Autowave solutions of a nonlocal model for geophysical media taking into account the hysteretic character of their deformation
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Інститут геофізики ім. С.I. Субботіна НАН України
2010
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| Назва видання: | Геофизический журнал |
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| Цитувати: | Autowave solutions of a nonlocal model for geophysical media taking into account the hysteretic character of their deformation / V. Danylenko, S. Skurativskyy // Геофизический журнал. — 2010. — Т. 32, № 4. — С. 30-33. — Бібліогр.: 3 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1013112016-06-03T03:02:17Z Autowave solutions of a nonlocal model for geophysical media taking into account the hysteretic character of their deformation Danylenko, V. Skurativskyy, S. 2010 Article Autowave solutions of a nonlocal model for geophysical media taking into account the hysteretic character of their deformation / V. Danylenko, S. Skurativskyy // Геофизический журнал. — 2010. — Т. 32, № 4. — С. 30-33. — Бібліогр.: 3 назв. — англ. 0203-3100 http://dspace.nbuv.gov.ua/handle/123456789/101311 en Геофизический журнал Інститут геофізики ім. С.I. Субботіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| format |
Article |
| author |
Danylenko, V. Skurativskyy, S. |
| spellingShingle |
Danylenko, V. Skurativskyy, S. Autowave solutions of a nonlocal model for geophysical media taking into account the hysteretic character of their deformation Геофизический журнал |
| author_facet |
Danylenko, V. Skurativskyy, S. |
| author_sort |
Danylenko, V. |
| title |
Autowave solutions of a nonlocal model for geophysical media taking into account the hysteretic character of their deformation |
| title_short |
Autowave solutions of a nonlocal model for geophysical media taking into account the hysteretic character of their deformation |
| title_full |
Autowave solutions of a nonlocal model for geophysical media taking into account the hysteretic character of their deformation |
| title_fullStr |
Autowave solutions of a nonlocal model for geophysical media taking into account the hysteretic character of their deformation |
| title_full_unstemmed |
Autowave solutions of a nonlocal model for geophysical media taking into account the hysteretic character of their deformation |
| title_sort |
autowave solutions of a nonlocal model for geophysical media taking into account the hysteretic character of their deformation |
| publisher |
Інститут геофізики ім. С.I. Субботіна НАН України |
| publishDate |
2010 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/101311 |
| citation_txt |
Autowave solutions of a nonlocal model for geophysical media taking into account the hysteretic character of their deformation / V. Danylenko, S. Skurativskyy // Геофизический журнал. — 2010. — Т. 32, № 4. — С. 30-33. — Бібліогр.: 3 назв. — англ. |
| series |
Геофизический журнал |
| work_keys_str_mv |
AT danylenkov autowavesolutionsofanonlocalmodelforgeophysicalmediatakingintoaccountthehystereticcharacteroftheirdeformation AT skurativskyys autowavesolutionsofanonlocalmodelforgeophysicalmediatakingintoaccountthehystereticcharacteroftheirdeformation |
| first_indexed |
2025-07-07T10:43:44Z |
| last_indexed |
2025-07-07T10:43:44Z |
| _version_ |
1836984592429481984 |
| fulltext |
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Autowave solutions of a nonlocal model for geophysical
media taking into account the hysteretic character
of their deformation
V. Danylenko, S. Skurativskyy, 2010
Division of Geodynamics of Explosion, Institute of Geophysics,
National Academy of Sciences of Ukraine, Kiev, Ukraine
skurserg@rambler.ru
Geophysical media as open thermodynamic systems actively display synergetic properties, ability to
creation of localized dissipative structures, and an order. Experimental investigations show that dynamics
of physical processes in nonequilibrium media is determined substantially by hierarchy and discreteness
of a media structure, the set of internal relaxing processes, the nonlocality of interaction between struc-
ture elements, the directed exchange of energy between the degrees of freedom. In the papers [Danylen-
ko, Skurativsky, 2007; Danylenko et al., 2008] it is proposed to take into account these features of internal
media structure in the dynamical equation of state. This leads to the nonlocal nonlinear mathematical
model for structured media:
0
u
u
dt
d
,
m
u
p
dt
du
,
211
1æ xxxxxxx
n pppp ��
���������������
3
2
2
21 66
2
h
p
h
ph , (1)
where is the density of a medium, u is the velocity, p is the pressure, m is the external mass force,
is the relaxing time, and h are parameters of spatial and temporal nonlocalities, the parameters æ and
are proportional to the squares of equilibrium and frozen speeds of the sound. The function 1 1 , p, � , p�
describes hysteretic reaction of a medium under the deformation, is the scale parameter.
Previous investigations of the wave solutions of model (1) in the form [Danylenko, Skurativsky, 2009]
R , p P , u 2 t U , x t2 (2)
shown that accounting the spatial and temporal nonlocal effects in the dynamical equation of state ex-
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pands substantially the class of solutions in comparison with the local model ( , , h are infinitesimal). In
particular, the set (2) contains periodic, quasiperiodic, multiperiodic, and chaotic regimes, which are
connected with each other by means of bifurcational scenarios. The solitary waves with the oscillating
asymptotics were discovered as well.
Thus, basing upon the results of investigations of models that do not take into account the hysteretic
character of media deformations, we shall study the influence of the hysteretic function �1 in the dyna-
mical equation of state on the structure of solutions (2). The function �1 describes the histeretic loop in the
plane ( ; p) under the harmonic loading. The form of this loop is defined by the following relation
,0/,,
1exp
1exp
1exp
,0/,,1exp
,0/,,
1exp
1exp
1exp
,0/,,1exp
0
0max1
01
0max2
002
0
0max1
01
0max2
002
1
dtd
C
C
C
dtdC
dtd
C
C
C
dtdC
(3)
where C2<C1. In the case, when C2<C 1, the area bounded by the loop is zero. We should note that the set
of enclosed loops appears in the plane ( ; p) instead of one loop, if we use the loading, distinct from
harmonic one. The elements of function (3) can be used for description of the simplest cases of enclosed
loops (Fig. 1).
Fig. 1. The construction of two (a) and three (b) encloset histeretic loops in the plane ( ; p).
Substituting solution (2) into system (1), we obtain the quadrature UR=S=const and the dynamical
system:
WR , W
R
S
RRP m
2
2
2 , ZW , 223
1
7
RSbS
RF
Z . (4)
Here dd /�� , ZWPRFF ,,, is the nonlinear function. The analitical expression for the function
F is omitted due to its length. Note that analitical expression (3) for the histeretic loop can also be written
in terms of invariant variables (2).
Nonlinear dynamical system (4) is investigated by means of the qualitative and numerical methods.
Equating the right parts of system (4) to zero, we get the coordinates of the fixed point
/#�)-(% '0%.�+,#(� #(%1
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1/1
0 /2 mR , nRP 00 æ , 00W , (5)
which do not depend on the parameters of the histeretic function 1 . Analyzing the stability of fixed point
(5) under the neglecting the histeretic function �1 we state that at = 16; S = 3.2; = 1; æ = 0.9; = 0.5;
h = 6.74; = 0; = 0.6; n = 4; m = 8; b = 3.8 the fixed point changes the type from unstable node-focus to
stable one. In the vicinity of fixed point (5) the unstable limit cycle appears at increasing the parameter h .
Bifurcational analysis of dynamical system (4) in the case when �1 0 and zero area of histeretic loop
shown that for C2 = C1 >0 fixed point (5) is a stable node-focus surrounded by both unstable and stable
cycles. Consider the case, when C1 C2 and function (3) describes the loop with nonzero area. Then as a
consequence of histeretic loop including the structure of the phase space of system (4) becomes more
complicated at increasing the parameter C1. In particular, at C2 = 7.5; C1 = 27 there are several localized
and separated regimes in the phase space of dynamical system (4), namely, fixed point (5), both stable
and unstable cycles surrounding it (Fig. 2, a), and the chaotic attractor in addition (Fig. 2, b). Note that
the chaotic attractor does not exist neither at C1 = C2 = 7.5 nor at C1<C2 = 27. So that the chaotic attractor
is created due to accounting the histeretic loop.
Fig. 2. The structure of the phase space of system (4) at C1=27 (a) and C2=7.5 (b).
Fig. 3. Poincare diagrames of the limit cycle development at increasing : a — C1=C2=9.5; b — C2=9.5, C1=19.
Another manifestation of the histeretic function �1 adding is regularization of chaotic oscillations. According
to the numerical experiments, at S = 3.8, C1 = C2 = 7.5, = 0,1 the complicated periodic trajectory exists in the
vicinity of the fixed point. Analyzing the development of the periodic trajectory by Poincare diagram we show
that the period doubling cascade with chaotic attractor creation is actualized at C1 [7.5; 8]. But if we fix C2 = 7.5
and vary C1 [7.5; 8], then the periodic regime exists only instead of the chaotic attractor.
The existence of new wave regimes for model (1) is connected with the effects of spatial nonlocality,
which are described by terms with the parameter . Analyzing the Poincare diagram of the limit cycle
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development at increasing and C1 = C2 (zero loop area) we see that the size of T-period cycle is growing
and there is an interval of corresponding to the existence of 2T-period cycle (Fig. 3, a). The structure of
solutions (2) is much more complicated in the case of nonzero histeretic loop area at C2 = 9,5, C1 = 19.
Studying the Poincare diagram in picture 3b we can distinguish several period doubling cascade, intervals
of the different type chaotic attractor existence, moment of the histeretic transition from one attractor to
other.
Thus, the accounting a histeretic loop in the dynamical equation of state causes new wave regimes
creation. The histeretic loop is the way of elastic energy utilization and in the same time it is the nonlinear
element of media, which can cause unstability generation in media and produce localized dissipative
structures.
References
Danylenko V. A., Danevich T. B., Skurativskyy S. I. Nonlinear mathematical models of media with tem-
poral and spatial nonlocalities. — Kiev: Institute of Geophysics NAS of Ukraine, 2008. — 86 p. (in
Russian).
Danylenko V. A., Skurativskyy S. I. Autowave solutions of a nonlocal model of geophysical media with
regard for the hysteretic character of their deformation // Rep. of the NAS of Ukraine. — 2009. — 1. —
P. 98—102 (in Ukrainian).
Danylenko V. A., Skurativskyy S. I. Invariant chaotic and quasi-periodic solutions of nonlinear nonlocal
models of relaxing media // Rep. on Math. Phys. — 2007. — 59. — P. 45—51.
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