共查询到20条相似文献,搜索用时 146 毫秒
1.
It is shown that in many cases globally defined, bounded solutions of evolution equations are as smooth (in time) as the corresponding operator, even if a general solution of the initial-value problem is much less smooth; i.e., initial values for bounded solutions are selected in such a way that optimal smoothness is attained. In particular, solutions which bifurcate from certain steady states, such as periodic orbits, almost-periodic orbits and also homo- and heteroclinic orbits, have this property. As examples, a neutral functional differential equation, a slightly damped non-linear wave equation, and a heat equation are considered. In the latter case the space variable is included into the discussion of smoothness. Finally, generalized Hopf bifurcation in infinite dimensions is considered. Here smoothness of the bifurcation function is discussed and known results on the order of a focus are generalized. 相似文献
2.
Nan Hu Zhengdong Du 《Communications in Nonlinear Science & Numerical Simulation》2013,18(12):3436-3448
We discuss bifurcation of periodic orbits in discontinuous planar systems with discontinuities on finitely many straight lines intersecting at the origin and the unperturbed system has either a limit cycle or an annulus of periodic orbits. Assume that the unperturbed periodic orbits cross every switching line transversally exactly once. For the first case we give a condition for the persistence of the limit cycle. For the second case, we obtain the expression of the first order Melnikov function and establish sufficient conditions on the number of limit cycles bifurcate from the periodic annulus. Then we generalize our results to systems with discontinuities on finitely many smooth curves. As an application, we present a piecewise cubic system with 4 switching lines and show that the maximum number of limit cycles bifurcate from the periodic annulus can be affected by the position of the switching lines. 相似文献
3.
In this work we consider the homoclinic bifurcations of expanding periodic points. After Marotto, when homoclinic orbits to expanding periodic points exist, the points are called snap-back-repellers. Several proofs of the existence of chaotic sets associated with such homoclinic orbits have been given in the last three decades. Here we propose a more general formulation of Marotto’s theorem, relaxing the assumption of smoothness, considering a generic piecewise smooth function, continuous or discontinuous. An example with a two-dimensional smooth map is given and one with a two-dimensional piecewise linear discontinuous map. 相似文献
4.
《Nonlinear Analysis: Theory, Methods & Applications》2005,61(6):1015-1030
The behavior of a periodically forced, linearly damped mass suspended by a linear spring is well known. In this paper we study the nature of periodic solutions to two nonlinear spring-mass equations; our nonlinear terms are similar to earlier models of motion in suspension bridges. We contrast the multiplicity, bifurcation, and stability of periodic solutions for a piecewise linear and smooth nonlinear restoring force. We find that while many of the qualitative properties are the same for the two models, the nature of the secondary bifurcations (period-doubling and quadrupling) differs significantly. 相似文献
5.
V. Carmona S. Fernández-García F. Fernández-Sánchez E. García-Medina A.E. Teruel 《Nonlinear Analysis: Theory, Methods & Applications》2012
The so-called noose bifurcation is an interesting structure of reversible periodic orbits that was numerically detected by Kent and Elgin in the well-known Michelson system. In this work we perform an analysis of the periodic behavior of a piecewise version of the Michelson system where this bifurcation also exists. This variant is a one-parameter three-dimensional piecewise linear continuous system with two zones separated by a plane and it is also a representative of a wide class of reversible divergence-free systems. 相似文献
6.
We prove the existence, uniqueness, and smoothness of weak solutions of a first-order differential-operator equation with
variable domains of nonself-adjoint piecewise smooth operators for which one has the corresponding majorant operators. We
analyze the well-posedness and smoothness of weak solutions of three new mixed problems with piecewise smooth (in time) coefficients
in the equations of finite and infinite order and in the boundary conditions. 相似文献
7.
In this paper, we study the bifurcation of limit cycles in piecewise smooth systems by perturbing a piecewise Hamiltonian system with a generalized homoclinic or generalized double homoclinic loop. We first obtain the form of the expansion of the first Melnikov function. Then by using the first coefficients in the expansion, we give some new results on the number of limit cycles bifurcated from a periodic annulus near the generalized (double) homoclinic loop. As applications, we study the number of limit cycles of a piecewise near-Hamiltonian systems with a generalized homoclinic loop and a central symmetric piecewise smooth system with a generalized double homoclinic loop. 相似文献
8.
Alberto Cabada Juan B Ferreiro 《Journal of Mathematical Analysis and Applications》2004,291(2):690-697
In this paper we obtain the expression of the Green's function related with a first order periodic differential equation with piecewise constant argument. We derive comparison results for the treated linear operator by studying the sign of the obtained Green's function. 相似文献
9.
S. V. Konyagin 《Mathematical Notes》1997,61(4):473-479
We prove that the integral of a sufficiently smooth odd conditionally periodic function with zero mean and incommensurable
frequencies recurs. Furthermore, we obtain the lower and upper bounds for smoothness guaranteeing the recurrence of the integral.
Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 570–577, April, 1997.
Translated by N. K. Kulman 相似文献
10.
In this paper, we study the existence of periodic orbits bifurcating from stationary solutions of a planar dynamical system
of Filippov type. This phenomenon is interpreted as a generalized Hopf bifurcation. In the case of smoothness, Hopf bifurcation
is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis. This method does not carry
over to nonsmooth systems, due to the lack of linearization at the origin which is located on the line of discontinuity. In
fact, generalized Hopf bifurcation is determined by interactions between the discontinuity of the system and the eigen-structures
of all subsystems. With the help of geometrical observations for a corresponding piecewise linear system, we derive an analytical
method to investigate the existence of periodic orbits that are
obtained by searching for the fixed points of return maps. 相似文献
11.
Temple H Fay 《International Journal of Mathematical Education in Science & Technology》2013,44(3):349-358
This paper compliments two recent articles by the author in this journal concerning solving the forced harmonic oscillator equation when the forcing is periodic. The idea is to replace the forcing function by its Fourier series and solve the differential equation term-by-term. Herein the convergence of such series solutions is investigated when the forcing function is bounded, piecewise continuous, and piecewise smooth. The series solution and its term-by-term derivative converge uniformly over the entire real line. The term-by-term differentiation produces a series for the second derivative that converges pointwise and uniformly over any interval not containing a jump discontinuity of the forcing function. 相似文献
12.
Green's function for second-order periodic boundary value problems with piecewise constant arguments
Juan J. Nieto 《Journal of Mathematical Analysis and Applications》2005,304(1):33-57
We obtain, under suitable conditions, the Green's function to express the unique solution for a second-order functional differential equation with periodic boundary conditions and functional dependence given by a piecewise constant function. This expression is given in terms of the solutions for certain associated problems. The sign of the solution is determined taking into account the sign of that Green's function. 相似文献
13.
We obtain conditions for branching (Malkin bifurcation) of periodic solutions of differential equations with a small parameter
from nonisolated equilibria of the averaged equation. An averaging principle is stated and proved. We define an abstract analog
of the Malkin bifurcation function. 相似文献
14.
The bifurcation methods of differential equations are employed to investigate traveling waves of the oceanic currents motion equations. The sufficient conditions to guarantee the existence of different kinds of bounded traveling wave solutions are rigorously determined. Further, due to the existence of a singular line in the corresponding traveling wave system, the smooth periodic traveling wave solutions gradually lose their smoothness and evolve to periodic cusp waves. The results of numerical simulation accord with theoretical analysis. 相似文献
15.
This paper provides a rigorous approach to the problem of bifurcation of periodic orbits of a chemical reactor system recently studied by Marek and Stuckl, and by Neu. Using transformation techniques and invariant manifold theory the problem is reduced to a two-parameter bifurcation equation in R. 相似文献
16.
In this paper, the Dullin-Gottwald-Holm equation is studied using semi-inverse method and integral bifurcation method. New periodic waves such as peakon-like periodic wave, compacton-like periodic wave and singular periodic wave are found and their dynamical behaviors and certain strange phenomena are explained using the proposed criterion. The exact parametric representations of these waves are also presented. 相似文献
17.
Normal form method is first employed to study the Hopf-pitchfork bifurcation in neutral functional differential equation (NFDE), and is proved to be an efficient approach to show the rich dynamics (periodic and quasi-periodic oscillations) around the bifurcation point. We give an algorithm for calculating the third-order normal form in NFDE models, which naturally arise in the method of extended time delay autosynchronization (ETDAS). The existence of Hopf-pitchfork bifurcation in a van der Pol’s equation with extended delay feedback is given and the unfoldings near this critical point is obtained by applying our algorithm. Some interesting phenomena, such as the coexistence of several stable periodic oscillations (or quasi-periodic oscillations) and the existence of saddle connection bifurcation on a torus, are found by analyzing the bifurcation diagram and are illustrated by numerical method. 相似文献
18.
Genqiang Wang 《Journal of Mathematical Analysis and Applications》2004,298(1):298-307
Existence criteria are proved for the periodic solutions of a first order nonlinear differential equation with piecewise constant arguments. 相似文献
19.
YUAN Rong 《中国科学A辑(英文版)》2000,43(4):371-383
A definition of pseudo almost periodic sequence is given and the existence of pseudo almost periodic sequence to difference
equation is studied. Based on these, the existence of pseudo almost periodic solutions to neutral delay differential equations
with piecewise constant argument is investigated 相似文献
20.
In this paper, we give a definition of pseudo almost periodic sequence and study the existence of pseudo almost periodic sequence to difference equation. Based on these, we investigate the existence of pseudo almost periodic solutions to differential equations with piecewise constant arguments which was considered by K. L. Cooke & J. Wiener and found applications in certain biomedical problems. 相似文献