Computer-Assisted Methods for the Study of Stationary Solutions in Dissipative Systems, Applied to the Kuramoto–Sivashinski Equation

We develop some computer-assisted techniques for the analysis of stationary solutions of dissipative partial differential equations, of their stability, and of their bifurcation diagrams. As a case study, these methods are applied to the Kuramoto–Sivashinski equation. This equation has been investigated extensively, and its bifurcation diagram is well known from a numerical point of view. Here, we rigorously describe the full graph of solutions branching off the trivial branch, complete with all secondary bifurcations, for parameter values between 0 and 80. We also determine the dimension of the unstable manifold for the flow at some stationary solution in each branch.     

Quasi-periodicity and chaos in a differentially heated cavity

Convective flows of a small Prandtl number fluid contained in a two-dimensional vertical cavity subject to a lateral thermal gradient are studied numerically. The chosen geometry and the values of the material parameters are relevant to semiconductor crystal growth experiments in the horizontal configuration of the Bridgman method. For increasing Rayleigh numbers we find a transition from a steady flow to periodic solutions through a supercritical Hopf bifurcation that maintains the centro-symmetry of the basic circulation. For a Rayleigh number of about ten times that of the Hopf bifurcation, the periodic solution loses stability in a subcritical Neimark–Sacker bifurcation, which gives rise to a branch of quasiperiodic states. In this branch, several intervals of frequency locking have been identified. Inside the resonance horns the stable limit cycles lose and gain stability via some typical scenarios in the bifurcation of periodic solutions. After a complicated bifurcation diagram of the stable limit cycle of the 1:10 resonance horn, a soft transition to chaos is obtained. PACS 44.25.+f, 47.20.Ky, 47.52.+j     

Secondary buckling and tertiary states of a beam on a non-linear elastic foundation

An axially compressed beam resting on a non-linear foundation undergoes a loss of stability (buckling) via a supercritical pitchfork bifurcation. In the post-buckled regime, it has been shown that under certain circumstances the system may experience a secondary bifurcation. This second bifurcation destablizes the primary buckling mode and the system “jumps” to a higher mode; for this reason, this phenomenon is often referred to as mode jumping. This work investigates two new aspects related to the problem of mode jumping. First, a three mode analysis is conducted. This analysis shows the usual primary and secondary buckling events. But it also shows stable solutions involving the third mode. However, for the cases studied here, there is no natural loading path that leads to this solution branch, i.e. only a contrived loading history would result in this solution. Second, the effect of an initial geometric imperfection is considered. This breaks the symmetry of the system and significantly complicates the bifurcation diagram.     

Hopf Bifurcations and Hysteresis in Flow-induced Vibrations of Cylinders

Flow-induced oscillations of rigid cylinders exposed to fully developed flow can be described by a fourth order autonomous system of ordinary differential equations. Its rest solution is the only equilibrium point which is unstable in the entire regime of parameters. It turns out that Hopf bifurcations from the trivial solution occur in regions of comparatively low damping. We found that a wind speed parameter, Ω, controls the bifurcations while the other parameters have been arranged into discrete sets. In the case of two bifurcating solutions with branches of amplitudes tending towards each other, hysteresis occurred. The bifurcating solutions are unstable close to their respective bifurcation points. The branch tending to the left-hand side changes its stability and exhibits high-level amplitudes of synchronized oscillations. This type of solution can also be analysed by means of asymptotic methods. Near the location of the bifurcation, the predictions of bifurcation theory, the multiple scales approach, and numerics are in quite good agreement. As opposed to this, the branch tending to the right-hand side represents synchronized oscillations of somewhat smaller period but much smaller cylinder amplitudes, and these vibrations remain unstable in the entire regime of parameters. This means that keeping the cylinder fixed, starting the wind tunnel, and releasing the cylinder at low wind speeds would lead to a jump of its displacement amplitude from the low, unstable to the comparatively high-stable values. It is shown that the theoretical predictions are in fairly good agreement with the experimental trends of flow-induced synchronized cylinder oscillations.     

Twisted and nontwisted bifurcations induced by diffusion

We discuss a diffusively perturbed predator-prey system. Freedman and Wolkowicz showed that the corresponding ODE can have a periodic solution that bifurcates from a homoclinic loop. When the diffusion coefficients are large, this solution represents a stable, spatially homogeneous time-periodic solution of the PDE. We show that when the diffusion coefficients become small, the spatially homogeneous periodic solution becomes unstable and bifurcates into spatially nonhomogeneous periodic solutions. The nature of the bifurcation is determined by the twistedness of an equilibrium/homoclinic bifurcation that occurs as the diffusion coefficients decrease. In the nontwisted case two spatially nonhomogeneous simple periodic solutions of equal period are generated, while in the twisted case a unique spatially nonhomogeneous double periodic solution is generated through period-doubling.     

QUALITATIVE ANALYSIS OF SPHERICAL CAVITY NUCLEATION AND GROWTH FOR INCOMPRESSIBLE GENERALIZED VALANIS-LANDEL HYPERELASTIC MATERIALS

I. INTRODUCTION In practice, cavity formulation in materials is recognized as precursors to failure. Thus void nucleationand growth in solid materials have a great in?uence on failure mechanism. Gent and Lindley[1] have observed experimentally the phe…     

Non-linear oscillations of a third-order differential equation

An investigation is conducted into the behavior of the solutions of a third-order non-linear differential equation which is characterized by a non-linearity depending solely upon the Euclidean norm of the associated phase space. The non-linearity represents a central restoring force, which has important applications in modern control theory. For small non-linearities, the existence of a limit cycle is established by a fixed point technique, the approach to the limit cycle is approximated by averaging methods, and the periodic solution is harmonically represented by perturbation. Computer solutions of the differential equation are provided in order to reinforce the analysis. Some related differential equations are discussed including one in which the periodic solution is explicitly prescribed.     

Flow Bifurcations in a Thin Gap Between Two Rotating Spheres

This paper presents the use of a parameter continuation method and a test function to solve the steady, axisymmetric incompressible Navier–Stokes equations for spherical Couette flow in a thin gap between two concentric, differentially rotating spheres. The study focuses principally on the prediction of multiple steady flow patterns and the construction of bifurcation diagrams. Linear stability analysis is conducted to determine whether or not the computed steady flow solutions are stable. In the case of a rotating inner sphere and a stationary outer sphere, a new unstable solution branch with two asymmetric vortex pairs is identified near the point of a symmetry-breaking pitchfork bifurcation which occurs at a Reynolds number equal to 789. This solution transforms smoothly into an unstable asymmetric 1-vortex solution as the Reynolds number increases. Another new pair of unstable 2-vortex flow modes whose solution branches are unconnected to previously known branches is calculated by the present two-parameter continuation method. In the case of two rotating spheres, the range of existence in the (Re ₁ , Re ₂ ) plane of the one and two vortex states, the vortex sizes as a function of both Reynolds numbers are identified. Bifurcation theory is used to discuss the origin of the calculated flow modes. Parameter continuation indicates that the stable states are accompanied by certain unstable states. Received 26 November 2001 and accepted 10 May 2002 Published online 30 October 2002 Communicated by M.Y. Hussaini     

Global dynamical behavior of a three-body system with flexible connection in the gravitational field

Summary In this paper, the global behavior of relative equilibrium states of a three-body satellite with flexible connection under the action of the gravitational torque is studied. With geometric method, the conditions of existence of nontrivial solutions to the relative equilibrium equations are determined. By using reduction method and singularity theory, the conditions of occurrence of bifurcation from trivial solutions are derived, which agree with the existence conditions of nontrivial solutions, and the bifurcation is proved to be pitchfork-bifurcation. The Liapunov stability of each equilibrium state is considered, and a stability diagram in terms of system parameters is presented. Received 10 March 1998; accepted for publication 21 July 1998     

Two-dimensional bifurcations of stokes waves

Bifurcation techniques are used to obtain a new class of small amplitude water waves of permanent form. This calculation illustrates an approach which can be applied to nonlinear waves of various types to generate new steady solutions from old.Stokes waves are used as a starting point, and the critical value of steepness at which bifurcation can occur is computed for various choices of modulation wavelength and angular orientation. It is found that, for two-dimensional surfaces, bifurcation can occur at small values of wave steepness.Second-order corrections to the wave amplitude, modulation, frequency, and speed, which apply when one moves off the bifurcation point onto a new branch of solutions, are also computed. Two types of new solutions are found, one symmetric with respect to the carrier wave propagation direction, and one asymmetric.The nonlinear Schrödinger equation is used to model water waves, and thus the calculation can be applied rather directly to other systems governed by the nonlinear Schrödinger equation.     

Local bifurcation theory of nonlinear systems with parametric excitation

This paper summarizes the authors' research on local bifurcation theory of nonlinear systems with parametric excitation since 1986. The paper is divided into three parts. The first one is the local bifurcation problem of nonlinear systems with parametric excitation in cases of fundamental harmonic, subharmonic and superharmonic resonance. The second one is the experiment investigation of local bifurcation solutions in nonlinear systems with parametric excitation. The third one is the universal unfolding study of periodic bifurcation solutions in the nonlinear Hill system, where the influence of every physical parameter on the periodic bifurcation solution is discussed in detail and all the results may be applied to engineering.     

An Alternative Approach to Study Bifurcation from a Limit Cycle in Periodically Perturbed Autonomous Systems

The goal of this paper is to present a new method to prove bifurcation of a branch of asymptotically stable periodic solutions of a T-periodically perturbed autonomous system from a T-periodic limit cycle of the autonomous unperturbed system. The method is based on a linear scaling of the state variables to convert, under suitable conditions, the singular Poincaré map (with two singularity conditions) associated to the perturbed autonomous system into an equivalent non-singular equation to which the classical implicit function theorem applies directly. As a result we obtain the existence of a unique branch of T-periodic solutions (usually found for bifurcations of co-dimension 2) as well as a relevant property of the spectrum of their derivatives. Finally, by a suitable representation formula of the classical Malkin bifurcation function, we show that our conditions are equivalent to the existence of a non-degenerate simple zero of the Malkin function. The novelty of the method is that it permits to solve the problem without explicit reduction of the dimension of the state space as it is usually done in the literature by the Lyapunov–Schmidt method.     

Bifurcations in the dynamics of an orthogonal double pendulum

The weakly nonlinear resonant response of an orthogonal double pendulum to planar harmonic motions of the point of suspension is investigated. The two pendulums in the double pendulum are confined to two orthogonal planes. For nearly equal length of the two pendulums, the system exhibits 1:1 internal resonance. The method of averaging is used to derive a set of four first order autonomous differential equations in the amplitude and phase variables. Constant solutions of the amplitude and phase equations are studied as a function of physical parameters of interest using the local bifurcation theory. It is shown that, for excitation restricted in either plane, there may be as many as six pitchfork bifurcation points at which the nonplanar solutions bifurcate from the planar solutions. These nonplanar motions can become unstable by a saddle-node or a Hopf bifurcation, giving rise to a new branch of constant solutions or limit cycle solutions, respectively. The dynamics of the amplitude equations in parameter regions of the Hopf bifurcations is then explored using direct numerical integration. The results indicate a complicated amplitude dynamics including multiple limit cycle solutions, period-doubling route to chaos, and sudden disappearance of chaotic attractors.     

Self-interrupted regenerative metal cutting in turning

A new approach is used to study the global dynamics of regenerative metal cutting in turning. The cut surface is modeled using a partial differential equation (PDE) coupled, via boundary conditions, to an ordinary differential equation (ODE) modeling the dynamics of the cutting tool. This approach automatically incorporates the multiple-regenerative effects accompanying self-interrupted cutting. Taylor's 3/4 power law model for the cutting force is adopted. Lower dimensional ODE approximations are obtained for the combined tool-workpiece model using Galerkin projections, and a bifurcation diagram computed. The unstable solution branch off the subcritical Hopf bifurcation meets the stable branch involving self-interrupted dynamics in a turning point bifurcation. The tool displacement at that turning point is estimated, which helps identify cutting parameter ranges where loss of stability leads to much larger self-interrupted motions than in some other ranges. Numerical bounds are also obtained on the parameter values which guarantee global stability of steady-state cutting, i.e., parameter values for which there exist neither unstable periodic motions nor self-interrupted motions about the stable equilibrium.     

Bifurcation Corresponding to the Onset of Asymmetric Periodic Structures and a Self-Induced Pressure Gradient in a Modified Taylor Flow

With reference to the example of a modified Taylor flow, the bifurcation of the loss of flow symmetry with the onset of a self-induced pressure gradient is studied theoretically and numerically. A linear analysis shows that the bifurcation is supercritical. It is necessarily accompanied by the appearance of a longitudinal pressure gradient and takes place at values of the parameters for which the solution of the linear system for the perturbations satisfies the condition of zero mass flow. It is established that, as a result of the bifurcation, two asymmetric solutions with oppositely directed pressure gradients are simultaneously generated. In the supercritical region, the symmetric branch of the solutions is also retained but becomes unstable. Bifurcation of the loss of symmetry and a self-induced pressure gradient can occur only in nonlinear systems.     

Double Hopf bifurcation of composite laminated piezoelectric plate subjected to external and internal excitations

The double Hopf bifurcation of a composite laminated piezoelectric plate with combined external and internal excitations is studied. Using a multiple scale method, the average equations are obtained in two coordinates. The bifurcation response equations of the composite laminated piezoelectric plate with the primary parameter resonance, i.e.,1:3 internal resonance, are achieved. Then, the bifurcation feature of bifurcation equations is considered using the singularity theory. A bifurcation diagram is obtained on the parameter plane. Different steady state solutions of the average equations are analyzed.By numerical simulation, periodic vibration and quasi-periodic vibration responses of the composite laminated piezoelectric plate are obtained.     

Stability and bifurcation of doubly curved shallow panels under quasi-static uniform load

The focus of this paper is on the investigation of the mathematical nature of buckling from the point of view of bifurcation theory. For the doubly curved orthotropic panels subjected to quasi-static uniform load and with hinged boundary conditions, the solution to the non-linear partial differential equation is partitioned into two parts and projected onto the complete space spanned by the eigenfunctions of the linear operator of the governing equation. Furthermore, the fundamental branch, from which a new solution will emanate, is approximated by the first single mode pair which is close to the real membrane state. Whereas the ensuing bifurcated branch is approximated by the other single mode pair, under the assumption that the coupling between modes can be neglected. The present analysis could give a deep insight into the mechanism of the instability of panel structures, and show that there exists a mode transition at the critical point and the snap-through, then results from saddle-node bifurcation on the bifurcated branch. As a conclusion, the buckling of the system studied can be stated as: a bifurcated branch emanates from the fundamental branch at a critical point, and a saddle-node bifurcation, behaving as jumping, then occurs on the ensuing bifurcated branch.     

The Study of a Parametrically Excited Nonlinear Mechanical System with the Continuation Method

The dynamic behavior of a one-degree-of-freedom, parametrically excited nonlinear system is investigated. The Galerkin method is applied to the principal and fundamental parameteric resonance of the system. The continuation method is used to study the change of harmonic oscillation with respect to the variation of excitation frequency. The numerical stability analysis of the trivial solution is carried out and the stable and unstable regions of the trivial solution are given. They are found to agree with the results obtained by the analytical method of Galerkin. Periodic solutions are traced and the coexistence of multi-periodic solutions is observed With the change of excitation frequency the large amplitude periodic-2 oscillation is found to be in the same closed branch with the small amplitude periodic-2 solution. In addition, the bifurcation pattern of the trivial solution is found to change from subcritical Hopf bifurcation into supercritical Hopf bifurcation with the increase of excitation amplitude. Combined with the conventional numerical integration method, new complex dynamic behavior is detected.     

Bifurcation method for solving multiple positive solutions to boundary value problem of <Emphasis Type="Italic">p</Emphasis>-Henon equation on the unit disk

In this paper, an algorithm is proposed to solve the 0（2） symmetric positive solutions to the boundary value problem of the p-Henon equation. Taking 1 in the p- Henon equation as a bifurcation parameter, the symmetry-breaking bifurcation point on the branch of the O（2） symmetric positive solutions is found via the extended systems. The other symmetric positive solutions are computed by the branch switching method based on the Liapunov-Schmidt reduction.     

Bifurcation method for solving multiple positive solutions to boundary value problem of p-Henon equation on the unit disk

In this paper, an algorithm is proposed to solve the O(2) symmetric positive solutions to the boundary value problem of the p-Henon equation. Taking l in the p-Henon equation as a bifurcation parameter, the symmetry-breaking bifurcation point on the branch of the O(2) symmetric positive solutions is found via the extended systems. The other symmetric positive solutions are computed by the branch switching method based on the Liapunov-Schmidt reduction.