Bifurcation Of Steady-state Solutions Research Articles

Stability and bifurcation in a reaction–diffusion model with nonlinear boundary conditions

In this paper, we investigate the existence, stability, local and global bifurcation of the steady state solutions for a diffusive logistic model with nonlinear boundary conditions. Supercritical and subcritical bifurcation of steady state solutions are obtained by virtue of the Crandall–Rabinowitz theorem and the Lyapunov–Schmidt reduction. The local stability and the possibility of obtaining an Allee effect are also analyzed. Furthermore, the result on global bifurcation is obtained as well by employing the Rabinowitz theorem.

On the Structure of the Space of States for a Thermal Model of Fluidized-Bed Reactor–Regenerator Units and Control Visualization Principles

A system of differential equations for the dynamics of thermal states in reactor–regenerator units with finely dispersed catalysts is proposed. Special attention is paid to the existence of positive feedback between thermal and chemical processes by two mutually connected channels, such as the temperature and the degree of residual cocking on a catalyst after regeneration. It has been shown that this system of model equations for the thermal dynamics of a reactor–regenerator unit is characterized by the multiplicity of steady states in the space of the mentioned variables. The three-dimensional phase patterns of the system are analyzed, and the bifurcations of steady-state solutions in the parametric region are studied. The problem of operative control based on state space structure visualization algorithms is considered.

Exploiting internal resonance for vibration suppression and energy harvesting from structures using an inner mounted oscillator

The flexural vibration of a symmetrically laminated composite cantilever beam carrying a sliding mass under harmonic base excitations is investigated. An internally mounted oscillator constrained to move along the beam is employed in order to fulfill a multi-task that consists of both attenuating the beam vibrations in a resonance status and harvesting this residual energy as a complementary subtask. The set of nonlinear partial differential equations of motion derived by Hamilton’s principle are reduced and semi-analytically solved by the successive application of Galerkin’s and the multiple-scales perturbation methods. It is shown that by properly tuning the natural frequencies of the system, internal resonance condition can be achieved. Stability of fixed points and bifurcation of steady-state solutions are studied for internal and external resonances status. It results that transfer of energy or modal saturation phenomenon occurs between vibrational modes of the beam and the sliding mass motion through fulfilling an internal resonance condition. This study also reveals that absorbers can be successfully implemented inside structures without affecting their functionality and encumbering additional space but can also be designed to convert transverse vibrations into internal longitudinal oscillations exploitable in a straightforward manner to produce electrical energy.

Steady Bifurcating Solutions of the Couette–Taylor Problem for Flow in a Deformable Cylinder

The classical Couette–Taylor problem is to describe the motion of a viscous incompressible fluid in the region between two rigid coaxial cylinders, which rotate at constant angular velocities. This paper treats a generalization of this problem in which the rigid outer cylinder is replaced by a deformable (nonlinearly elastic) cylinder. The inner cylinder is rigid and rotates at a prescribed angular velocity. We study steady rotationally symmetric motions of the fluid coupled with steady axisymmetric motions of the deformable outer cylinder in which it rotates at a prescribed constant angular velocity, typically different from that of the inner cylinder. The motion of the outer cylinder is governed by a geometrically exact theory of shells and the motion of the liquid by the Navier–Stokes equations, with the domain occupied by the liquid depending on the deformation of the outer cylinder. The nonlinear fluid-solid system admits a (trivial) steady solution, termed the Couette solution, which can be found explicitly. This paper treats the global (multiparameter) bifurcation of steady-state solutions from the Couette solution. This problem exhibits technical mathematical difficulties directly due to the fluid-solid interaction: The smoothness of the shell’s configuration restricts the smoothness of the fluid variables, and their boundary values on the shell determine the smoothness of the shell’s configuration. It is essential to ensure that this cycle of implications is consistent.

Nonlocal generalized models of predator-prey systems

The method of generalized modeling has been applied successfully in many different contexts, particularly in ecology and systems biology. It can be used to analyze the stability and bifurcations of steady-state solutions. Although many dynamical systems in mathematical biology exhibit steady-state behaviour one also wants to understand nonlocal dynamics beyond equilibrium points. In this paper we analyze predator-prey dynamical systems and extend the method of generalized models to periodic solutions. First, we adapt the equilibrium generalized modeling approach and compute the unique Floquet multiplier of the periodic solution which depends upon so-called generalized elasticity and scale functions. We prove that these functions also have to satisfy a flow on parameter (or moduli) space. Then we use Fourier analysis to provide computable conditions for stability and the moduli space flow. The final stability analysis reduces to two discrete convolutions which can be interpreted to understand when the predator-prey system is stable and what factors enhance or prohibit stable oscillatory behaviour. Finally, we provide a sampling algorithm for parameter space based on nonlinear optimization and the Fast Fourier Transform which enables us to gain a statistical understanding of the stability properties of periodic predator-prey dynamics.

Non-linear vibrations and stability analysis of tensioned pipes conveying fluid with variable velocity

In this study, the non-linear transverse vibrations of highly tensioned pipes with vanishing flexural stiffness and conveying fluid with variable velocity are investigated. The pipe is on fixed supports and the immovable end conditions result in the extension of the pipe during vibration and hence introduce further non-linear terms to the equation of motion. The velocity is assumed to be a harmonic function about a mean velocity. These systems experience a Coriolis acceleration component which renders such systems gyroscopic. The equation of motion is solved analytically by direct application of the method of multiple scales (a perturbation technique). Principal parametric resonance cases are investigated in detail. Non-linear frequencies are derived depending on amplitude. For frequencies close to two times the natural frequency, stability and bifurcations of steady-state solutions are analyzed. For frequencies close to zero, it is shown that the amplitudes are bounded in time.

Non-linear vibrations and stability of an axially moving beam with time-dependent velocity

Non-linear vibrations of an axially moving beam are investigated. The non-linearity is introduced by including stretching effect of the beam. The beam is moving with a time-dependent velocity, namely a harmonically varying velocity about a constant mean velocity. Approximate solutions are sought using the method of multiple scales. Depending on the variation of velocity, three distinct cases arise: (i) frequency away from zero or two times the natural frequency, (ii) frequency close to zero, (iii) frequency close to two times the natural frequency. Amplitude-dependent non-linear frequencies are derived. For frequencies close to two times the natural frequency, stability and bifurcations of steady-state solutions are analyzed. For frequencies close to zero, it is shown that the amplitudes are bounded in time.

On the Nonlinear Transverse Vibrations and Stability of an Axially Accelerating Beam

Nonlinear vibrations and stability analysis of an axially moving Euler-Bernoulli type beam are investigated. The beam is on fixed supports and moving with a harmonically varying velocity about a constant mean value. The method of multiple scales is used in the analysis. Nonlinear frequencies depending on vibration amplitudes are obtained. Stability and bifurcations of steady-state solutions are analyzed for frequencies close to two times any natural frequency. It is shown that the amplitudes are bounded in time for frequencies close to zero. The effect of fixed supports is discussed.

Numerical computation of stability and detection of Hopf bifurcations of steady state solutions of delay differential equations

The characteristic equation of a system of delay differential equations (DDEs) is a nonlinear equation with infinitely many zeros. The stability of a steady state solution of such a DDE system is determined by the number of zeros of this equation with positive real part. We present a numerical algorithm to compute the rightmost, i.e., stability determining, zeros of the characteristic equation. The algorithm is based on the application of subspace iteration on the time integration operator of the system or its variational equations. The computed zeros provide insight into the system’s behaviour, can be used for robust bifurcation detection and for efficient indirect calculation of bifurcation points.

Stability and Hopf Bifurcation of Steady State Solutions of a Singularly Perturbed Reaction-Diffusion System

This paper presents a stability analysis of steady state solutions of a singularly perturbed reaction-diffusion system which arises as a model for predator-prey interactions. This is the first illustration of the application of a topological invariant for systems of boundary value problems called the stability index, recently introduced by C. Jones and the author, which counts the multiplicity of unstable eigenvalues of the linearized equations. The main results are as follows: (i) It is shown that the spectrum $\sigma $ of the linearized operator is approximated by the spectrum $\sigma _R $ of a certain reduced operator defined in the small parameter limit. (ii) Under additional assumptions on the parameters $\sigma _R $ is characterized in two cases. In the first case stability is obtained for all interval sizes L, while in the second case, countably many Hopf bifurcations occur as L is increased.

Global Bifurcation of Steady-State Solutions on a Biochemical System

The differential equation $u'' - ({u /{k\beta ^2 }}(u^2 + 2\varepsilon u + 4k\varepsilon ^2 )) = 0$ with the boundary conditions $u(0) = u(1) = 1$ governs the steady-state solutions from a mono-enz...

Bifurcation of steady-state solutions in predator-prey and competition systems

SynopsisWe discuss steady-state solutions of systems of semilinear reaction-diffusion equations which model situations in which two interacting species u and v inhabit the same bounded region. It is easy to find solutions to the systems such that either u or v is identically zero; such solutions correspond to the case where one of the species is extinct. By using decoupling and global bifurcation theory techniques, we prove the existence of solutions which are positive in both u and v corresponding to the case where the populations can co-exist.

The group combustion of a spherical cloud of monodisperse fuel droplets

The vaporization and combustion of a spherical, uniform monodisperse cloud of fuel drops in equilibrium with a quiescent atmosphere is considered. For typical fuel sprays in which the drops are separated by five to ten drop diameters only the droplets within a thin inwardly propagating vaporization wave at the edge of the cloud will vaporize. In analogy with single drop theory the cloud radius is found to decrease acording to a “ d 2 law,” although with a modified vaporization constant for both purely vaporizing and burning clouds. The cloud interior in all cases remains in saturated, non vaporizing equilibrium at a temperature determined by ambient conditions. Burning occurs at a spherical diffusion flame front outside the cloud. The flame radius to cloud radius ratio is found to be constant and independent of the cloud radius and the droplet radius and number density within the cloud. An external ignition temperature is determined by considering the bifurcation of steady state solutions for single step, irreversible Arrhenius kinetics. At ignition the cloud interior reaches a new equilibrium which is still too cool and rich for single drop ignition. Under conditions of the present analysis a fuel cloud, if it burns at all, will do so with an external diffusion flame.