Solid State Electrochemistry (Cambridge University Press, Cambridge, 1995); F. M. Gray, Polymer Electrolytes (RSC Materials Monographs, The Royal Society of Chemistry, Cambridge, 1997); J.-M. Tarascon and M. Armand, Nature 414, 359 (2001); A. Barnes et al., Smart Mater. Struct. 7, 752 (1998). 2
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 209.183.185.254 On: Sun, 23 Nov 2014 03:53:10
043121-7 17
Ray, Baura, and Bag
C. A. Angell, C. Liu, and E. Sanchez, Nature 362, 137 (1993); A. M. Christie, S. J. Lilley, E. Staunton, Y. G. Andreev and P. G. Bruce, ibid. 433, 50 (2005). 18 P. K. Ghosh, F. Marchesoni, S. E. Savel’ev, and F. Nori, Phys. Rev. Lett. 104, 020601 (2010). 19 A. Pototsky, P. H€anggi, F. Marchesoni, and S. E. Savelev, Phys. Rev. E 84, 011107 (2011). 20 M. Das, D. Mondal, and D. S. Ray, J. Chem. Phys. 136, 114104 (2012). 21 P. K. Ghosh, V. R. Misko, F. Marchesoni, and F. Nori, Phys. Rev. Lett. 110, 268301 (2013). 22 A. Pototsky, F. Marchesoni, F. V. Kusmartsev, P. H€anggi, and S. E. Savelev, Eur. Phys. J. B 85, 356 (2012). 23 A. Holle, G. Wiebusch, J. Main, B. Hager, H. Rottke, and K. H. Welge, Phys. Rev. Lett. 56, 2594 (1986). 24 J. Main, G. Wiebusch, A. Holle, and K. H. Welge, Phys. Rev. Lett. 57, 2789 (1986).
Chaos 23, 043121 (2013) 25
H. Friedrich and D. Wintgen, Phys. Rep. 183, 37 (1989). B. C. Bag, S. Chaudhuri, J. Ray Chaudhuri, and D. S. Ray, Physica D 125, 47 (1999); B. C. Bag and D. S. Ray, J. Stat. Phys. 96, 271 (1999); Phys. Rev. E 61, 3223 (2000); B. C. Bag, S. Chaudhuri, J. Ray Chaudhuri and D. S. Ray, J. Phys. A: Math. Gen. 33, 8331 (2000); B. C. Bag and D. S. Ray, Phys. Rev. E 62, 1927 (2000); 62, 4409 (2000). 27 M. Casartelli, E. Diana, L. Galgani, and A. Scotti, Phys. Rev. A 13, 1921 (1976). 28 G. Benettin, L. Galgani, and J. Strelcyn, Phys. Rev. A 14, 2338 (1976). 29 A. J. Lichtenberg and M. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983). 30 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, 2007). 31 A. Saha and A. M. Jayannavar, Phys. Rev. E 77, 022105 (2008). 26
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 209.183.185.254 On: Sun, 23 Nov 2014 03:53:10
