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1.
M. H. B. M. Shariff 《Journal of Elasticity》2013,110(2):237-241
It is often declared in the literature that the seven classical invariants used to characterize the strain energy of a compressible orthotropic elastic solid are independent. In this paper, we show that only six of the seven classical invariants are independent, and a syzygy exists between the classical invariants. Consequently, all other sets of seven invariants, proposed in the literature, that are uniquely related to the set of classical invariants, have only six independent invariants. 相似文献
2.
David J. Unger 《Journal of Elasticity》2002,66(1):93-97
The two-dimensional point load problem on a linear elastic half plane is solved using a one-parameter group transformation of the biharmonic equation. This solution technique allows the incorporation of the singular load directly through use of the Dirac delta function. The resulting fourth-order ordinary differential equation, while formidable in appearance, is exact and is readily solved through simple integration. 相似文献
3.
We investigate a semiexplicit Cauchy problem for a system of ordinary differential equations in the case of a variable pencil of matrices. We determine sufficient conditions of existence of solutions and consider the problem of their number. 相似文献
4.
The group method of solving ODEs without any quadrature goes back to Lie. In order to apply it, the number of symmetries admitted by a given ODE has to be greater by one than the number of arbitraryconstants in the general solution of the equation. In this paper, we use thetechnique of canonical Lie–Bäcklund symmetries that makes theproof of the statement concerning integrals of ODEs more evident. The methodis extended to the solution of system of ODEs with a small parameter of arbitrary order. 相似文献
5.
Invariants of a Remarkable Family of Nonlinear Equations 总被引:1,自引:0,他引:1
In classical literature, invariants of families of differentialequations were considered for linear equations only, e.g. the renownedLaplace invariants for linear hyperbolic partial differential equationsand invariants of linear ordinary differential equations with variablecoefficients. The restriction to linear equations was essential inpioneering works of Cockle, Laguerre, Halphen, andForsyth for tackling the problem of invariants of differentialequations. Lie regretted that these authors did not use advantagesprovided by his theory of infinite continuous groups, but he himself didnot undertake further developments in this direction.Recently, the present author considered the possibility hinted byLie's remark and introduced the infinitesimal technique in thetheory of invariants of families of differential equations thatwas lacking in old methods. In consequence, a simple unifiedapproach was developed for calculation of invariants of algebraicand differential equations independent on the assumption oflinearity of equations. It was employed recently for calculationof Laplace type invariants for parabolic equations. Here, themethod is applied to calculation of invariants for the family ofnonlinear equations appearing in the problem on linearization ofnonlinear ordinary differential equations. 相似文献
6.
Experimental Techniques - 相似文献
7.
In the last 30 years, some authors have been studying several classes of boundary value problems (BVP) for partial differential
equations (PDE) using the method of reduction to obtain a difference equation with continuous argument which behavior is determined
by the iteration of a one-dimensional (1D) map (see, for example, Romanenko, E. Yu. and Sharkovsky, A. N., International Journal of Bifurcation and Chaos 9(7), 1999, 1285–1306; Sharkovsky, A. N., International Journal of Bifurcation and Chaos 5(5), 1995, 1419–1425; Sharkovsky, A. N., Analysis Mathematica Sil 13, 1999, 243–255; Sharkovsky, A. N., in “New Progress in Difference Equations”, Proceedings of the ICDEA'2001, Taylor and Francis, 2003, pp. 3–22; Sharkovsky, A. N., Deregel, Ph., and Chua, L. O., International Journal of Bifurcation and Chaos 5(5), 1995, 1283–1302; Sharkovsky, A. N., Maistrenko, Yu. L., and Romanenko, E. Yu., Difference Equations and Their Applications, Kluwer, Dordrecht, 1993.). In this paper we consider the time-delayed Chua's circuit introduced in (Sharkovsky, A. N., International Journal of Bifurcation and Chaos 4(5), 1994, 303–309; Sharkovsky, A. N., Maistrenko, Yu. L., Deregel, Ph., and Chua, L. O., Journal of Circuits, Systems and Computers 3(2), 1993, 645–668.) which behavior is determined by properties of one-dimensional map, see Sharkovsky, A. N., Deregel, Ph.,
and Chua, L. O., International Journal of Bifurcation and Chaos 5(5), 1995, 1283–1302; Maistrenko, Yu. L., Maistrenko, V. L., Vikul, S. I., and Chua, L. O., International Journal of Bifurcation and Chaos 5(3), 1995, 653–671; Sharkovsky, A. N., International Journal of Bifurcation and Chaos 4(5), 1994, 303–309; Sharkovsky, A. N., Maistrenko, Yu. L., Deregel, Ph., and Chua, L. O., Journal of Circuits, Systems and Computers 3(2), 1993, 645–668. To characterize the time-evolution of these circuits we can compute the topological entropy and to distinguish
systems with equal topological entropy we introduce a second topological invariant. 相似文献
8.
O. Yu. Volkova 《Nonlinear Oscillations》2005,8(2):157-162
We study monotonicity properties of the kneading invariant for one-parameter families of piecewise-linear unimodal maps and
prove a theorem on the violation of monotonicity of the kneading invariant for maps that are symmetric and convex and consist
of four linear pieces. The fact that such maps cannot be approximated by smooth mappings with negative Schwarzian is proved
using a dynamics argument.
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Published in Neliniini Kolyvannya, Vol. 8, No. 2, pp. 159–164, April–June, 2005. 相似文献
9.
N. Kh. Ibragimov 《Journal of Applied Mechanics and Technical Physics》2004,45(2):158-166
This paper gives a solution of the Laplace problem, which consists of finding all invariants of the hyperbolic equations and constructing a basis of the invariants. Three new invariants of the first and second orders are found, and invariantdifferentiation operators are constructed. It is shown that the new invariants, together with the two invariants detected by Ovsyannikov, form a basis such that any invariant of any order is a function of the basis invariants and their invariant derivatives. 相似文献
10.
We establish the local symmetry group of the dynamically and kinematically exact theory of elastic shells. The group consists of an ordered triple of tensors which make the shell strain energy density invariant under change of the reference placement. Definitions of the fluid shell, the solid shell, and the membrane shell are introduced in terms of members of the symmetry group. Within solid shells we discuss in more detail the isotropic, hemitropic, and orthotropic shells and corresponding invariant properties of the strain energy density. For the physically linear shells, when the density becomes a quadratic function of the shell strain and bending tensors, reduced representations of the density are established for orthotropic, cubic-symmetric, and isotropic shells. The reduced representations contain much less independent material constants to be found from experiments. 相似文献
11.
12.
First, some linear techniques in multivariate time-series analysis in EEG research are reviewed to highlight the problem of estimating the dimensionality of the state space (embedding dimension), the reconstruction of an attractor, and the evaluation of invariant properties of the attractor. The traditional linear techniques included the usual spectral and cospectral measures of power, phase, and coherence to which stepwise discriminant analysis was applied for canonical representation of the attractor. Then, some traditional nonlinear techniques of attractor reconstruction and dimensional analysis which use the time-lagged univariate approach of Ruelle and Takens (Takens, 1981) are reviewed. Next, updates and multivariate generalizations that use singular-value decomposition (Broomhead & King, 1986) are reviewed. Finally, Stewart's (1995, 1996) multivariate generalization of the method of false nearest neighbors (Abarbanel, Brown, Sidorowich, & Tsimring, 1993; Kennel, Brown, & Abarbanel, 1992) is reviewed. These are particularly relevant for evaluating multivariate coherence in research on the complex cooperative dynamical systems found in neuroscience, psychology, and social science when time series of sufficient length are investigated. 相似文献
13.
IntroductionThefamousAsgeirssonmeanvaluetheoremhasansweredthattheCauchyproblemsareill_posedtotheultra_hyperbolicpartialdifferentialequationsofthesecond_order(Δ2 x-Δ2 y)u=0 . ( 1 )Theresultcanbeusedtoprovethecontinuationofthesolutionsofthisequation ( 1 ) .Soitisi… 相似文献
14.
We investigate necessary conditions for the absolute exponential stability of a system of linear parabolic differential equations with one delay. 相似文献
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16.
Pavel Nesterov 《Journal of Dynamics and Differential Equations》2018,30(4):1797-1816
We propose an asymptotic integration method for certain class of functional differential systems. This class includes the delay differential equations with oscillatory decreasing coefficients and variable delays that are close to constants at infinity. Both the ideas of the centre manifold theory and the averaging method together with some classical asymptotic theorems are used to construct the asymptotics for solutions. We illustrate the asymptotic integration method by constructing the asymptotics for solutions of scalar differential equation with variable delay. 相似文献
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18.
Local Bifurcation Control of a Forced Single-Degree-of-Freedom Nonlinear System: Saddle-Node Bifurcation 总被引:3,自引:0,他引:3
It is well known that saddle-node bifurcations can occur in the steady-state response of a forced single-degree-of-freedom (SDOF) nonlinear system in the cases of primary and superharmonic resonances. This discontinuous or catastrophic bifurcation can lead to jump and hysteresis phenomena, where at a certain interval of the control parameter, two stable attractors exist with an unstable one in between. A feedback control law is designed to control the saddle-node bifurcations taking place in the resonance response, thus removing or delaying the occurrence of jump and hysteresis phenomena. The structure of candidate feedback control law is determined by analyzing the eigenvalues of the modulation equations. It is shown that three types of feedback – linear, nonlinear, and a combination of linear and nonlinear – are adequate for the bifurcation control. Finally, numerical simulations are performed to verify the effectiveness of the proposed feedback control. 相似文献
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20.
V. Ya. Ternovoi K. V. Khishchenko A. A. Charakhch’yan 《Journal of Applied Mechanics and Technical Physics》2009,50(3):371-379
This paper considers the interaction between an absolutely rigid wall or a steel plate and the rarefaction wave arising in
solid deuterium when a 30–150 GPa shock wave arrives at the free surface. It is shown that, in the entropy trace near the wall or interface with the plate,
a high-temperature plasma arises, in which a thermonuclear fusion is possible, at least, for shock-wave pressures above 70 GPa. The dimension of the plasma region and the time of its establishment are proportional to the distance between the free surface
and the wall. Estimates of the proportionality coefficients are given. It is noted that, in this case, unlike in other methods
of high-temperature plasma generation, the time of existence of the plasma may not depend on the sound velocity in it. It
is shown that, by using a conical solid-state target wit an exit hole, the shock-wave pressure in solid deuterium can be increased
from 10 to 100 GPa.
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Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 15–24, May–June, 2009. 相似文献