Four-Body Central Configurations¶with some Equal Masses

We prove firstly that any convex non-collinear central configuration of the planar 4-body problem with equal opposite masses β >α > 0, such that the diagonal corresponding to the mass α is not shorter than that corresponding to the mass β, must possess a symmetry and therefore must be a kite. Then by a recent result of Bernat, Llibre and Perez-Chavela, this kite is actually a rhombus. Secondly we prove that a convex non-collinear planar 4-body central configuration with three equal masses must be a kite too. We also prove that the concave central configuration with three equal masses forming a triangle and the fourth one with any given mass in the interior must be either an equilateral triangle with the fourth mass at its geometric center, or an isosceles triangle with the fourth mass on the symmetry axis.     

Convex but not Strictly Convex Central Configurations

Central configurations of the n-body problem have been studied for more than 200 years since the pioneer works of Euler and Lagrange. In this article we study convex central configurations which are not strictly convex. We give explicit examples of such configurations in both planar and spatial n-body problems. Particularly, in the spatial case, we consider regular polyhedra with bodies of same mass m at the vertices and bodies of same mass M at the middle points of each edge. In this setting we prove that the cube is the unique regular polyhedron such that this construction leads to a convex central configuration which is not strictly convex.     

Convex Four-Body Central Configurations with Some Equal Masses

We prove that there is an unique convex noncollinear central configuration of the planar Newtonian four-body problem when two equal masses are located at opposite vertices of a quadrilateral and, at most, only one of the remaining masses is larger than the equal masses. Such a central configuration possesses a symmetry line and it is a kite-shaped quadrilateral. We also show that there is exactly one convex noncollinear central configuration when the opposite masses are equal. Such a central configuration also possesses a symmetry line and it is a rhombus.     

Morse Theory and Central Configurations in the Spatial N-body Problem

In the spirit of Palmore and Pacella, Morse Theory is used to obtain a lower bound for the number of central configurations in the spatial N-body problem. The homology of the configuration ellipsoid with the collision and collinear manifolds removed and the SO(3) symmetry quotiented out is calculated. As intermediate steps, homology calculations are carried out for several additional manifolds naturally arising in the N-body problem.     

Collinear Central Configurations and Singular Surfaces in the Mass Space

For a given m=(m₁,...,m_n)(R⁺)ⁿ, let p and q(R³)ⁿ be two central configurations for m. Then we call p and q equivalent and write pq if they differ by an SO(3) rotation followed by a scalar multiplication as well as by a permutation of bodies. Denote by L(n,m) the set of equivalent classes of n-body collinear central configurations in R³ for any given mass vector m=(m₁,...,m_n)(R⁺)ⁿ. The main discovery in this paper is the existence of a union H₃ of three non-empty algebraic surfaces in the mass half space (m₁,m₂–m₁,m₃–m₂)R⁺×R² besides the planes generated by equal masses, which decreases the number of collinear central configurations. The union H₃ in R⁺×R ² is explicitly constructed by three 6-degree homogeneous polynomials in three variables such that, for any mass vector m=(m₁,m₂,m₃)(R⁺)³, ^# L(3,m)=3, if m₁, m₂, and m₃ are mutually distinct and (m₁,m₂–m₁,m₃–m₂)H₃, ^# L(3,m)=2, if m₁, m₂, and m₃ are mutually distinct and (m₁,m₂–m₁,m₃–m₂)H₃, ^# L(3,m)=2, if two of m₁, m₂, and m₃ are equal but not the third, ^# L(3,m)=1, if m₁=m₂=m₃. We give also a sharp upper bound on ^#L(n,m) for any positive mass vector m(R⁺)ⁿ.     

Convex Central Configurations of the n-Body Problem Which are not Strictly Convex

It is well-known that if a planar central configuration for the Newtonian 4-body problem is convex, then it must be strictly convex. In some literature, same conclusion was believed to hold for the case of five or even more bodies but rigorous treatments are absent. With the help of some numerical calculations, in this paper we provide concrete examples of central configurations which are convex but not strictly convex. Our examples include planar central configurations with five bodies and spatial central configurations with seven bodies.     

On Co-Circular Central Configurations in the Four and Five Body-Problems for Homogeneous Force Law

We study the central configurations (cc for short) for four masses arranged on a common circle (called co-circular cc) in two different situations, namely with no mass inside and later adding a fifth mass at the center of the circle. In the former, we focus the kite shape configurations by proving the existence of a one-parameter family of cc which goes from the kite containing an equilateral triangle up to the square shape. After, by putting a fifth mass at the center, we feature the planar cc of five bodies as a tensor of corange two see, “Albouy and Chenciner (Invent Math 131:151–184, 1998)” and we prove that cc is stacked see, “Hampton (Nonlinearity 18:2299–2304, 2005b)” in a such way that the center of mass of the four bodies should be the center of the circle. We emphasize that our approach includes not only the Newtonian force law, but the homogeneous ones with exponent $a\le -1$ a ≤ ? 1 .     

Bifurcations of Heteroclinic Orbits

The search for traveling wave solutions of a semilinear diffusion partial differential equation can be reduced to the search for heteroclinic solutions of the ordinary differential equation ü − cu̇ + f(u) = 0, where c is a positive constant and f is a nonlinear function. A heteroclinic orbit is a solution u(t) such that u(t) → γ ₁ as t → −∞ and u(t) → γ ₂ as t → ∞ where γ ₁, γ ₂ are zeros of f. We study the existence of heteroclinic orbits under various assumptions on the nonlinear function f and their bifurcations as c is varied. Our arguments are geometric in nature and so we make only minimal smoothness assumptions. We only assume that f is continuous and that the equation has a unique solution to the initial value problem. Under these weaker smoothness conditions we reprove the classical result that for large c there is a unique positive heteroclinic orbit from 0 to 1 when f(0) = f(1) = 0 and f(u) > 0 for 0 < u < 1. When there are more zeros of f, there is the possibility of bifurcations of the heteroclinic orbit as c varies. We give a detailed analysis of the bifurcation of the heteroclinic orbits when f is zero at the five points −1 < −θ < 0 < θ < 1 and f is odd. The heteroclinic orbit that tends to 1 as t → ∞ starts at one of the three zeros, −θ, 0, θ as t → −∞. It hops back and forth among these three zeros an infinite number of times in a predictable sequence as c is varied.     

Bifurcations and chaos of an inclined cable

The nonlinear behavior of an inclined cable subjected to a harmonic excitation is investigated in this paper. The Galerkin’s method is applied to the partial differential governing equations to obtain a two-degree-of-freedom nonlinear system subjected to harmonic excitation. The nonlinear systems in the presence of both external and 1:1 internal resonances are transformed to the averaged equations by using the method of averaging. The averaged equations are numerically examined to obtain the steady-state responses and chaotic solutions. Five cascades of period-doubling bifurcations leading to chaotic solutions, 3-periodic solutions leading to chaotic solution, boundary crisis phenomena, as well as the Shilnikov mechanism for chaos, are observed. In order to study the global dynamics of an inclined cable, after determining the averaged equations of motion in a suitable form, a new global perturbation technique developed by Kova?i? and Wiggins is used. This technique provides analytical results for the critical parameter values at which the dynamical system, through the Shilnikov type homoclinic orbits, possesses a Smale horseshoe type of chaos.     

Bifurcations and catastrophes in aerodynamic characteristics

The results of an experimental investigation of the longitudinal stability of a model maneuvering aircraft are presented. The results for a wide angle-of-attack range are obtained in a wind tunnel flow on an aerodynamic setup of free oscillations with a single degree of freedom. It is shown that the static aerodynamic dependences of the normal force and pitch moment coefficients on the angle of attack include catastrophic transitions from one steady state into another. The salient features of these transitions are established. It is experimentally found that the loss of the longitudinal stability of the model aircraft in a flow with variation in the deflection angles of stabilizers is softly realized via the Hopf bifurcation. At high angles of attack the flow regimes are found to exist in which steady motion represents a strange attractor.     

Bifurcations and Attractor-Repeller Splittings of Non-Saddle Sets

This paper is devoted to the study of some aspects of the stability theory of flows. In particular, we study Morse decompositions induced by non-saddle sets, including their corresponding Morse equations, attractor-repeller splittings of non-saddle sets and bifurcations originated by implosions of the basin of attraction of asymptotically stable fixed points. We also characterize the non-saddle sets of the plane in terms of the Euler characteristic of their region of influence.     

Cellular and Tulip Flame Configurations

Flame propagation in a plane channel with the formation of tulip and cellular configurations of the combustion front is simulated. The near-flame flow structure and the thermal flow structure are determined. An analogy is found between the tulip configuration and flame inflections at cell interfaces.     

Symmetry-Breaking Bifurcations of Charged Drops

It has been observed experimentally that an electrically charged spherical drop of a conducting fluid becomes nonspherical (in fact, a spheroid) when a dimensionless number X inversely proportional to the surface tension coefficient is larger than some critical value (i.e., when <_c). In this paper we prove that bifurcation branches of nonspherical shapes originate from each of a sequence of surface-tension coefficients ), where ₂=_c. We further prove that the spherical drop is stable for any >₂, that is, the solution to the system of fluid equations coupled with the equation for the electrostatic potential created by the charged drop converges to the spherical solution as t provided the initial drop is nearly spherical. We finally show that the part of the bifurcation branch at =₂ which gives rise to oblate spheroids is linearly stable, whereas the part of the branch corresponding to prolate spheroids is linearly unstable.     

Bifurcations of Umbilic Points and Related Principal Cycles

The simplest patterns of qualitative changes on the configurations of lines of principal curvature around umbilic points on surfaces whose immersions into ³ depend smoothly on a real parameter (codimension one umbilic bifurcations) are described in this paper.Global effects, due to umbilic bifurcations, on these configurations such as the appearance and annihilation of periodic principal lines, called also principal cycles, are also studied here.     

Bifurcations and dynamical analysis of Coriolis-stabilized spherical lagging pendula

Nonlinear Dynamics - The dynamics of a spherical pendulum with a horizontally rotating support, exhibiting regimes of intriguing lagging behaviour in the presence of aerodynamic drag, is analysed...     

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.     

Bifurcations of a Thin Plate-Strip Excited Transversally and Axially

The complex vibrations and bifurcations of plates modeled as systemswith infinite degrees-of-freedom are considered. Both theBubnov–Galerkin with high-order approximations and finite differencemethods with approximation O(h ⁴)are applied. In addition, the calculation ofthe Lyapunov exponents of the system is performed, and the results arecompared to those derived by Bennetin's method. Some examples of newnonlinear phenomena exhibited by the considered systems are reported.     

Bifurcations and stability of a discrete singular bioeconomic system

The paper analyzes the stability and bifurcations of a discrete singular bioeconomic system in the closed first quadrant $R_{+}^{3}$ . First, applying the Poincaré scheme to a differential-algebraic predator–prey system where the economic interest of harvesting is taken into account, a discrete singular bioeconomic system is proposed. Then, local stability and the existing conditions of the flip bifurcation and Neimark–Sacker bifurcation around the interior equilibria of the proposed model are discussed by using the normal form of the discrete singular bioeconomic system, the center manifold theorem and the bifurcation theory, when choosing the step size δ as the parameter of the bifurcation. Finally, the results are illustrated and the complex dynamical behaviors are exhibited by computer numerical simulations.