Bifurcations and a chaos strip in states of long Josephson junctions
Stationary and nonstationary, in particular, chaotic states in long Josephson junctions are investigated. Bifurcation lines on the parametric bias current-external magnetic field plane are calculated. The chaos strip along the bifurcation line is observed. It is shown that transitions between statio...
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irk-123456789-1378632018-06-18T03:10:07Z Bifurcations and a chaos strip in states of long Josephson junctions Yugay, K.N. Blinov, N.V. Shirokov, I.V. Электpонные свойства металлов и сплавов Stationary and nonstationary, in particular, chaotic states in long Josephson junctions are investigated. Bifurcation lines on the parametric bias current-external magnetic field plane are calculated. The chaos strip along the bifurcation line is observed. It is shown that transitions between stationary states are the transitions from metastable to stable states and that the thermodynamical Gibbs potential of these stable states may be larger than for some metastable states. The definition of a dynamical critical magnetic field characterizing the stability of the stationary states is given. 1999 Article Bifurcations and a chaos strip in states of long Josephson junctions / K.N. Yugay, N.V. Blinov, I.V. Shirokov // Физика низких температур. — 1999. — Т. 25, № 7. — С. 712-717. — Бібліогр.: 15 назв. — англ. 0132-6414 http://dspace.nbuv.gov.ua/handle/123456789/137863 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Электpонные свойства металлов и сплавов Электpонные свойства металлов и сплавов |
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Электpонные свойства металлов и сплавов Электpонные свойства металлов и сплавов Yugay, K.N. Blinov, N.V. Shirokov, I.V. Bifurcations and a chaos strip in states of long Josephson junctions Физика низких температур |
| description |
Stationary and nonstationary, in particular, chaotic states in long Josephson junctions are investigated. Bifurcation lines on the parametric bias current-external magnetic field plane are calculated. The chaos strip along the bifurcation line is observed. It is shown that transitions between stationary states are the transitions from metastable to stable states and that the thermodynamical Gibbs potential of these stable states may be larger than for some metastable states. The definition of a dynamical critical magnetic field characterizing the stability of the stationary states is given. |
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Article |
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Yugay, K.N. Blinov, N.V. Shirokov, I.V. |
| author_facet |
Yugay, K.N. Blinov, N.V. Shirokov, I.V. |
| author_sort |
Yugay, K.N. |
| title |
Bifurcations and a chaos strip in states of long Josephson junctions |
| title_short |
Bifurcations and a chaos strip in states of long Josephson junctions |
| title_full |
Bifurcations and a chaos strip in states of long Josephson junctions |
| title_fullStr |
Bifurcations and a chaos strip in states of long Josephson junctions |
| title_full_unstemmed |
Bifurcations and a chaos strip in states of long Josephson junctions |
| title_sort |
bifurcations and a chaos strip in states of long josephson junctions |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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1999 |
| topic_facet |
Электpонные свойства металлов и сплавов |
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http://dspace.nbuv.gov.ua/handle/123456789/137863 |
| citation_txt |
Bifurcations and a chaos strip in states of long Josephson junctions / K.N. Yugay, N.V. Blinov, I.V. Shirokov // Физика низких температур. — 1999. — Т. 25, № 7. — С. 712-717. — Бібліогр.: 15 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT yugaykn bifurcationsandachaosstripinstatesoflongjosephsonjunctions AT blinovnv bifurcationsandachaosstripinstatesoflongjosephsonjunctions AT shirokoviv bifurcationsandachaosstripinstatesoflongjosephsonjunctions |
| first_indexed |
2025-07-10T04:38:35Z |
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2025-07-10T04:38:35Z |
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1837233414703415296 |
| fulltext |
Fizika Nizkikh Temperatur, 1999, v. 25, No 7, p. 712–717Yugay K. N., Blinov N. V., and Shirokov I. V.Bifurcations and a chaos strip in states of long Josephson junctionsYugay K.
N., Blinov N. V., and Shirokov I. V.Bifurcations and a chaos strip in states of long Josephson junctions
Bifurcations and a chaos strip in states of long
Josephson junctions
K. N. Yugay, N. V. Blinov, and I. V. Shirokov
Omsk State University, 55a Mira Ave., 644077, Omsk, Russia
E-mail: yugay@univer.omsk.su
Received January 6, 1999
Stationary and nonstationary, in particular, chaotic states in long Josephson junctions are investi-
gated. Bifurcation lines on the parametric bias current-external magnetic field plane are calculated. The
chaos strip along the bifurcation line is observed. It is shown that transitions between stationary states
are the transitions from metastable to stable states and that the thermodynamical Gibbs potential of
these stable states may be larger than for some metastable states. The definition of a dynamical critical
magnetic field characterizing the stability of the stationary states is given.
PACS: 74.50.+r, 05.45.+b
Introduction
Dynamical chaos in long Josephson junctions is
of great interest because it can be a source of a
dynamical noise in devices based on them, in par-
ticular, in SQUIDs, limiting the sensitivity of these
devices. Furthermore, dynamical chaos in long
Josephson junctions (LJJ) is a very interesting
physical phenomenon taking place in nonlinear sys-
tems in the absence of an external stochastic
force [1–9]. Dynamical chaos in a LJJ is easily
excited and therefore it may also be investigated
experimentally rather easily [10,11].
In our previous works [12,13] we have shown
that among a set of solutions of the Ferrell-Prange
equation describing stationary states of the LJJ in
an external magnetic field [14] are both stable and
unstable ones. At the same time, these stationary
states are asymptotic solutions of the nonstationary
sine-Gordon equation and we have also shown that
a selection of the stable solutions can be governed
by the initial perturbation damping rapidly in time
and entering into the nonstationary sine-Gordon
equation through the boundary conditions. Chang-
ing the intensity of this perturbation at fixed shape,
we can obtain various stationary states for the LJJ
without a bias current or three clusters of states
(stationary, and time dependent regular and cha-
otic) in the presence of a bias current. It turned out
that asymptotic states are very sensitive to an exter-
nal perturbation, its value and shape define the
state (stationary, regular or chaotic) to which the
system will tend at t → ∞ (we have called this
influence on the selection of asymptotic states of the
small initial perturbation damping rapidly in time
an effect of memory). The fact of coexistence of all
these three characteristic asymptotic states selected
only by the form of the initial perturbation seems to
be astonishing. It is evidently enough that the
Ferrell-Prange equation will not have solutions at a
large bias current β. Therefore the question arises:
at which values of β do stationary states of a LJJ
disappear or what will be a boundary in the para-
metric β − H0 plane (H0 is an external magnetic
field) that separates this plane on the regions where
stationary states do and do not exist? Since the
number of solutions of the Ferrell-Prange equation
changes at variation of the parameters (H0 , β),
another question arises: what is the form of bifurca-
tion lines in the plane β − H0 that separate the
parametric plane on the regions with a different
number of stationary solutions of the Ferrell-Prange
equation?
The existence of several stable solutions of the
Ferrell-Prange equation is equivalent to the fact
that thermodynamical Gibbs potential G associated
with the distribution of the magnetic field along the
junction has minima, and each minimum corre-
sponds to a certain solution of the Ferrell-Prange
equation. Does a global minimum of G correspond
to the most stable state (e.g., in the Lyapunov
sense)? In the case of the junction of the finite
length both Meissner and one-fluxon states are
© K. N. Yugay, N. V. Blinov, and I. V. Shirokov, 1999
thermodynamically advantageous simultaneously,
so it is interesting to investigate dynamical proper-
ties of these states. Answering this question, we
introduce a dynamical critical field that describes
the stability characteristic of the junctions.
In Sec. 1 bifurcation lines on the parametric
β − H0 plane are calculated. In Sec. 2 the definition
of the dynamical critical magnetic field is given and
the dependence of this field on β and the length of
the junction L is calculated. In Sec. 3 transitions
between states are described. It is shown in Sec. 4
that a chaos strip arises along the bifurcation line
on the parametric β−H0 plane. The last Sec. 5
contains the discussion of our calculation and brief
conclusions.
1. Bifurcation lines
Stationary states of a LJJ are investigated using
the numerical integration of the Ferrell-Prange
equation:
ϕ
xx
(x) = sin ϕ(x) − β , (1)
where ϕ(x) is the stationary Josephson phase vari-
able, β is the dc bias current density normalized to
the critical current jc , x is the distance along the
junction normalized to the Josephson penetration
length λJ = √CΦ0/8π2jc d, Φ0 is the flux quantum,
d = 2λL + b, λL is the London penetration length,
b is the thickness of the dielectric barrier. The
boundary conditions for Eq. (1) have the form
ϕ
x
(x)|
x=0 = ϕ
x
(x)|
x=L
= H0 , (2)
where L is the total length of the junction norma-
lized to λJ and H0 is the external magnetic field
perpendicular to the junction and normalized to
H~ = Φ0/2πλJ d.
Numerical integration of Eqs. (1)–(2) allows us
to find the regions with a certain number of solu-
tions on the parametric β−H0 plane (Fig. 1). It is
easy to show that the set of points corresponding to
the even number of solutions forms two-dimensional
domains on this plane, whereas the set correspond-
ing to the odd ones may form just one-dimensional
curves. Mostly, the lines corresponding to the odd
number of the solutions of the Ferrell-Prange
boundary problem coincide with the bifurcation
lines. Using the shooting method for solving of the
boundary problem one can prove that the 2π-peri-
odicity of the function H(ϕ0) expressing the depen-
dence of the magnetic field at the right side of the
junction (x = L) on the phase taken at the left side
(x = 0) results in the appearing of the odd number
of solutions only when the H(ϕ0) touches the line
H = H0 in an extreme point, i.e., ∂H(ϕ0)/∂ϕ0 = 0.
As an illustration, we have plotted in Fig. 2 the
function H(ϕ0) at H0 = 0.5, L = 5, β = 0.25 and
β = 0.45.
Boundaries between the regions — bifurcation
lines — define an essential modification of the
system. The bifurcation lines in Fig. 1 are obtained
for L = 5; here a step by β is equal to 5⋅10−3 and a
step by H0 is equal to 2.5⋅10−3. In this figure the
numbers of solutions of Eq. (1)–(2) are pointed
out, the numbers of stable solutions are given in the
brackets, and M and 1f denote a stable Meissner
and one-fluxon states, respectively. It is seen that a
Meissner state is stable at small values of H0 and at
large values of H0 a one fluxon state is stable. It
should be noted that the region where there are no
Fig. 1. Bifurcation lines. The number of solution of the Ferrell-
Prange equation (1)–(2) are pointed out. The number of stable
states is indicated in brackets. M denotes a stable Meissner
state and 1f denotes a stable one-fluxon state. L = 5.
Fig. 2. Dependence of the magnetic field at x = L on the phase
taken at the left side of junction x = 0 at H0 = 0.5, L = 5,
β = 0.25 and β = 0.45.
Bifurcations and a chaos strip in states of long Josephson junctions
Fizika Nizkikh Temperatur, 1999, v. 25, No 7 713
stationary solutions (region 0) bounds with the
region having a minimum of stationary solutions,
being equal to 2 (region 2). In approaching the
boundary of region 0 and 2 the number of stationary
solutions decreases: 6 → 4 → 2 → 0, on the other
hand, a number of nonstationary states which are
the asymptotic solutions of the sine-Gordon equa-
tion, increases. Our calculations have shown that
one of two stationary solutions in region 2 is stable,
and another is unstable (metastable). We noted
earlier [12] that the stable states are symmetrical.
The presence of bias current β leads to a symmetry
violation that results, evidently, in the instability
of the states.
The problem of the stability of stationary states
ϕ(x) was solved in the following way [13]: the
sine-Gordon equation was linearized in the vicinity
of stationary solution: ϕ(x, t) = ϕ(x) + θ(x, t),
where θ(x, t) is the infinitesimal perturbation. The
equation for θ(x, t) — the linearized sine-Gordon
equation — we can solve by means of the expansion
of this function in terms of a complete system of
eigenfunctions of the Schro..dinger operator with
potential cos [ϕ(x)]:
θ(x, t) = ∑
n
eλ
n
tun(x) , (3)
where un(x) are eigenfunctions of the Schro..dinger
operator of the problem:
− u
xx
(x) + u(x) cos ϕ(x) = Eu(x) , (4)
u
x
(x)|
x=0 = u
x
(x)|
x=L
= 0 ,
and
λ
n
= − γ ± √γ2 − En , (5)
where γ is the dissipative coefficient in the sine-Gor-
don equation. We note that values of λ coincide
with corresponding values of Lyapunov exponents
in the case when perturbations are considered with
respect to the stationary solutions. In general case,
Lyapunov exponents are calculated in the same way
as in Ref. 13. Thus, in the presence of a bias current
we have the different picture of a LJJ states than at
β = 0 (this case has been examined in Ref. 12). For
example, at H0 = 1.9 the increasing of β from 0 to
0.22 leads to the changing of the stationary states
number 6 → 4 → 2 → 0, i.e., to a consecutive lo-
sing of the stationary solutions. Simultaneously, an
increasing of the number of nonstationary states
occurs that we found by directly solving the nonsta-
tionary sine-Gordon equation.
2. Dynamic critical field
In the literature the critical magnetic field Hc1 in
a LJJ is defined as a field value, at which an
existence of a Josephson vortex (fluxon, soliton)
becomes advantageous thermodynamically for the
first time (see, for example, Refs. 10,11). In the
case of an infinitely long junction the critical field
is Hc1(∞) = 4/π −~ 1.274. Essentially, this field cor-
responds to the global minimum of the thermody-
namic Gibbs potential for the one-fluxon state.
However, in a junction of finite length there are
some local minima that coexist with the global one
and the every minimum corresponds to the solution
of Eqs. (1)–(2). Some of these solutions are stable,
another unstable in the sense discussed in Sec. 1.
We write down the thermodynamic Gibbs poten-
tial in the form
G =∫
0
L
dx [
1
2 ϕx
2(x) + 1 − cos ϕ(x) − βϕ(x) − H0 ϕx
(x)].
(6)
Here G is the thermodynamic Gibbs potential per
unit length along an external magnetic field and
normalized to G~ = Φ0/16π3λj d. The Ferrell-Prange
equation is an extremal of the functional (6). An
investigation of the second variation of G shows
that all extrema of this functional satisfy to the
necessary and sufficient conditions of a strong mi-
nimum [15]. Thus, all solutions of Eqs. (1)–(2)
(both stable and unstable ones) correspond to mi-
nima of the thermodynamic Gibbs potential; one of
them is global, the others are local. Our calcula-
tions of the thermodynamic Gibbs potential (6)
show that, for example, at β = 0, L = 5 and
H0 = 0.67 the Meissner state has a global minimum
(GM = − 0.44), but the stable one-fluxon state has
a local one (G1f = 4.03). The one-fluxon state has
a global minimum of G starting at H0 = 1.57
(G1f = − 2.582) and at the same value of β and L.
At this value of H0 a Meissner state has a local
minimum GM = − 2.58. At H0 ≥ 2.09 the Meissner
state disappears. Thus, at a field less than the
critical one Hc1, the stable one-fluxon state exists.
We shall further call the minimum value of a
magnetic field at given L and β, at which the stable
one-fluxon state appeares for the first time and
which corresponds to the local minimum of the
thermodynamic Gibbs potential as the dynamical
critical field Hdc . It is interesting that the dynami-
cal critical field Hdc makes up on the parametric
plane a line that coincides with the bifurcation line
BC (see Fig. 1). Our calculations show that the
K. N. Yugay, N. V. Blinov, and I. V. Shirokov
714 Fizika Nizkikh Temperatur, 1999, v. 25, No 7
bias current increases the dynamical critical field
Hdc . Evidently, it is connected with a symmetry
violation of a state by the bias current β. In Fig. 3
two stable one-fluxon states at β = 0 and β = 0.1
(L = 5, H0 = 1.4) are shown. It is seen that the
state with β = 0.1 is asymmetric. The dynamical
critical field at L = 5 are Hdc = 0.67 at β = 0 and
Hdc = 1.4 at β = 0.1. Upon increasing L the value
of Hdc is changed (β = 0): Hdc(L = 5) = 0.66,
Hdc(L = 6) = 0.4, Hdc(L = 7) = 0.26, Hdc(L = 8) =
= 0.15, Hdc(L = 10) = 0.06, i.e. the Hdc decreases.
In this case the critical field Hc1 has the values:
Hc1(L = 5) = 1.57, Hc1(L = 6) = 1.45, Hc1(L = 7) =
= 1.38, Hc1(L = 8) = 1.34, Hc1(L = 10) = 1.28, i.e.,
the Hc1 decreases also approaching to the value of
Hc1(L = ∞) = 1.274.
3. Transitions between states
As it has been shown in the previous section,
every stationary state of LJJ, i.e., the solution of
Eqs. (1)–(2), corresponds to a minimum of the
thermodynamic Gibbs potential and these minima
are not equivalent with respect to the problem of
instability. For example, in Fig. 4 stationary states
of LJJ at H0 = 2.035, β = 0.001 and L = 5 are
shown. The values of the Gibbs potential calculated
using Eq. (6) are as follows: G1 = − 5.03, G2 =
= − 4.52, G3 = − 4.61, G4 = − 4.64, G5 = − 4.61,
G6 = − 6.7. States 4 (Meissner) and 6 (one-fluxon)
are stable, the other ones are metastable. It should
be noted that unstable state 1 corresponds to deeper
minimum than the stable state 4. This property
contradicts the naive idea that more stable states
occur at deeper minima. Now we shall consider this
question in detail.
The sine-Gordon equation with dissipation and
bias current describing an evolution of the initial
state has the form:
ϕ
tt
(x, t) + 2γϕt(x, t) − ϕ
xx
(x, t) = − sin ϕ(x, t) + β ,
(7)
where t is a time normalized to the inverse of the
Josephson plasma frequency ωJ = √2πcjc/CΦ0 , C is
the junction capacitance per unit area,
γ = Φ0 ωJ /4πcRjc is the dissipative coefficient per
unit area, R is the resistance of junction per unit
area. We write down the boundary conditions for
Eq. (7) in the form
ϕ
x
(x, t)
x=0 ≡ H(0, t) = ϕ
x
(x, t)|
x=L
≡
≡ H(L, t) = H0(1 − a e−t/2t
0 cos 0.5t) . (8)
The integration of Eqs. (7)–(8) for H0 = 2.035,
β = 0.001, L = 5 (the same as in Fig. 4) and
γ = 0.26 gives: the metastable state 1 passes to the
stable state 6 at any values of perturbation parame-
ter a, 2 → 4 at a = 0, 2 → 6 at a = 1, 3 → 4 at
a = 0.05, 3 → 6 at a = 0.07, 4 → 6 at a = 0.5 and so
on. Every transition from the metastable state to
the stable one, m → n, is a transition from the state
with the certain value of local minimum Gm to
other state with smaller value of minimum Gn .
These transitions m → n with Gm > Gn are realized
by certain values of the parameter of the initial
perturbation a in expression (8). One can say that
the local minima of Gl are connected with each
other by a certain disintegration channel along the
coordinate a. From this point of view one can say
also that stationary states contain a specific «la-
Fig.3. One-fluxon states at H0 = 1.4 and L = 5 for β = 0 and
β = 0.1.
Fig. 4. Stationary states of LJJ at H0 = 2.035, β = 0.001 and
L = 5. States 1, 2, 3, 5 are unstable, states 4 and 6 are stable.
Bifurcations and a chaos strip in states of long Josephson junctions
Fizika Nizkikh Temperatur, 1999, v. 25, No 7 715
tent» parameter, by which a connection with diffe-
rent local minima Gl may be realized. In particular,
the perturbation parameter a appears here as a «la-
tent» parameter. It is possible, there are several
«latent» parameters connecting the stationary
states. One of the most important characteristics of
«latent» parameters is that the stationary state does
not depend on them directly; however, the form of
the asymptotic state and the rate of disintegration
depend essentially on them. The presence of a «la-
tent» parameter apparently explains, a nonequiva-
lence of the different local minima with respect to
the stability, especially in the case when a stable
local minimum is above a nonstable local one. In
Table results of the integration of Eqs. (1), (2) and
the calculation of G for the every of these solutions
at H0 = 1.174, β = 0, L = 8 are represented. The
transitions between states m → n are defined as
follows: the m-th solution of the stationary Ferrell-
Prange equation (1)–(2) was taken as an initial
condition of the sine-Gordon Eqs. (7), (8). If this
m-th state was unstable so it fell into the n-th stable
state.
Table
Transitions between states
Number
of state
Stability G Transitions
m → n
Sort of
stable states
1 unstable 2.34 1 → 10
2 unstable 2.78 2 → 8
3 unstable 0.64 3 → 8
4 unstable 14.69 4 → 10
5 unstable 14.98 5 → 10
6 unstable 14.69 6 → 10
7 unstable 13.53 7 → 10
8 stable –1.42 8 → 8 Meissner
9 unstable 0.64 9 → 8
10 stable –0.44 10 → 10 1 fluxon
11 unstable 2.34 11 → 10
12 stable 2.29 12 → 12 2 fluxon
The scheme of the transitions between states
m → n is represented in Fig. 5. It is seen that
Gm > Gn for the all transitions (we note that G3
and G9 for the metastable states 3 and 9 are less
than G12; the state 12 is stable). The stable states
— Meissner, one-fluxon, and two-fluxon — are
shown in Fig. 6 at the same parameters as in Fig. 5.
4. Chaos strip
As we noted above, the number of stationary
states decreases with approach to the bifurcation
line 0–2, but the number of nonstationary asymp-
totic states is increased simultaneously. Changing
the perturbation parameter a we can obtain three
sorts of typical states: stationary, regular and cha-
otic [13]. These states are distinguished not only by
a form of the field distribution in the junction and
a variation in time, but also by values of the
Lyapunov exponent λ: for the stationary states
λ < 0, for the regular states λ ≤ 0 and for the chaos
states λ > 0. The Lyapunov exponents were calcu-
lated in the same way as in Ref. 13. However, as
the calculations have shown, chaotic states may be
excited not in the whole region 2 (see Fig. 1), but
only in the bounded region in close to the bifurca-
tion line 0–2. This region is extended in the form of
Fig. 5. The scheme of transitions between states m → n. States
8, 10 and 12 are stable (8 — Meissner, 10 — one fluxon, 12 —
two fluxon), others are unstable. H0 = 1.174, L = 8, β = 0.
Fig. 6. The stable states: M — Meissner, 1f — one-fluxon, and
2f — two-fluxon at the same parameters as those in Fig. 5.
K. N. Yugay, N. V. Blinov, and I. V. Shirokov
716 Fizika Nizkikh Temperatur, 1999, v. 25, No 7
a narrow strip along the bifurcation line 0–2 ap-
proximately from 0.7 to 1.6 in H0 and in the range
of 0.002–0.015 in β. We note that the chaos strip is
arranged mostly under the bifurcation line in the
region 2, but not in the region 0, as it may be
expected because of all states in the region 0 are
nonstationary. The chaos strip is outlined on the
parametric β−H0 plane in Fig. 1.
This chaos strip along the bifurcation line 0–2
calls to mind (to a certain extent) the separatrix of
a nonlinear oscillator, where a chaos motion is
observed.
5. Discussion and conclusions
In the present work we have shown that the
parametric β−H0 plane of a LJJ is separated on
series of regions with the different number of solu-
tions of the stationary Ferrell-Prange equation. The
boundaries between these regions — bifurcation
lines — characterize an essential modification of the
system. A chaos strip arises along the bifurcation
line 0–2. We have found that the chaos strip is
arranged in the main below the bifurcation line
0–2, where stationary states take place.
We have introduced the definition of a dynami-
cal critical field as the lowest field at which the
one-fluxon state becomes stable for the first time in
the Lyapunov sense. In addition, the Meissner state
may also be stable at same parameters. Because
both the Meissner and the one-fluxon states may be
thermodynamically advantageous simultaneously,
our definition based on the stability in the
Lyapunov sense characterizes an important feature
of the stationary states of the LJJ.
We have shown that disintegration of the meta-
stable states and the transition to some stable states
m → n occur for Gm > Gn . A metastable state
corresponds to the local minimum of the Gibbs
potential, and also this minimum may be lower than
this one of a stable state. A nonequivalence of these
local minima we explain by means of existing of a
«latent» parameter not detecting in a stationary
state, by which, for example, two local minima may
be connected and a channel of the disintegration of
the upper state may arise. In our case the perturba-
tion parameter plays a role of a «latent» parameter,
however, the number of these parameters may be
much greater. We note the analogy between the
quantum transitions and the transitions mentioned
above, although the system is described by the
classical Ferrell-Prange and sine-Gordon equations.
We are aware that we could not touch upon all
questions concerning the properties of a LJJ. We
hope to return to the problems of a LJJ in our next
work.
These investigations are supported by the
Russian Foundation for Basic Research (project
No 96-02-19321).
1. W. J. Yeh, O. G. Symko, and D. J. Zheng, Phys. Rev.
B42, 4080 (1990).
2. N. Gronbech-Jensen, P. S. Lomdahl, and M. R. Samuelsen,
Phys. Rev. B43, 1279 (1991).
3. N. Gronbech-Jensen, Phys. Rev. B45, 7315 (1992).
4. G. F. Eriksen and J. B. Hansen, Phys. Rev. B41, 4189
(1990).
5. X. Yao, J. Z. Wu, and C. S. Ting, Phys. Rev. B42, 244
(1990).
6. L. E. Guerrero and M. Ostavio, Physica B165–166, 11657
(1990).
7. I. H. Dalsgaard, A. Larsen, and M. Mygind, Physica
B165–166, 1661 (1990).
8. S. Rajasekar and M. Lakshmanan, Physica A167, 793
(1990).
9. S. Rajasekar and M. Lakshmanan, Phys. Lett. A147, 264
(1990).
10. A. Barone and G. Paterno, Physics and Applications of the
Josephson Effect, Wiley-Interscience, New-York (1982).
11. K. K. Likharev, Introduction to Dynamics of Josephson
Junctions, Nauka, Moscow (1985) (in Russian).
12. K. N. Yugay, N. V. Blinov, and I. V. Shirokov, Phys. Rev.
B49, 12036 (1994).
13. K. N. Yugay, N. V. Blinov, and I. V. Shirokov, Phys. Rev.
B51, 12737 (1995).
14. C. S. Owen and D. J. Scalapino, Phys. Rev. 164, 538
(1967).
15. I. M. Gelfand and S. V. Fomin, Calculus of Variations,
Fizmatgiz, Moscow (1961) (in Russian).
Bifurcations and a chaos strip in states of long Josephson junctions
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