Elastic stability of DNA configurations. I. General theory

Results are presented in the theory of the elastic rod model for DNA, among which are criteria enabling one to determine whether a calculated equilibrium configuration of a DNA segment is stable in the sense that it gives a local minimum to the sum of the segment's elastic energy and the potential of forces acting on it. The derived stability criteria are applicable to plasmids and to linear segments subject to strong anchoring end conditions. Their utility is illustrated with an example from the theory of configurations of the extranucleosomal loop of a DNA miniplasmid in a mononucleosome, with emphasis placed on the influence that nicking and ligation on one hand, and changes in the ratio of elastic coefficients on the other, have on the stability of equilibrium configurations. In that example, the configurations studied are calculated using an extension of the method of explicit solutions to cases in which the elastic rod modeling a DNA segment is considered impenetrable, and hence excluded volume effects and forces arising from self-contact are taken into account.     

From low-dimensional synchronous chaos to high-dimensional desynchronous spatiotemporal chaos in coupled systems

The dynamic behavior of coupled chaotic oscillators is investigated. For small coupling, chaotic state undergoes a transition from a spatially disordered phase to an ordered phase with an orientation symmetry breaking. For large coupling, a transition from full synchronization to partial synchronization with translation symmetry breaking is observed. Two bifurcation branches, one in-phase branch starting from synchronous chaos and the other antiphase branch bifurcated from spatially random chaos, are identified by varying coupling strength epsilon. Hysteresis, bistability, and first-order transitions between these two branches are observed.     

Bifurcation,stability and symmetry of nonlinear waves

The bifurcation of wave-like spatio-temporal structures due to a hard-mode instability at non-zero wave number is investigated for a simple class of driven systems in one space dimension. We find generically a bifurcation of two branches of waves, travelling waves and standing waves, characterized by nontrivial subgroups of the symmetry group of the system. If both branches are supercritical, the wave with the larger amplitude is found to be stable. In all other cases, both waves are unstable for small amplitudes. At the common boundary of the stability regions of the two wave types in parameter space we find a bifurcation of a branch of modulated waves involving two independent frequencies, connecting the branches of travelling waves and standing waves.Work supported by the Swiss National Science Foundation     

Bistability transition in the self-trapping of an electron in a one-dimensional molecular chain

Semi-classical approach to study the self-trapping of an electron in a linear molecular chain of finite number of sites N reveals a bistability reflecting two degenerate ground state configurations of the electron-lattice system. This bistability emerges as a result of symmetry breaking which occurs only for even N when the electron-phonon coupling exceeds a threshold. As a result, bifurcation diagrams for site displacements are obtained.     

Fixed equilibria of point vortices in symmetric multiply connected domains

The paper provides symmetric fixed configurations of point vortices in multiply connected domains in the unit circle with many circular obstacles. When the circular domain is invariant with respect to rotation around the origin by a degree of 2π/M, a regular M-polygonal ring configuration of identical point vortices becomes a fixed equilibrium. On the other hand, when we assume a special symmetry, called the folding symmetry, on the circular domain, we find a fixed equilibrium in which M point vortices with the positive unit strength and M point vortices with the negative unit strength are arranged alternately at the vertices of a 2M-polygon. We also investigate the stability of these fixed equilibria and their bifurcation for a special circular domain with the rotational symmetry as well as the folding symmetry. Furthermore, we discuss fixed equilibria in non-circular multiply connected domains with the same symmetries. We give sufficient conditions for the conformal mappings, by which fixed equilibria in the circular domains are mapped to those in the general multiply connected domains. Some examples of such conformal mappings are also provided.     

Conservative spatial chaos of buckled elastic linkages

Buckling of an elastic linkage under general loading is investigated. We show that buckling is related to an initial value problem, which is always a conservative, area-preserving mapping, even if the original static problem is nonconservative. In some special cases, we construct the global bifurcation diagrams, and argue that their complicated structure is a consequence of spatial chaos. We characterize spatial chaos by the associated initial value problem's topological entropy, which turns out to be related to the number of buckled configurations.     

Bifurcation symmetries

The bifurcation of one mode into several branches frequently generates symmetries among the branches. These symmetries are “naturally broken”. After a look at some simple examples, we study a model where the bifurcation symmetry has some features of a flavor symmetry.     

An efficient method for switching branches of period-doubling bifurcations of strongly non-linear autonomous oscillators with many degrees of freedom

Popular algorithms for switching branches at a bifurcation point of strongly non-linear oscillators are generally quite involved as they require the computation of the tangent of a new branch and second derivatives. In this paper, a simple but efficient algorithm is presented by using a perturbation-incremental method for switching branches at a period-doubling bifurcation of strongly non-linear autonomous oscillators with many degrees of freedom. To switch to a new branch at a bifurcation point, a parameter is simply turned on from zero to a small positive value so as to obtain an initial solution on the emanating branch for subsequent continuation. The parametric value at a period-doubling bifurcation can also be determined accurately. Furthermore, limit cycles of period 2k(k?1) can be calculated to any desired degree of accuracy.     

Contour instabilities in early tumor growth models

Recent tumor growth models are often based on the multiphase mixture framework. Using bifurcation theory techniques, we show that such models can give contour instabilities. Restricting to a simplified but realistic version of such models, with an elastic cell-to-cell interaction and a growth rate dependent on diffusing nutrients, we prove that the tumor cell concentration at the border acts as a control parameter inducing a bifurcation with loss of the circular symmetry. We show that the finite wavelength at threshold has the size of the proliferating peritumoral zone. We apply our predictions to melanoma growth since contour instabilities are crucial for early diagnosis. Given the generality of the equations, other relevant applications can be envisaged for solving problems of tissue growth and remodeling.     

一类三次方对称离散混沌系统的分岔控制

通过分析对称性破缺分岔机制, 采用了一个直接的、有效的线性控制器, 精确控制了一类三次方对称离散混沌系统发生对称性破缺分岔和倍周期分岔时分岔点的位置. 进而分析了系统对初始值的敏感性和对称性, 选择合适的吸引域, 将对称性破缺分岔进行进一步控制, 从而使得对称性破缺分岔所缺解枝得以恢复. 数值结果表明了该控制器的有效性. 关键词： 离散混沌系统 对称性破缺 倍周期 分岔控制     

Double Loops and Pitchfork Symmetry Breaking Bifurcations of Optical Solitons in Nonlinear Fractional Schrödinger Equation with Competing Cubic-Quintic Nonlinearities

Symmetry breaking bifurcations of solitons are investigated in framework of a nonlinear fractional Schrödinger equation (NLFSE) with competing cubic-quintic nonlinearity. Some prototypical characteristics of the symmetry breaking, featured by transformations of symmetric and antisymmetric soliton families into asymmetric ones, are found. Stable asymmetric solitons emerge from unstable symmetric and antisymmetric ones by way of two different symmetry breaking scenarios. A twisting branch, featured with double loops bifurcation, bifurcates off from the base branch of symmetric soliton solutions and crosses it, then merges into the base branch driven by the competitive nonlinear effect. A supercritical pitchfork bifurcation is bifurcated from the branch of antisymmetric soliton solutions and gives rise to a supercritical pitchfork bifurcation. Stability of the soliton families is explored by linear stability analysis. With the increase of the Lévy index, stability region induced by the twisting loops bifurcation is expanded. However, stability region of the pitchfork bifurcation is shrunk on the parameter plane of the Lévy index and the soliton power.     

液晶混合光学双稳态的分叉图

本文报道液晶混合光学双稳态中分叉图的计算结果。给出x-A和x-x_B两类分叉图以及A-x_B相图。观察到由于液晶透过率双峰函数引起的分叉图对称性,混沌区突变的滞后现象以及对混沌区结构的影响。 关键词：     

Structurally stable phase portraits for the five-dimensional Lorenz equations

We discuss the universal unfolding of a planar codimension four singularity which occurs in the five dimensional Lorenz equations. All structurally stable phase portraits are given and some representative bifurcation diagrams are displayed. These phase portraits have a rich structure including up to four limit cycles. The bifurcation sets in unfolding space — where the phase portraits undergo a qualitative change — are determined and new types of saddle loops are found. We show that the codimension four singularity occurs stably in a model for the laser with saturable absorber. Solution branches indicating birhythmicity and saddle loops for the pulsed mode of laser operation are found in bifurcation diagrams corresponding to the universal unfolding of the codimension four singularity. This explains numerical solutions of other authors which so far have not been related to a bifurcation analysis. Hints to other Lorenz models are given.     

斜拉索风雨振的静态分岔研究

利用达朗伯原理建立了斜拉索风雨振连续体理论模型,实现了比截断模型能更好地体现连续体斜拉索完整的动力学特性.利用奇异性理论,对系统关于Z2对称的余维一分岔问题进行了分析,建立了分岔参数与物理参数的对应关系,得到了转迁集和分岔图,同时对平衡点附近的稳定性进行了讨论.从而有利于进一步开展斜拉索风雨振分岔行为的分析,为斜拉索桥的抗风雨振设计提供理论依据,便于拓展和应用到实际工程中.     

Symmetric Soliton Configurations of Bosonic String Theories

We obtain a reduction of the symmetry holographic principle for symmetric configurations of Nambu–Goto–Polyakov string theories in a semi-Riemannian space. The argument reduces the search of string configurations with a certain degree of symmetry to that for elastic curves in a corresponding orbit space. These solutions are solitons which are holographically related to particles that evolve along elastic worldlines in the orbit space. We also exhibit examples and applications to obtain soliton string shapes with cylindrical, rotational, toroidal etc. symmetry. In most of the cases we can determine the whole moduli space of symmetric solitons.     

High-codimensional static bifurcations of strongly nonlinear oscillator

The static bifurcation of the parametrically excited strongly nonlinear oscillator is studied. We consider the averaged equations of a system subject to Duffing--van der Pol and quintic strong nonlinearity by introducing the undetermined fundamental frequency into the computation in the complex normal form. To discuss the static bifurcation, the bifurcation problem is described as a 3-codimensional unfolding with $Z_{2}$ symmetry on the basis of singularity theory. The transition set and bifurcation diagrams for the singularity are presented, while the stability of the zero solution is studied by using the eigenvalues in various parameter regions.     

A new type of global bifurcation in Hénon map

A simple branch of solution on a bifurcation diagram, which begins at static bifurcation and ends at boundary crisis (or interior crisis in a periodic window), is generally a period-doubling cascade. A domain of solution in parameter space, enclosed by curves of static bifurcation and that of boundary crisis (or the interior of a periodic window), is the trace of branches of solution. A P-n branch of solution refers to the one starting from a period-n (n≥1) solution, and the corresponding domain in parameter space is named the P-n domain of solution. Because of the co-existence of attractors, there may be several branches within one interval on a bifurcation diagram, and different domains of solution may overlap each other in some areas of the parameter space. A complex phenomenon, concerned both with the co-existence of attractors and the crises of chaotic attractors, was observed in the course of constructing domains of steady state solutions of the Hénon map in parameter space by numerical methods. A narrow domain of period-m solutions firstly co-exists with (lies on) a big period-n (m≥3n) domain. Then it enters the chaotic area of the big domain and becomes period-m windows. The co-existence of attractors disappears and is called the landing phenomenon. There is an interaction between the two domains in the course of landing: the chaotic area in the big domain is enlarged, and there is a crisis step near the landing area.     

Symmetry chaotic attractors and bursting dynamics of semiconductor lasers subjected to optical injection

This paper presents the nonlinear dynamics and bifurcations of optically injected semiconductor lasers in the frame of relative high injection strength. The behavior of the system is explored by means of bifurcation diagrams; however, the exact nature of the involved dynamics is well described by a detailed study of the dynamics evolutions as a function of the effective gain coefficient. As results, we notice the different types of symmetry chaotic attractors with the riddled basins, supercritical pitchfork and Hopf bifurcations, crisis of attractors, instability of chaos, symmetry breaking and restoring bifurcations, and the phenomena of the bursting behavior as well as two connected parts of the same chaotic attractor which merge in a periodic orbit.     

Symmetries and pattern selection in Rayleigh-Bénard convection

This paper describes the process of pattern selection between rolls and hexagons in Rayleigh-Bénard convection with reflectional symmetry in the horizontal midplane. This symmetry is a consequence of the Boussinesq approximation, provided the boundary conditions are the same on the top and bottom plates. All possible local bifurcation diagrams (assuming certain non-degeneracy conditions) are found using only group theory. The results are therefore applicable to other systems with the same symmetries. Rolls, hexagons, or a new solution, regular triangles, can be stable depending on the physical system. Rolls are stable in ordinary Rayleigh-Bénard convection. The results are compared to those of Buzano and Golubitsky [1] without the midplane reflection symmetry. The bifurcation behavior of the two cases is quite different, and a connection between them is established by considering the effects of breaking the reflectional symmetry. Finally, the relevant experimental results are described.     

Classification and unfolding of degenerate Hopf bifurcations with O(2) symmetry: No distinguished parameter

The Hopf bifurcation in the presence of O(2) symmetry is considered. When the bifurcation breaks the symmetry, the critical imaginary eigenvalues have multiplicity two and generically there are two primary branches of periodic orbits which bifurcate simultaneously. In applications these correspond to rotating (traveling) waves and standing waves. Using equivariant singularity theory a classification of all such bifurcations up to and including codimension three is presented. No distinguished parameter is assumed. The universal unfoldings reveal the existence of both 2-tori and 3-tori; corresponding to quasiperiodic waves with two and three independent frequencies, respectively.