Internal gravity–capillary solitary waves in finite depth

We consider a two‐dimensional inviscid irrotational flow in a two layer fluid under the effects of gravity and interfacial tension. The upper fluid is bounded above by a rigid lid, and the lower fluid is bounded below by a rigid bottom. We use a spatial dynamics approach and formulate the steady Euler equations as a Hamiltonian system, where we consider the unbounded horizontal coordinate x as a time‐like coordinate. The linearization of the Hamiltonian system is studied, and bifurcation curves in the (β,α)‐plane are obtained, where α and β are two parameters. The curves depend on two additional parameters ρ and h, where ρ is the ratio of the densities and h is the ratio of the fluid depths. However, the bifurcation diagram is found to be qualitatively the same as for surface waves. In particular, we find that a Hamiltonian‐Hopf bifurcation, Hamiltonian real 1:1 resonance, and a Hamiltonian 0²‐resonance occur for certain values of (β,α). Of particular interest are solitary wave solutions of the Euler equations. Such solutions correspond to homoclinic solutions of the Hamiltonian system. We investigate the parameter regimes where the Hamiltonian‐Hopf bifurcation and the Hamiltonian real 1:1 resonance occur. In both these cases, we perform a center manifold reduction of the Hamiltonian system and show that homoclinic solutions of the reduced system exist. In contrast to the case of surface waves, we find parameter values ρ and h for which the leading order nonlinear term in the reduced system vanishes. We make a detailed analysis of this phenomenon in the case of the real 1:1 resonance. We also briefly consider the Hamiltonian 0²‐resonance and recover the results found by Kirrmann. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamics of infection with nonlinear incidence in a simple vaccination model

We develop and analyze a simple SIV epidemic model including susceptible, infected and imperfectly vaccinated classes, with a nonlinear incidence rate. We investigate the interaction of the nonlinear incidence and partial immunity. Our main results show that nonlinear incidence rate could induce the forward bifurcation with hysteresis except for the backward bifurcation. The plausible effects of vaccination program have been demonstrated by two models with nonlinear incidence rate. Vaccination program may contribute to disease spread, depending on which transmission term involves nonlinear incidence rate.     

Bifurcation and chaos in a system simulating magnetic field configurations

In the present article, the behaviour of a nonlinear dynamical system has been analysed using the approach of bifurcation theory. The system is important due to the fact that it can simulate the magnetic field configurations in various situations. The nature of bifurcation has been explored in the parameter space with the help of continuation algorithm. The various limit and bifurcation points (BPs) are classified. In the second part, we have studied the temporal evolution of the system which also shows a chaotic behaviour. The system under consideration shows instability both with respect to parameter variation and evolution of time. Lastly, some mechanisms have been studied to control such chaotic scenario.     

Periodic orbit of the pendulum with a small nonlinear damping

We study the pendulum with a small nonlinear damping, which can be expressed by a Hamiltonian system with a small perturbation. We prove that a unique periodic orbit exists for any initial position between the equilibrium point and the heteroclinic orbit of the unperturbed system, depending on the choice of the bifurcation parameter in the damping. The main tools are bifurcation theory and Abelian integral technique, as well as the Zhang''s uniqueness theorem on Li\''enard equations.     

Asymptotic Stability of Large-amplitude Oscillations in Systems with Hysteresis

We study depending on a parameter periodic systems with the main linear part and a hysteresis nonlinearity; the linearized at infinity system has a one-dimensional subspace of periodic solutions for the critical parameter value. We prove theorems on a number, localization, and asymptotic stability of large-amplitude periodic solutions for the nonlinear system. Received March 1998     

Twistless Tori Near Low-Order Resonances

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.     

Bifurcation of critical points for continuous families of C 2 functionals of Fredholm type

Given a continuous family of C ² functionals of Fredholm type, we show that the nonvanishing of the spectral flow for the family of Hessians along a known (trivial) branch of critical points not only entails bifurcation of nontrivial critical points but also allows to estimate the number of bifurcation points along the branch. We use this result for several parameter bifurcation, estimating the number of connected components of the complement of the set of bifurcation points in the parameter space and apply our results to bifurcation of periodic orbits of Hamiltonian systems. By means of a comparison principle for the spectral flow, we obtain lower bounds for the number of bifurcation points of periodic orbits on a given interval in terms of the coefficients of the linearization.     

Singular Hopf Bifurcation in Systems with Fast and Slow Variables

Summary. We study a general nonlinear ODE system with fast and slow variables, i.e., some of the derivatives are multiplied by a small parameter. The system depends on an additional bifurcation parameter. We derive a normal form for this system, valid close to equilibria where certain conditions on the derivatives hold. The most important condition concerns the presence of eigenvalues with singular imaginary parts, by which we mean that their imaginary part grows without bound as the small parameter tends to zero. We give a simple criterion to test for the possible presence of equilibria satisfying this condition. Using a center manifold reduction, we show the existence of Hopf bifurcation points, originating from the interaction of fast and slow variables, and we determine their nature. We apply the theory, developed here, to two examples: an extended Bonhoeffer—van der Pol system and a predator-prey model. Our theory is in good agreement with the numerical continuation experiments we carried out for the examples. Received October 24, 1996; revised October 31, 1997; accepted November 3, 1997     

Chaotic motions for a perturbed nonlinear Schrödinger equation with the power-law nonlinearity in a nano optical fiber

Perturbed nonlinear Schrödinger (NLS) equation with the power-law nonlinearity in a nano optical fiber is studied with the help of its equivalent two-dimensional planar dynamic system and Hamiltonian. Via the bifurcation theory and qualitative theory, equilibrium points for the two-dimensional planar dynamic system are obtained. With the external perturbation taken into consideration, chaotic motions for the perturbed NLS equation with the power-law nonlinearity are derived based on the equilibrium points.     

Bifurcation of a three-unit neural network

Considered is a system of delay differential equations modeling a time-delayed connecting network of three neurons without self-feedback. Discussing the change of the number of eigenvalues with zero real part, we locate the boundary of the stability region and finally determine the largest stability region of trivial solution. We investigate the existence of bifurcation phenomena of codimension one/two of the trivial equilibrium by considering the intersections of some parameter curves, which, in the aτ-half parameter plane, correspond to zero root or pure imaginary roots. In particular, the equivariant bifurcation is studied because of the equivariance of the system. We also present numerical simulations to demonstrate the rich dynamical behavior near the equivariant Pitchfork-Hopf bifurcation points, Hopf-Hopf bifurcation points, and some higher codimension bifurcation points.     

On the topological structure of singular attractors of nonlinear systems of differential equations

We study the topological structure of singular (in the sense of the Feigenbaum-Sharkovskii-Magnitskii theory) attractors of nonlinear dissipative systems of differential equations. We show that any such attractor is a stable nonperiodic trajectory lying on a two-dimensional infinitely folded heteroclinic separatrix manifold generated by the unstable two-dimensional invariant manifold of the original singular cycle as the bifurcation parameter of the system varies. The results obtained for two-dimensional nonautonomous and three-dimensional autonomous dissipative systems are generalized to autonomous multi- and infinite-dimensional dissipative systems as well as to conservative (in particular, Hamiltonian) systems.     

BISTAB: A portable bifurcation and stability analysis package

A portable bifurcation and stability analysis package, called BISTAB, is described. The package is written in FORTRAN V and can follow the connected set of equilibrium curves for a system of nonlinear ordinary differential equations in the state × parameter space by varying a bifurcation parameter. The curves are traced from an initial point with the continuation method of Kubicek, and the tangent method of Keller is used to find initial points on bifurcating curves near simple bifurcation points. Linearized stability analysis, location of Hopf bifurcation points, and sorting of points for plotting are also supported. While the package contains no new numerical methods, the lack of a requirement for any derivative information higher than the Jacobian makes BISTAB computationally efficient and useful for applied problems where nonnumerical bifurcation analysis may be difficult.     

Nonlinear rotating convection in a sparsely packed porous medium

We investigate linear and weakly nonlinear properties of rotating convection in a sparsely packed Porous medium. We obtain the values of Takens–Bogdanov bifurcation points and co-dimension two bifurcation points by plotting graphs of neutral curves corresponding to stationary and oscillatory convection for different values of physical parameters relevant to rotating convection in a sparsely packed porous medium near a supercritical pitchfork bifurcation. We derive a nonlinear two-dimensional Landau–Ginzburg equation with real coefficients by using Newell–Whitehead method [16]. We investigate the effect of parameter values on the stability mode and show the occurrence of secondary instabilities viz., Eckhaus and Zigzag Instabilities. We study Nusselt number contribution at the onset of stationary convection. We derive two nonlinear one-dimensional coupled Landau–Ginzburg type equations with complex coefficients near the onset of oscillatory convection at a supercritical Hopf bifurcation and discuss the stability regions of standing and travelling waves.     

Nonlinear vibration of a quarter-car model excited by the road surface profile

We examine the Melnikov criterion for a transition to chaos in case of a single–degree–of–freedom nonlinear oscillator with the Duffing potential with a nonlinear hard stiffness and a kinematic excitation term caused by the road profile. Using the new effective Hamiltonian we have examined appearance of homoclinic orbits in a quarter car model. Cross–sections of stable and unstable manifolds defined the condition of transition to chaos through a homoclinic bifurcation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On stationary solutions of a nonlinear system of reactor dynamic equations with distributed parameters

We analyze a nonlinear stationary model of reactor dynamics with distributed parameters. We find sufficient conditions for the existence of bifurcation points in this system and study the behavior of solutions in a neighborhood of the bifurcation points. We prove the existence of countably many bifurcation points in the case of a homogeneous medium and obtain constructive estimates for the distance between the bifurcation points.     

Effect of bifurcation on the semi-active optimal control problem

For nonlinear dynamic systems near bifurcation, the basins of attraction of fixed points as well as the steady-state responses can change considerably with a small variation of the bifurcating parameter. This paper studies the effect of bifurcation on the semi-active optimal control problem with fixed final state by using the cell mapping method. A system parameter is taken as the control. The admissible control values considered encompass a bifurcation point of the system. Global changes in the optimal control solution for different targets are studied. Saddle node, supercritical pitchfork and subcritical Hopf bifurcations are considered in the examples. It has been found that the global topology of the optimal control solution is strongly dependent on the state of the target.     

On the Number of Unbounded Solution Branches in a Neighborhood of an Asymptotic Bifurcation Point

We suggest a method for studying asymptotically linear vector fields with a parameter. The method permits one to prove theorems on asymptotic bifurcation points (bifurcation points at infinity) for the case of double degeneration of the principal linear part. We single out a class of fields that have more than two unbounded branches of singular points in a neighborhood of a bifurcation point. Some applications of the general theorems to bifurcations of periodic solutions and subharmonics as well as to the two-point boundary value problem are given.     

An efficient algorithm for the determination of certain bifurcation points

Many practical problems require information about a branch of solutions of a system of nonlinear equations dependent upon a scalar parameter. We discuss some techniques for following such a branch through a turning point and describe an efficient method, with second order convergence, for finding the turning point. We also show that, if extra information is available about the solution branch, the method can be successfully applied to finding simple bifurcation points.     

Dynamic Hopf bifurcations generated by nonlinear terms

We consider the so-called delayed loss of stability phenomenon for singularly perturbed systems of differential equations in case that the associated autonomous system with a scalar parameter undergoes the Hopf bifurcation at the zero equilibrium point. It is assumed that the linearization of the associated system is independent of the parameter and the next terms in the expansion of the right-hand parts at zero are positive homogeneous of order α>1. Simple formulas are presented to estimate the asymptotic delay for the delayed loss of stability phenomenon. More precisely, we suggest sufficient conditions which ensure that zeros of a simple function ψ defined by the positive homogeneous nonlinear terms are the Hopf bifurcation points of the associated system, the sign of ψ at other points determines stability of the zero equilibrium, and the asymptotic delay equals the distance between the bifurcation point and a zero of some primitive of ψ.     

小波Galerkin法在非线性分岔问题求解中的应用

通过一个典型的Bratu问题,研究了小波Galerkin法(WGM)在非线性分岔问题求解方面的应用.首先,利用基于Coiflet的小波Galerkin法,对一维和二维Bratu方程进行离散;然后针对单参数问题,推导了追踪解曲线的伪弧长格式和直接计算极值型分岔点的扩展方程;针对双参数问题,推导了追踪稳定边界的伪弧长格式和求解尖点型分岔点的扩展方程.数值结果表明,基于小波Galerkin法的非线性分岔计算不仅具有更高的计算精度,而且能够有效地捕捉双参数分岔问题的折迭线和尖点突变曲面.该算例展示了基于小波Galerkin法数值分岔计算的具体过程及其求解多参数分岔问题复杂行为的应用潜力.