共查询到20条相似文献,搜索用时 15 毫秒
1.
Dag Nilsson 《Mathematical Methods in the Applied Sciences》2019,42(12):4113-4145
We consider three‐dimensional inviscid‐irrotational flow in a two‐layer fluid under the effects of gravity and surface tension, where the upper fluid is bounded above by a rigid lid and the lower fluid is bounded below by a flat bottom. We use a spatial dynamics approach and formulate the steady Euler equations as an infinite‐dimensional Hamiltonian system, where an unbounded spatial direction x is considered as a time‐like coordinate. In addition, we consider wave motions that are periodic in another direction z. By analyzing the dispersion relation, we detect several bifurcation scenarios, two of which we study further: a type of 00(is)(iκ0) resonance and a Hamiltonian Hopf bifurcation. The bifurcations are investigated by performing a center‐manifold reduction, which yields a finite‐dimensional Hamiltonian system. For this finite‐dimensional system, we establish the existence of periodic and homoclinic orbits, which correspond to, respectively, doubly periodic travelling waves and oblique travelling waves with a dark or bright solitary wave profile in the x direction. The former are obtained using a variational Lyapunov‐Schmidt reduction and the latter by first applying a normal form transformation and then studying the resulting canonical system of equations. 相似文献
2.
P. Grinfeld 《Studies in Applied Mathematics》2010,125(3):223-264
Two‐dimensional models for hydrodynamic systems, such as soap films, have been studied for over two centuries. Yet there has not existed a fully nonlinear system of dynamic equations analogous to the classical Euler equations. We propose the following exact system for the dynamics of a fluid film Here δ/δ t is the invariant time derivative, ρ is the two‐dimensional density of the film, C is the normal component of the velocity field, Vα are the tangential components, Bαβ is the curvature tensor, and ?α is the covariant surface derivative. The surface energy density e(ρ) is a generalization of the common surface tension and eρ is its derivative. The Laplace model corresponds to e(ρ) =σ/ρ , where σ is the surface tension density. The proper choice of e(ρ) in paramount in capturing particular effects displayed by fluid films. The proposed system is exact in the sense that neither velocities nor deviation from the equilibrium are assumed small. The system is derived in the classical Hamiltonian framework. The assumption that e is a function of ρ alone can be relaxed in practical physical and biological applications. This leads to more complicated systems, briefly discussed in the text. 相似文献
3.
Ali. Elamin. M. Saeed Luo Dingjun 《高校应用数学学报(英文版)》2005,20(4):431-440
To continue the discussion in (Ⅰ ) and ( Ⅱ ),and finish the study of the limit cycle problem for quadratic system ( Ⅲ )m=0 in this paper. Since there is at most one limit cycle that may be created from critical point O by Hopf bifurcation,the number of limit cycles depends on the different situations of separatrix cycle to be formed around O. If it is a homoclinic cycle passing through saddle S1 on 1 +ax-y = 0,which has the same stability with the limit cycle created by Hopf bifurcation,then the uniqueness of limit cycles in such cases can be proved. If it is a homoclinic cycle passing through saddle N on x= 0,which has the different stability from the limit cycle created by Hopf bifurcation,then it will be a case of two limit cycles. For the case when the separatrix cycle is a heteroclinic cycle passing through two saddles at infinity,the discussion of the paper shows that the number of limit cycles will change from one to two depending on the different values of parameters of system. 相似文献
4.
This article presents a rigorous existence theory for
three-dimensional gravity-capillary water waves which are uniformly
translating and periodic in one spatial direction x and have the
profile of a uni- or multipulse solitary wave in the other z. The
waves are detected using a combination of Hamiltonian spatial dynamics
and homoclinic Lyapunov-Schmidt theory.
The hydrodynamic problem is formulated as an infinite-dimensional
Hamiltonian system in which z is the timelike variable, and a
family of points Pk,k+1, k = 1,2,... in its two-dimensional
parameter space is identified at which a Hamiltonian 0202
resonance takes place (the zero eigenspace and generalised eigenspace
are respectively two and four dimensional). The point Pk,k+1 is
precisely that at which a pair of two-dimensional periodic linear
travelling waves with frequency ratio k:k+1 simultaneously exist
(Wilton ripples). A reduction principle is applied to demonstrate
that the problem is locally equivalent to a four-dimensional
Hamiltonian system near Pk,k+1.
It is shown that a Hamiltonian real semisimple 1:1 resonance, where
two geometrically double real eigenvalues exist, arises along a
critical curve Rk,k+1 emanating from Pk,k+1. Unipulse
transverse homoclinic solutions to the reduced Hamiltonian system at
points of Rk,k+1 near Pk,k+1 are found by a scaling and
perturbation argument, and the homoclinic Lyapunov-Schmidt method is
applied to construct an infinite family of multipulse homoclinic
solutions which resemble multiple copies of the unipulse solutions. 相似文献
5.
Thomas J. Bridges 《Journal of Differential Equations》2008,244(7):1629-1674
A fundamental class of solutions of symmetric Hamiltonian systems is relative equilibria. In this paper the nonlinear problem near a degenerate relative equilibrium is considered. The degeneracy creates a saddle-center and attendant homoclinic bifurcation in the reduced system transverse to the group orbit. The surprising result is that the curvature of the pullback of the momentum map to the Lie algebra determines the normal form for the homoclinic bifurcation. There is also an induced directional geometric phase in the homoclinic bifurcation. The backbone of the analysis is the use of singularity theory for smooth mappings between manifolds applied to the pullback of the momentum map. The theory is constructive and generalities are given for symmetric Hamiltonian systems on a vector space of dimension (2n+2) with an n-dimensional abelian symmetry group. Examples for n=1,2,3 are presented to illustrate application of the theory. 相似文献
6.
In this paper, we investigate the behavior of the twist near low-order resonances of a periodic orbit or an equilibrium of a Hamiltonian system with two degrees of freedom. Namely, we analyze the case where the Hamiltonian has multiple eigenvalues (the Hamiltonian Hopf bifurcation) or a zero eigenvalue near the equilibrium and the case where the system has a periodic orbit whose multipliers are equal to 1 (the saddle-center bifurcation) or −1 (the period-doubling bifurcation). We show that the twist does not vanish at least in a small neighborhood of the period-doubling bifurcation. For the saddle-center bifurcation and the resonances of the equilibrium under consideration, we prove the existence of a “twistless” torus for sufficiently small values of the bifurcation parameter. An explicit dependence of the energy corresponding to the twistless torus on the bifurcation parameter is derived. Bibliography: 6 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 135–144. 相似文献
7.
Xuanliang Liu 《Applied mathematics and computation》2011,218(5):2300-2309
A predator-prey system with disease in the prey is considered. Assume that the incidence rate is nonlinear, we analyse the boundedness of solutions and local stability of equilibria, by using bifurcation methods and techniques, we study Bogdanov-Takens bifurcation near a boundary equilibrium, and obtain a saddle-node bifurcation curve, a Hopf bifurcation curve and a homoclinic bifurcation curve. The Hopf bifurcation and generalized Hopf bifurcation near the positive equilibrium is analyzed, one or two limit cycles is also discussed. 相似文献
8.
《Chaos, solitons, and fractals》2006,27(1):75-86
This paper characterizes some complex dynamics of Chen’s system. Some conditions of existence for pitchfork bifurcation and Hopf bifurcation are derived by using bifurcation theory and the center manifold theorem. Numerical simulation results not only show consistence with the theoretical analysis but also display some new and interesting dynamical behaviors including homoclinic bifurcation and the coexistence of two stable limit cycles and one chaotic attractor as well as some periodic solutions emerging from Hopf bifurcation but ending in homoclinic bifurcation, which are different from those reported in the literature before. All these show that Chen’s system has very rich nonlinear dynamics. 相似文献
9.
Xiaoxiao Zheng Yadong Shang Xiaoming Peng 《Mathematical Methods in the Applied Sciences》2017,40(7):2623-2633
This paper investigates the orbital stability of solitary waves for the coupled Klein–Gordon–Zakharov (KGZ) equations where α ≠ 0. Firstly, we rewrite the coupled KGZ equations to obtain its Hamiltonian form. And then, we present a pair of sech‐type solutions of the coupled KGZ equations. Because the abstract orbital stability theory presented by Grillakis, Shatah, and Strauss (1987) cannot be applied directly, we can extend the abstract stability theory and use the detailed spectral analysis to obtain the stability of the solitary waves for the coupled KGZ equations. In our work, α = 1,β = 0 are advisable. Hence, we can also obtain the orbital stability of solitary waves for the classical KGZ equations which was studied by Chen. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
10.
It is well-known that on a versal deformation of the Takens–Bogdanov bifurcation is possible to find dynamical systems that
undergo saddle-node, Hopf, and homoclinic bifurcations. In this document a nonlinear control system in the plane is considered,
whose nominal vector field has a double-zero eigenvalue, and then the idea is to find under which conditions there exists
a scalar control law such that be possible establish a priori, that the closed-loop system undergoes any of the three bifurcations: saddle-node, Hopf or homoclinic. We will say then that
such system undergoes the controllable Takens–Bogdanov bifurcation. Applications of this result to the averaged forced van der Pol oscillator, a population dynamics, and adaptive control systems
are discussed. 相似文献
11.
We consider the following mean field equation: where M is a compact Riemann surface with volume 1, h* is a positive C1 function on M, and ρ and αj are constants satisfying αj > ?1. In this paper, we derive the topological‐degree‐counting formula for noncritical values of ρ. We also give several applications of this formula, including existence of the curvature + 1 metric with conic singularities, doubly periodic solutions of electroweak theory, and a special case of self‐gravitating strings. © 2015 Wiley Periodicals, Inc. 相似文献
12.
This paper deals with the analysis of Hamiltonian Hopf as well as saddle-center bifurcations in 4-DOF systems defined by perturbed
isotropic oscillators (1:1:1:1 resonance), in the presence of two quadratic symmetries Ξ and L
1. When we normalize the system with respect to the quadratic part of the energy and carry out a reduction with respect to
a three-torus group we end up with a 1-DOF system with several parameters on the thrice reduced phase space. Then, we focus
our analysis on the evolution of relative equilibria around singular points of this reduced phase space. In particular, dealing
with the Hamiltonian Hopf bifurcation the ‘geometric approach’ is used, following the steps set up by one of the authors in
the context of 3-DOF systems. In order to see the interplay between integrals and physical parameters in the analysis of bifurcations,
we consider as a perturbation a one-parameter family, which in particular includes one of the classical Stark–Zeeman models
(parallel case) in three dimensions. 相似文献
13.
Slow Passage Through the Nonhyperbolic Homoclinic Orbit of the Saddle-Center Hamiltonian Bifurcation
Slowly varying Hamiltonian systems, for which action is a well-known adiabatic invariant, are considered in the case where the system undergoes a saddle center bifurcation. We analyze the situation in which the solution slowly passes through the nonhyperbolic homoclinic orbit created at the saddle-center bifurcation. The solution near this homoclinic orbit consists of a large sequence of homoclinic orbits surrounded by near approaches to the autonomous nonlinear nonhyperbolic saddle point. By matching this solution to the strongly nonlinear oscillations obtained by averaging before and after crossing the homoclinic orbit, we determine the change in the action. If one orbit comes sufficiently close to the nonlinear saddle point, then that one saddle approach instead satisfies the nonautonomous first Painlevé equation, whose stable manifold of the unstable saddle (created in the saddle-center bifurcation) separates solutions approaching the stable center from those involving sequences of nearly homoclinic orbits. 相似文献
14.
On homoclinic bifurcation emanating from Takens-Bogdanov points in Hamiltonian systems 总被引:1,自引:0,他引:1
The paper is devoted to studying the bifurcation of periodic and homoclinic orbits in a 2n-dimensional Hamiltonian system with 1 parameter from a TB-point (Hamiltonian saddle node). In addition to the proof of existence, the paper gives an expansion formula of the bifurcating homoclinic orbits. With the help of center manifold reduction and a blow up transformation, the problem is focused on studying a planar Hamiltonian system, the proof for the perturbed homoclinic and periodic orbits is elementary in the sense that it uses only implicit function arguments. Two applications to travelling waves in PDEs are shown. 相似文献
15.
Gierer–Meinhardt system as a molecularly plausible model has been proposed to formalize the observation for pattern formation. In this paper, the Gierer–Meinhardt model without the saturating term is considered. By the linear stability analysis, we not only give out the conditions ensuring the stability and Turing instability of the positive equilibrium but also find the parameter values where possible Turing–Hopf and spatial resonance bifurcation can occur. Then we develop the general algorithm for the calculations of normal form associated with codimension-2 spatial resonance bifurcation to better understand the dynamics neighboring of the bifurcating point. The spatial resonance bifurcation reveals the interaction of two steady state solutions with different modes. Numerical simulations are employed to illustrate the theoretical results for both the Turing–Hopf bifurcation and spatial resonance bifurcation. Some expected solutions including stable spatially inhomogeneous periodic solutions and coexisting stable spatially steady state solutions evolve from Turing–Hopf bifurcation and spatial resonance bifurcation respectively. 相似文献
16.
In this communication we deal with the analysis of Hamiltonian Hopf bifurcations in 4-DOF systems defined by perturbed isotropic
oscillators (1-1-1-1 resonance), in the presence of two quadratic symmetries I
1 and I
2. As a perturbation we consider a polynomial function with a parameter. After normalization, the truncated normal form gives
rise to an integrable system which is analyzed using reduction to a one degree of freedom system. The Hamiltonian Hopf bifurcations
are found using the ‘geometric method’ set up by one of the authors.
相似文献
17.
Pablo S. Casas Àngel Jorba 《Communications in Nonlinear Science & Numerical Simulation》2012,17(7):2864-2882
This paper studies various Hopf bifurcations in the two-dimensional plane Poiseuille problem. For several values of the wavenumber α, we obtain the branch of periodic flows which are born at the Hopf bifurcation of the laminar flow. It is known that, taking α ≈ 1, the branch of periodic solutions has several Hopf bifurcations to quasi-periodic orbits. For the first bifurcation, calculations from other authors seem to indicate that the bifurcating quasi-periodic flows are stable and subcritical with respect to the Reynolds number, Re. By improving the precision of previous works we find that the bifurcating flows are unstable and supercritical with respect to Re. We have also analysed the second Hopf bifurcation of periodic orbits for several α, to find again quasi-periodic solutions with increasing Re. In this case the bifurcated solutions are stable to superharmonic disturbances for Re up to another new Hopf bifurcation to a family of stable 3-tori. The proposed numerical scheme is based on a full numerical integration of the Navier-Stokes equations, together with a division by 3 of their total dimension, and the use of a pseudo-Newton method on suitable Poincaré sections. The most intensive part of the computations has been performed in parallel. We believe that this methodology can also be applied to similar problems. 相似文献
18.
Analyses of Bifurcations and Stability in a Predator-prey System with Holling Type-Ⅳ Functional Response 总被引:9,自引:0,他引:9
Ji-caiHuang Dong-meiXiao 《应用数学学报(英文版)》2004,20(1):167-178
In this paper the dynamical behaviors of a predator-prey system with Holling Type-Ⅳfunctionalresponse are investigated in detail by using the analyses of qualitative method,bifurcation theory,and numericalsimulation.The qualitative analyses and numerical simulation for the model indicate that it has a unique stablelimit cycle.The bifurcation analyses of the system exhibit static and dynamical bifurcations including saddle-node bifurcation,Hopf bifurcation,homoclinic bifurcation and bifurcation of cusp-type with codimension two(ie,the Bogdanov-Takens bifurcation),and we show the existence of codimension three degenerated equilibriumand the existence of homoclinic orbit by using numerical simulation. 相似文献
19.
Kazuyuki Yagasaki 《Journal of Differential Equations》2002,185(1):1-24
Using a Melnikov-type technique, we study codimension-two bifurcations called the Bogdanov-Takens bifurcations for subharmonics in periodic perturbations of planar Hamiltonian systems. We give a criterion for the occurrence of the Bogdanov-Takens bifurcations and present approximate expressions for saddle-node, Hopf and homoclinic bifurcation sets near the Bogdanov-Takens bifurcation points. We illustrate the theoretical result with an example. 相似文献
20.
In this paper, we consider the discretization of parameter-dependent delay differential equation of the formIt is shown that if the delay differential equation undergoes a Hopf bifurcation at τ=τ*, then the discrete scheme undergoes a Hopf bifurcation at τ(h)=τ*+O(hp) for sufficiently small step size h, where p1 is the order of the Runge–Kutta method applied. The direction of numerical Hopf bifurcation and stability of bifurcating invariant curve are the same as that of delay differential equation. 相似文献