On the stability of periodic trajectories of a planar Birkhoff billiard

The inertial motion of a material point is analyzed in a plane domain bounded by two curves that are coaxial segments of an ellipse. The collisions of the point with the boundary curves are assumed to be absolutely elastic. There exists a periodic motion of the point that is described by a two-link trajectory lying on a straight line segment passed twice within the period. This segment is orthogonal to both boundary curves at its endpoints. The nonlinear problem of stability of this trajectory is analyzed. The stability and instability conditions are obtained for almost all values of two dimensionless parameters of the problem.     

Efficient invariant-manifold, low-thrust planar trajectories to the Moon

Two-impulse trajectories as well as mixed invariant-manifold, low-thrust efficient transfers to the Moon are discussed. Exterior trajectories executing ballistic lunar capture are formalized through the definition of special attainable sets. The coupled restricted three-body problems approximation is used to design appropriate first guesses for the subsequent optimization. The introduction of the Moon-perturbed Sun-Earth restricted four-body problem allows us to formalize the idea of ballistic escape from the Earth, and to take explicitly advantage of lunar fly-by. Accurate first guess solutions are optimized, through a direct method approach and multiple shooting technique.     

Fractal analysis of spiral trajectories of some planar vector fields

We study box dimension, Minkowski content and Minkowski measurability of nonrectifiable, smooth spiral trajectories of some dynamical systems in the plane. From this point of view we consider a standard model of Hopf-Takens bifurcation and study the behaviour of trajectories near singular points and limit cycles.     

On the existence of overtaking trajectories

A study is made of the conditions sufficient for the existence of overtaking trajectories for a class of infinite time-horizon, time-variant optimal control systems. Nonautonomy is restricted to disturbances with limits at infinity. The convergence property of the overtaking trajectories to the optimal steady-state limit is shown.     

Dispersion of Lagrangian trajectories in a random large-scale velocity field

We study the distribution of the distance R(t) between two Lagrangian trajectories in a spatially smooth turbulent velocity field with an arbitrary correlation time and a non-Gaussian distribution. There are two dimensionless parameters, the degree of deviation from the Gaussian distribution α and β=τD, where τ is the velocity correlation time and D is a characteristic velocity gradient. Asymptotically, R(t) has a lognormal distribution characterized by the mean runaway velocity and the dispersion Δ. We use the method of higher space dimensions d to estimate and Δ for different values of α and β. It was shown previously that for β≪ 1 and for β≫ 1. The estimate of Δ is then nonuniversal and depends on details of the two-point velocity correlator. Higher-order velocity correlators give an additional contribution to Δ estimated as αD²τ for β≪1 and αβ/τ for β≫1. For α above some critical value σ_cr, the values of and Δ are determined by higher irreducible correlators of the velocity gradient, and our approach loses its applicability. This critical value can be estimated as for β≪1 and for β≫1. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 3, pp. 456–467, March, 2000.     

On the parity of planar covers

A covering is a graph map ?: G → H that is an isomorphism when restricted to the star of any vertex of G. If H is connected then |?^?1(v)| is constant. This constant is called the fold number. In this paper we prove that if G is a planar graph that covers a nonplanar H, then the fold number must be even.     

On orthogonal trajectories and optimization

This note discusses a generalization of the trapezoidal method for numerical integration of the differential equations of the orthogonal trajectories of a function as a means of finding maximum or minimum values of the function. Connections between the choice of a parameter in the integration scheme and various modifications of Newton's method are indicated.     

Smoothness of the trajectories of ideal fluid particles with Yudovich vorticities in a planar bounded domain

We consider the incompressible Euler equations in a (possibly multiply connected) bounded domain Ω of R², for flows with bounded vorticity, for which Yudovich (1963) proved in [29] global existence and uniqueness of the solution. We prove that if the boundary ∂Ω of the domain is C^∞ (respectively Gevrey of order M?1) then the trajectories of the fluid particles are C^∞ (respectively Gevrey of order M+2). Our results also cover the case of “slightly unbounded” vorticities for which Yudovich (1995) extended his analysis in [30]. Moreover if in addition the initial vorticity is Hölder continuous on a part of Ω then this Hölder regularity propagates smoothly along the flow lines. Finally we observe that if the vorticity is constant in a neighborhood of the boundary, the smoothness of the boundary is not necessary for these results to hold.     

On the planar integer two-flow problem

We consider the two-commodity flow problem and give a good characterization of the optimum flow if the augmented network (with both source-sink edges added) is planar. We show that max flow ≧ min cut −1, and describe the structure of those networks for which equality holds.     

On the planar integrable differential systems

In this paper, we prove that the C ¹ planar differential systems that are integrable and non-Hamiltonian roughly speaking are C ¹ equivalent to the linear differential systems ${\dot u= u}$ , ${\dot v= v}$ . Additionally, we show that these systems have always a Lie symmetry. These results are improved for the class of polynomial differential systems defined in ${\mathbb{R}^2}$ or ${\mathbb{C}^2}$ .     

On the depth of a planar graph

In this paper, we have defined the concept of the depth of a planar graph. We show that, if G is a simple finite planar graph with p vertices and q edges and q > 3(p ? 1) ? p/2^s?1, then the depth of G is at least equal to s.     

On the connectivity of maximal planar graphs

Let d₁ ? d₂ ? … ? d_p be the vertex degrees of a maximal planar graph G. Etourneau has shown that if d₁ ? 6 and d_p = 5, then G is 5-connected. We generalize Etourneau's result by giving sufficient conditions in terms of the vertex degrees for G to be d_p -connected.     

On the linear arboricity of planar graphs

The linear arboricity la(G) of a graph G is the minimum number of linear forests that partition the edges of G. Akiyama, Exoo, and Harary conjectured that for any simple graph G with maximum degree Δ. The conjecture has been proved to be true for graphs having Δ = 1, 2, 3, 4, 5, 6, 8, 10. Combining these results, we prove in the article that the conjecture is true for planar graphs having Δ(G) ≠ 7. Several related results assuming some conditions on the girth are obtained as well. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 129–134, 1999     

On the toughness index of planar graphs

The toughness indexτ(G) of a graph G is defined to be the largest integer t such that for any S ? V(G) with |S| > t, c(G - S) < |S| - t, where c(G - S) denotes the number of components of G - S. In particular, 1-tough graphs are exactly those graphs for which τ(G) ≥ 0. In this paper, it is shown that if G is a planar graph, then τ(G) ≥ 2 if and only if G is 4-connected. This result suggests that there may be a polynomial-time algorithm for determining whether a planar graph is 1-tough, even though the problem for general graphs is NP-hard. The result can be restated as follows: a planar graph is 4-connected if and only if it remains 1-tough whenever two vertices are removed. Hence it establishes a weakened version of a conjecture, due to M. D. Plummer, that removing 2 vertices from a 4-connected planar graph yields a Hamiltonian graph.