Stability and bifurcation in a harmonic oscillator with delays

We consider a harmonic oscillator with delays. Linear stability is investigated by analyzing the associated characteristic transcendental equation. The bifurcation analysis of the equation shows that Hopf bifurcation can occur as the delay τ (taken as a parameter) crosses some critical values. The direction and stability of the Hopf bifurcation are considered by using the normal form theory due to Faria and Magalhães. An example is given to explain the results. Numerical simulations support our results.     

Falkner-Skan equation for flow past a moving wedge with suction or injection

The characteristics of steady two-dimensional laminar boundary layer flow of a viscous and incompressible fluid past a moving wedge with suction or injection are theoretically investigated. The transformed boundary layer equations are solved numerically using an implicit finite-difference scheme known as the Keller-box method. The effects of Falkner-Skan power-law parameter (m), suction/injection parameter (f0) and the ratio of free stream velocity to boundary velocity parameter (λ) are discussed in detail. The numerical results for velocity distribution and skin friction coefficient are given for several values of these parameters. Comparisons with the existing results obtained by other researchers under certain conditions are made. The critical values off ₀,m and λ are obtained numerically and their significance on the skin friction and velocity profiles is discussed. The numerical evidence would seem to indicate the onset of reverse flow as it has been found by Riley and Weidman in 1989 for the Falkner-Skan equation for flow past an impermeable stretching boundary.     

On the critical amplitudes of acceleration wave to shock wave transition in white noise random media

The problem of transition of acceleration waves into shock waves is addressed in the context of weakly random media. A class of random media is modeled by a vector white noise random process representing two material coefficients appearing in the Bernoulli equation governing the evolution of acceleration waves. The problem of shock formation, which involves a stochastic competition of dissipation and elastic nonlinearity, is treated using a diffusion formulation for the Markov process of the inverse amplitude. The first four moments of the critical inverse amplitude are derived explicitly as functions of the means and crosscorrelations of the underlying vector random process. It is found that the Stratonovich as well as the Itô interpretation of the stochastic Bernoulli equation lead to an increase of the average critical amplitude of the random medium problem over the critical amplitude of the deterministic homogeneous medium problem. Probability distribution of the critical inverse amplitude is found to be, in general, of Pearson's Type IV.     

Local dynamics of an equation with distributed delay

We study the properties of the local dynamics of a differential equation with a distributed delay. We consider two forms of distribution functions, exponential and linear. We indicate parameters for which critical cases take place. It is shown that critical cases have an infinite dimension, and special equations describing the dynamics of the original problem (analogs of normal forms) are constructed in each critical case. The results on the correspondence of solutions of quasinormal forms and the original equation are represented.     

Bifurcation analysis and chaos control of the modified Chua’s circuit system

From the view of bifurcation and chaos control, the dynamics of modified Chua’s circuit system are investigated by a delayed feedback method. Firstly, the local stability of the equilibria is discussed by analyzing the distribution of the roots of associated characteristic equation. The regions of linear stability of equilibria are given. It is found that there exist Hopf bifurcation and Hopf-zero bifurcation when the delay passes though a sequence of critical values. By using the normal form method and the center manifold theory, we derive the explicit formulas for determining the direction and stability of Hopf bifurcation. Finally, chaotic oscillation is converted into a stable equilibrium or a stable periodic orbit by designing appropriate feedback strength and delay. Some numerical simulations are carried out to support the analytic results.     

Bifurcation and Stability Theory of Periodic Solutions For Integrodifferential Equations

This paper is concerned with a nonlinear integrodifferential equation (the delay logistic equation) governing the growth dynamics of a single species N(t) for time t > 0. This equation contains a positive parameter λ. Suppose that there exists a positive equilibrium solution N = c which is stable for all small values of λ. Assume also that this solution loses stability as λ is increased past a critical value λ*. This will correspond to a simple pure imaginary conjugate pair of roots of a characteristic equation associated with the linearized stability of N = c at λ = λ*. Then we will construct a unique bifurcating time periodic solution of the equation as a Taylor series in a parameter ε. Furthermore this solution exists either for supercritical values of the parameter (λ > λ*) or for subcritical values (λ < λ*). The stability behavior of this small periodic solution can be characterized according to whether the bifurcation is supercritical or subcritical-supercritical solutions are stable, but subcritical solutions are unstable. Therefore these results are analogous to Hoprs bifurcation theorem for autonomous systems of differential equations.     

Some applications of functional analysis to problems of neutron transport

Ritz method is used to obtain an approximate solution of the stationary neutron transport Boltzmann equation in its integral form in plane and spherical symmetry. Such a method is based on the maximum property of the quadratic form corresponding to a symmetric transformation in a finite dimensional subspace spanned by the firstn functions of a complete orthonormal set. In order to justify some numerical results, we also show that the transport operators are compact as acting both on the spaceC and on the spaceL ₂. Moreover, we investigate some properties of the solution and we prove that the neutron distribution is not increasing as a function of the spatial coordinate. Finally, a series of calculations have been performed for various values of the system-dimensions measured in mean-free-paths and the number of secondaries per conllision to maintain the system critical has been found.     

Divergence symmetries of semilinear polyharmonic equations involving critical nonlinearities

It is shown that a Lie point symmetry of the semilinear polyharmonic equations involving nonlinearities of power or exponential type is a variational/divergence symmetry if and only if the equation parameters assume critical values. The corresponding conservation laws for critical polyharmonic semilinear equations are established.     

Solitary waves in dusty plasmas with variable dust charge and two temperature ions

Propagation of nonlinear waves in dusty plasmas with variable dust charge and two temperature ions is analyzed. The Kadomtsev–Petviashivili (KP) equation is derived by using the reductive perturbation theory. A Sagdeev potential for this system has been proposed. This potential is used to study the stability conditions and existence of solitonic solutions. Also, it is shown that a rarefactive soliton can be propagates in most of the cases. The soliton energy has been calculated and a linear dispersion relation has been obtained using the standard normal-modes analysis. The effects of variable dust charge on the amplitude, width and energy of the soliton and its effects on the angular frequency of linear wave are discussed too. It is shown that the amplitude of solitary waves of KP equation diverges at critical values of plasma parameters. Solitonic solutions of modified KP equation with finite amplitude in this situation are derived.     

Disturbed critical surface waves in a channel of arbitrary cross section

Long waves in a current of an inviscid fluid of constant density flowing through a channel of arbitrary cross section under disturbances of pressure distribution on free surface and obstructors on the wall of the channel are considered. The first order asymptotic approximation of the elevation of the free surface satisfies a forced Korteweg-de Vries equation when the current is near its critical state. To determine the coefficients of the forced Korteweg-de Vries equation, we need to solve a linear Neumann problem of an elliptic partial differential equation, of which analytical solutions are found for constant current and rectangular or triangular cross section of the channel. It is proved that the forced Korteweg-de Vries equation has at least two solutions when the current is supercritical and the parameter is greater than a critical value _c >0. It is also proved that there do not exist solitary waves in a current exactly at its critical state. Numerical solutions of steady state are obtained for both supercritical and subcritical currents.     

Gaussian Unitary Ensemble with Boundary Spectrum Singularity and σ‐Form of the Painlevé II Equation

We consider the Gaussian unitary ensemble perturbed by a Fisher–Hartwig singularity simultaneously of both root type and jump type. In the critical regime where the singularity approaches the soft edge, namely, the edge of the support of the equilibrium measure for the Gaussian weight, the asymptotics of the Hankel determinant and the recurrence coefficients, for the orthogonal polynomials associated with the perturbed Gaussian weight, are obtained and expressed in terms of a family of smooth solutions to the Painlevé XXXIV equation and the σ‐form of the Painlevé II equation. In addition, we further obtain the double scaling limit of the distribution of the largest eigenvalue in a thinning procedure of the conditioning Gaussian unitary ensemble, and the double scaling limit of the correlation kernel for the critical perturbed Gaussian unitary ensemble. The asymptotic properties of the Painlevé XXXIV functions and the σ‐form of the Painlevé II equation are also studied.     

Functional classical mechanics and rational numbers

The notion of microscopic state of the system at a given moment of time as a point in the phase space as well as a notion of trajectory is widely used in classical mechanics. However, it does not have an immediate physical meaning, since arbitrary real numbers are unobservable. This notion leads to the known paradoxes, such as the irreversibility problem. A “functional” formulation of classical mechanics is suggested. The physical meaning is attached in this formulation not to an individual trajectory but only to a “beam” of trajectories, or the distribution function on phase space. The fundamental equation of the microscopic dynamics in the functional approach is not the Newton equation but the Liouville equation for the distribution function of the single particle. The Newton equation in this approach appears as an approximate equation describing the dynamics of the average values and there are corrections to the Newton trajectories. We give a construction of probability density function starting from the directly observable quantities, i.e., the results of measurements, which are rational numbers.     

Phase transitions of McKean–Vlasov processes in double-wells landscape

The aim of this work is to establish the results for a particular class of inhomogeneous processes, the McKean–Vlasov diffusions. Such diffusions correspond to the hydrodynamical limit of an interacting particle system. In convex landscapes, existence and uniqueness of the invariant probability is a well-known result. However, previous results state the nonuniqueness of the invariant probabilities under nonconvexity assumptions. Here, we prove that there exists a phase transition. Below a critical value, there are exactly three invariant probabilities and above another critical value, there is exactly one. Under simple assumptions, these critical values coincide and it is characterized by a simple implicit equation. We also investigate other cases in which phase transitions occur. Finally, we provide numerical estimations of the critical values.     

A perturbation solution for forced convection in a porous-saturated duct

Fully developed forced convection through a porous medium bounded by two isoflux parallel plates is investigated analytically on the basis of a Brinkman–Forchheimer model. The matched asymptotic expansion method is applied for small values of the Darcy number. For the case of large Darcy number the solution for the Brinkman–Forchheimer momentum equation is found in terms of an asymptotic expansion. With the velocity distribution determined, the energy equation is solved using the same asymptotic technique. The results for limiting cases are found to be in good agreement with those available in the literature and the numerical results obtained here.     

Stability and bifurcation analysis in an amplitude equation with delayed feedback

An amplitude equation is considered. The linear stability of the equation with direct control is investigated, and hence a bifurcation set is provided in the appropriate parameter plane. It is found that there exist stability switches when delay varies, and the Hopf bifurcation occurs when delay passes through a sequence of critical values. Furthermore, the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, some numerical simulations are performed to illustrate the obtained results.     

Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response

We consider a delayed predator-prey system with Beddington-DeAngelis functional response. The stability of the interior equilibrium will be studied by analyzing the associated characteristic transcendental equation. By choosing the delay τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay τ crosses some critical values. The direction and stability of the Hopf bifurcation are investigated by following the procedure of deriving normal form given by Faria and Magalhães. An example is given and numerical simulations are performed to illustrate the obtained results.     

Numerical simulation of thermal hydraulics in an ADSS model

Accelerator Driven Sub-critical nuclear reactor Systems (ADSS) are envisaged to enhance neutronics of reactors as well as safety physics. The spallation target module is the most innovative and key component for an ADSS. A conjugate heat transfer analysis is accomplished to mimic the physical operating condition of an ADSS in a more realistic way. The conduction equation of the beam window is solved in conjunction with the energy equation using the paradigm of domain decomposition parallelization method and the temperature distribution along the beam window is found. Finally, the thermal and mechanical stresses along the radial direction on the beam window is determined using temperature and pressure values. The stress values are found to increase with increasing Reynolds number (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Buckling of a nonlinear elastic rod

The buckling of a pin-ended slender rod subjected to a horizontal end load is formulated as a nonlinear boundary value problem. The rod material is taken to be governed by constitutive laws which are nonlinear with respect to both bending and compression. The nonlinear boundary value problem is converted to a suitable integral equation to allow the application of bounded operator methods. By treating the integral equation as a bifurcation problem, the branch points (critical values of load) are determined and the existence and form of nontrivial solutions (buckled states) in the neighborhood of the branch points is established. The integral equation also affords a direct attack upon the question of uniqueness of the trivial solution (unbuckled state). It is shown that, under certain conditions on the material properties, only the trivial solution is possible for restricted values of the load. One set of conditions gives uniqueness up to the first branch point.     

A Simple Stochastic Model of Air–Naval Combat

Consideration is given to a scenario where a fleet of ships, each armed with the same anti-aircraft weapon, is attacked by identical aircraft in line-ahead formation. It is assumed that each aircraft uses its weapon against one ship, but all ships use their weapons against each aircraft as it attacks in turn. A difference equation for the bivariate probability distribution of ship and aircraft casualties is obtained, and some analytic results are derived by decomposition of the problem. Using data from the sinking of the HMS Prince of Wales and HMS Repulse in World War II to estimate parameter values, a numerical solution of the difference equation is provided. A separate method for computing the ultimate or terminal form of the distribution is given.