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1.
The limit-cycle phenomenon in the Lorenz system is studied with considering bifurcation slates of a dynamic system. It is established that the trajectory has a complex structure and includes intervals of periodic solutions of different kinematics and an interval of saddle-node solution 相似文献
2.
Alberto Castro Ortega Ernesto A. Lacomba 《Journal of Dynamics and Differential Equations》2012,24(1):85-100
Consider three charged masses moving along the line. For this model we study the solutions near total collision using blow
up techniques obtaining that for given masses and charges the vector field on the collision manifold has a non-hyperbolic
equilibrium point. To study this situation the vector field is written in normal form and the center manifold theory is used
obtaining that all nonzero solutions near the origin escape to infinity. 相似文献
3.
We develop a general stability theory for equilibrium points of Poisson dynamical systems and relative equilibria of Hamiltonian systems with symmetries, including several generalisations of the Energy-Casimir and Energy-Momentum Methods. Using a topological generalisation of Lyapunovs result that an extremal critical point of a conserved quantity is stable, we show that a Poisson equilibrium is stable if it is an isolated point in the intersection of a level set of a conserved function with a subset of the phase space that is related to the topology of the symplectic leaf space at that point. This criterion is applied to generalise the energy-momentum method to Hamiltonian systems which are invariant under non-compact symmetry groups for which the coadjoint orbit space is not Hausdorff. We also show that a G-stable relative equilibrium satisfies the stronger condition of being A-stable, where A is a specific group-theoretically defined subset of G which contains the momentum isotropy subgroup of the relative equilibrium. The results are illustrated by an application to the stability of a rigid body in an ideal irrotational fluid.Acknowledgement This work was partially supported by an EPSRC Visiting Fellowship (GR/L57074) and an NSERC individual research grant for GWP, an EPSRC Research Grant (GR/K99893), a Scheme Four grant from the London Mathematical Society, a European Community Marie Curie Fellowship (HPMF-CT-2000-00542) for CW, and by European Community funding for the Research Training Network MASIE (HPRN-CT-2000-00113). We thank the University of Warwick Mathematics Institute for its hospitality during several visits when parts of the paper were written. We are also grateful to TUDOR RATIU for some very helpful remarks. 相似文献
4.
The investigation of codimension-2 bifurcations, in particular in systems with cylindric symmetry, enables us to deduce new
types of secondary regimes branching-off from the symmetric regimes. This investigation also allows us the unique possibility
of a rigorous treatment of chaotic solutions to Navier–Stokes and other nonlinear PDE’s. The central manifold approach combined
with the reduction to the normal form lead to the so-called amplitude systems. These ODE systems describe the nonlinear interaction
between the neutral modes, and always include several nonlinear terms due to so-called intrinsic resonances. However, sometimes
additional resonances appear. In this paper we present the complete list of all possible resonances in dynamic systems with
cylindric symmetry and the corresponding forms of the amplitude equations. Further, we present the results of extensive numerical
investigation of the resonant codimension-2 bifurcations in the Couette–Taylor problem, thus creating an intriguing subject
for further investigation. 相似文献
5.
Antonio Russo 《Journal of Mathematical Fluid Mechanics》2009,11(3):407-414
We consider the stationary motion of a viscous incompressible fluid in a two-dimensional exterior domain; we prove that the
problem has a solution for small values of the flux of the boundary datum through the boundary. 相似文献
6.
D. A. Nield 《Transport in Porous Media》2006,64(2):185-188
The problem of fully developed forced convection in a parallel-plates channel filled with a saturated porous medium (involving
a Brinkman model for the momentum equation), with the effect of viscous dissipation (involving a Brinkman number), is discussed.
Some general matters relating to the possibility of fully developed convection are also discussed. 相似文献
7.
In the second part a practically important problem, namely the stability of relative equilibria of a dumbell satellite on an orbit around the Earth is treated by means of the reduced energy-momentum method. The dumbell satellite is used to emphasize the advantages of the reduced energy-momentum method which did not become obvious in the simple example of the rotating pendulum treated in Part I, as well as, to discuss some of the finer technical details. 相似文献
8.
V. Thümmler 《Journal of Dynamics and Differential Equations》2008,20(2):425-477
In this paper we prove nonlinear stability results for the numerical approximation of relative equilibria of equivariant parabolic
partial differential equations in one space dimension. Relative equilibria are solutions which are equilibria in an appropriately
comoving frame and occur frequently in systems with underlying symmetry. By transforming the PDE into a corresponding PDAE
via a freezing ansatz [2] the relative equilibrium can be analyzed as a stationary solution of the PDAE. The main result is
the fact that nonlinear stability properties are inherited by the numerical approximation with finite differences on a finite
equidistant grid with appropriate boundary conditions. This is a generalization of the results in [14] and is illustrated
by numerical computations for the quintic complex Ginzburg Landau equation.
相似文献
9.
DNA molecules in the familiar Watson–Crick double helical B form can be treated as though they have rod-like structures obtained
by stacking dominoes one on top of another with each rotated by approximately one-tenth of a full turn with respect to its
immediate predecessor in the stack. These “dominoes” are called base pairs. A recently developed theory of sequence-dependent DNA elasticity (Coleman, Olson, & Swigon, J. Chem. Phys. 118:7127–7140, 2003) takes into account the observation that the step from one base pair to the next can be one of several
distinct types, each having its own mechanical properties that depend on the nucleotide composition of the step. In the present
paper, which is based on that theory, emphasis is placed on the fact that, as each base in a base pair is attached to the
sugar-phosphate backbone chain of one of the two DNA strands that have come together to form the Watson–Crick structure, and
each phosphate group in a backbone chain bears one electronic charge, two such charges are associated with each base pair,
which implies that each base pair is subject to not only the elastic forces and moments exerted on it by its neighboring base
pairs but also to long range electrostatic forces that, because they are only partially screened out by positively charged
counter ions, can render the molecule’s equilibrium configurations sensitive to changes in the concentration c of salt in the medium. When these electrostatic forces are taken into account, the equations of mechanical equilibrium for
a DNA molecule with N + 1 base pairs are a system of μN non-linear equations, where μ, the number of kinematical variables describing the relative displacement and orientation of adjacent pairs is in general
6; it reduces to 3 when base-pair steps are assumed to be inextensible and non-shearable. As a consequence of the long-range
electrostatic interactions of base pairs, the μN × μN Jacobian matrix of the equations of equilibrium is full. An efficient numerically stable computational scheme is here presented
for solving those equations and determining the mechanical stability of the calculated equilibrium configurations. That scheme
is employed to compute and analyze bifurcation diagrams in which c is the bifurcation parameter and to show that, for an intrinsically curved molecule, small changes in c can have a strong effect on stable equilibrium configurations. Cases are presented in which several stable configurations
occur at a single value of c.
相似文献
10.
In this first part of the paper, we review methods for the investigation of stability of relative equilibria of symmetric Hamiltonian systems and explain them by means of the model problem of a rotating pendulum. For this example the modern approaches, known as energy momentum methods are compared with stability assessment by linearization and by the classical method of Routh. 相似文献
11.
Bifurcations in impact oscillations 总被引:3,自引:0,他引:3
Models of impact oscillators using an instantaneous impact law are by their very nature discontinuous. These discontinuities geve rise to bifurcations which cannot be classified using the usual tools of bifurcation analysis. However, we present numerical evidence which suggests that these discontinuous bifurcations are just the limits (in some sense) of standard bifurcations of smooth dynamical systems as the impact is hardened. Finally we show how one dimensional maps of the interval with essentially similar characteristics can exhibit the same kinds of bifurcational behaviour, and how these bifurcations are related to standard bifurcations. 相似文献
12.
The problem of the determination at any point P in a body of that pair of infinitesimal material line elements which suffers the maximum shear in a deformation has been
solved [1]. Here that problem is revisited and a short proof, of geometrical type, of the result is presented.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
13.
We consider the regularized Ericksen model of an elastic bar on an elastic foundation on an interval with Dirichlet boundary
conditions as a two-parameter bifurcation problem. We explore, using local bifurcation analysis and continuation methods,
the structure of bifurcations from double zero eigenvalues. Our results provide evidence in support of Müller’s conjecture
(Müller, Calc. Var. 1:169–204, 1993) concerning the symmetry of local minimizers of the associated energy functional and describe in detail the structure of
the primary branch connections that occur in this problem. We give a reformulation of Müller’s conjecture and suggest two
further conjectures based on the local analysis and numerical observations. We conclude by analysing a “loop” structure that
characterizes (k,3k) bifurcations.
相似文献
14.
Philippe Souplet 《Journal of Dynamics and Differential Equations》2007,19(3):819-823
The goal of paper is to give a simpler proof and some extensions of a result of Weinberger [3] concerning diffusion induced
blow-up. The result states that, for certain systems of two parabolic equations with equal diffusion and homogeneous Neumann
boundary conditions, some blowing-up solutions exist, although the corresponding system of ODE’s has only global bounded solutions.
To Professor Pavol Brunovsky, on the occasion of his 70th birthday. 相似文献
15.
We obtain a uniqueness theorem for linear elastostatics, for a homogeneous isotropic body, under the condition of strong ellipticity,
in the case of “mixed–mixed” boundary conditions. The theorem applies to bodies of restricted local concavity and convexity.
Some example domains are illustrated.
相似文献
16.
A simplified optimal control method is presented for controlling or suppressing homoclinic bifurcations of general nonlinear oscillators with one degree-of-freedom. The simplification is based on the addition of an adjustable parameter and a superharmonic excitation in the force term. By solving an optimization problem for the optimal amplitude coefficients of the harmonic and superharmonic excitations to be used as the controlled parameters, the force term as the controller can be designed. By doing so, the control gain and small optimal amplitude coefficients can be obtained at lowest cost. As the adjustable parameter decreases, a gain of some amplitude coefficient ratio is increased to the highest degree, which means that the region where homoclinic intersection does not occur will be enlarged as much as possible, leading to the best possible control performance. Finally, it is shown that the theoretical analysis is in agreement with the numerical simulations on several concerned issues including the identification of the stable and unstable manifolds and the basins of attraction. 相似文献
17.
Florin Diacu Juan Manuel Sánchez-Cerritos Shuqiang Zhu 《Journal of Dynamics and Differential Equations》2018,30(1):209-225
We prove that the fixed points of the curved 3-body problem and their associated relative equilibria are Lyapunov stable if the solutions are restricted to \({\mathbb {S}}^1\), but unstable if the bodies are considered in \({\mathbb {S}}^2\). 相似文献
18.
Bifurcations in Nonlinear Discontinuous Systems 总被引:7,自引:0,他引:7
This paper treats bifurcations of periodic solutions indiscontinuous systems of the Filippov type. Furthermore, bifurcations offixed points in non-smooth continuous systems are addressed. Filippov'stheory for the definition of solutions of discontinuous systems issurveyed and jumps in fundamental solution matrices are discussed. It isshown how jumps in the fundamental solution matrix lead to jumps of theFloquet multipliers of periodic solutions. The Floquet multipliers canjump through the unit circle causing discontinuous bifurcations.Numerical examples are treated which show various discontinuousbifurcations. Also infinitely unstable periodic solutions are addressed. 相似文献
19.
The paper studies the bifurcations of limit cycles in a rather general class of nonlinear dynamic systems. Relying on the classical harmonic balance approach as applied in control engineering neat frequency conditions for such bifurcations are derived. These results, approximate in nature, make clear the structural mechanism of the considered phenomena and can be applied to predict the occurrence of bifurcations as a function of system parameters. The application to several examples of different complexity shows the simplicity and accuracy of the proposed method for solving complicated problems of nonlinear dynamics. 相似文献
20.
The equivariant dynamics near relative equilibria to actions of noncompact, finite‐dimensional Lie groups G can be described by a skew‐product flow on a center manifold: with , with v in a slice transverse to the group action, and a(v) in the Lie algebra of G. We present a normal form theory near relative equilibria in this general case. For the specific case of the Euclidean groups the skew product takes the form with . We give a precise meaning to the intuitive idea of tip motion of a meandering spiral: it corresponds to the dynamics of
. This clarifies the notion of meander radii and drift resonance in the plane . For illustration, we discuss the unbounded tip motions associated with a weak focus in v, on the verge of Hopf bifurcation, in the case of resonant Hopf and rotation frequencies of the spiral, and study resonant
relative Hopf bifurcation. We also encounter random Brownian tip motions for trajectories which become homoclinic for . We conclude with some comments on the homoclinic tip shifts and drift resonance velocities in the Bogdanov‐Takens bifurcation,
which turn out to be small beyond any finite order.
(Accepted March 30, 1998) 相似文献