Transversal Intersection Research Articles

Advances on similarity via transversal intersection of manifolds

Let A be an n×n real matrix. As shown in the recent paper S.M. Fallat, H.T. Hall, J.C.-H. Lin, and B.L. Shader (2022) [5], if the manifolds MA={G−1AG:G∈GL(n,R)} and Q(sgn(A)) (consisting of all real matrices having the same sign pattern as A), both considered as embedded submanifolds of Rn×n, intersect transversally at A, then every superpattern of sgn(A) also allows a matrix similar to A. Those authors introduced a condition on A (in terms of certain linear matrix equations) equivalent to the above transversality, called the nonsymmetric strong spectral property (nSSP). In this paper, this transversality property of A is characterized using an alternative, more direct and convenient condition, called the similarity-transversality property (STP). Let X=[xij] be a generic matrix of order n whose entries are independent variables. The STP of A is defined as the full row rank property of the Jacobian matrix of the entries of AX−XA at the zero entry positions of A with respect to the nondiagonal entries of X. This new approach makes it possible to take better advantage of the combinatorial structure of the matrix A, and provides theoretical foundation for constructing matrices similar to a given matrix while the entries have certain desired signs. In particular, several important classes of zero-nonzero patterns and sign patterns that require or allow this transversality property are identified. Examples illustrating many possible applications (such as diagonalizability, number of distinct eigenvalues, nilpotence, idempotence, semi-stability, eigenvalues and their algebraic and geometric multiplicities, Jordan canonical form, minimal polynomial, and rank) are provided.

The Plykin and Solenoid attractor are homoclinic classes

A homoclinic class is the closure of the transverse intersection points of the stable and unstable manifolds of a hyperbolic periodic orbit. In this paper, we prove, using the techniques presented in [1], that the Plykin and the Solenoid attractors are a homoclinic class.

Application of a meshed artificial dermal scaffold and negative-pressure wound therapy in the treatment of full-thickness skin defects: a prospective in vivo study.

Artificial dermal scaffolds (ADSs) have great value in repairing deep skin defects. However, problems such as unsatisfactory angiogenesis and local dropsy or empyema often occur, resulting in delayed or even failed wound healing. Negative pressure wound therapy (NPWT) is an effective therapy to promote wound healing or shorten wound bed preparation time. Studies on whether it can improve the effects of ADSs have never been interrupted, and no consensus has been reached. In this study, an improved ADS was prepared by mesh technology, physicochemical experiments were conducted, cell adhesion and proliferation were assessed with the meshed ADS, and in vivo experiments were conducted to investigate the effects of meshed ADS or ADS combined with NPWT in repairing full-thickness skin defects. The results showed that the meshed ADS showed through-layer channels arranged in parallel longitudinal and transverse intersections. The cell experiments confirmed the good cytocompatibility. The in vivo experiments showed that there were no differences in the take rate or contraction of grafted skin among all experiment groups. The meshed ADS exhibited good histocompatibility, and there were no differences in tissue inflammation, dermal angiogenesis, or degradation among all groups. In addition, necrosis, dropsy, or empyema of the dermal scaffold were found in all experiment groups except for the meshed ADS + NPWT group, which showed better wound repair results, including fewer scaffold-related complications and satisfactory skin graft survival and wound contraction. In conclusion, this novel meshed ADS, which has a regular through-layer mesh structure and possesses stable physicochemical properties and good biocompatibility, combined with NPWT can ensure adequate subdermal drainage and reduce the risk of scaffold-related complications, thereby improving the quality and efficiency of wound repair, promoting a broader application of biomaterials, and helping physicians and readers implement more effective wound management.

Implicit-implicit and parametric-implicit surface intersections in Euclidean n-space

In this paper, we study the intersection problem of two surfaces in which at least one surface is given by its implicit equation in Euclidean n-space. We first obtain the curvatures of the transversal intersection curve of two implicit surfaces. Later, the intersection problem is studied for a parametric surface and an implicit surface in n-dimensional Euclidean space. Finally, we present examples as applications of the given methods.

Proof of the transverse intersection of the stable and unstable manifolds in the two-dimensional area preserving map

Abstract In this paper, we study the area preserving map T included in the Hénon family of maps. The map T defined on the plane is characterized by a parameter a (≥ 0). There exist the saddle fixed point P and the elliptic point Q at a > 0. Let u be the primary homoclinic point at which the unstable manifold Wu starting from the saddle fixed point P intersects the symmetry axis denoted by the dominant axis of Q. The stable manifold Ws also intersects Wu at u. At u, let ξu(u) be the inclination of Wu and ξs(u) be that of Ws. Using a new geometric method, we prove the transversality of the intersection for any value of a (> 0). Here, the transversality means that ξu(u) < ξs(u) holds.

Local terms for transversal intersections

The goal of this note is to show that in the case of ‘transversal intersections’ the ‘true local terms’ appearing in the Lefschetz trace formula are equal to the ‘naive local terms’. To prove the result, we extend the strategy used in our previous work, where the case of contracting correspondences is treated. Our new ingredients are the observation of Verdier that specialization of an étale sheaf to the normal cone is monodromic and the assertion that local terms are ‘constant in families’. As an application, we get a generalization of the Deligne–Lusztig trace formula.

Elementary proof for the existence of periodic points with real and simple spectrum for diffeomorphisms in any dimension

Using only elementary techniques we prove that for any $ C^r $ diffeomorphism, $ f $, of a compact manifold of dimension $ d>2 $, $ 1\leq r\leq \infty $, admitting a transverse homoclinic intersection, we can find a $ C^1 $-open neighborhood of $ f $ containing a $ C^1 $-open and $ C^r $-dense set of $ C^r $ diffeomorphisms which have a periodic point with real and simple spectrum. We use this fact to prove that $ C^r $-generically among $ C^r $ diffeomorphisms with horseshoes, we have density of periodic points with real and simple spectrum inside the horseshoe. As a corollary, we obtain that generically in the $ C^1 $-topology the unique obstruction to the existence of periodic points with real and simple spectrum are the Morse-Smale diffeomorphisms with all the periodic points admitting non-real eigenvalues.

Nonlocal chaotic analysis of double layered viscoelastic nanoplates embedded in Winkler–Pasternak elastic foundation and under in‐plane biaxial loads

AbstractIn this paper, based on the extended Melnikov method the homoclinic analysis for chaotic vibration of a double layered viscoelastic nanoplates (DLNPs) embedded in Winkler–Pasternak elastic foundation and under in‐plane excited biaxial loads using the Eringen's nonlocal elasticity theory has been investigated. The governing nonlinear coupled partial differential equations (PDEs) of motion for DLNP considering viscoelastic Kelvin–Voigt model in the Kirchhoff's thin plate theory in conjunction with the von‐Karman geometrical nonlinear strain–displacement relations have been derived. Then, applying the Galerkin method the governing nonlocal coupled PDEs of motion of DLNP have been converted into the nonlinear coupled ordinary differential equations (ODEs) using mode summation technique and discretized in terms of time. The criteria for the homoclinic transverse intersection for synchronous and asynchronous types of buckling are studied to predict the homoclinic orbits and chaotic motions applying the generalized Melnikov method. Besides, the critical buckling load has been determined for asynchronous and synchronous types of buckling. The obtained results are validated with comparing them with the results presented in the literature for a special case. Then, the effect of changes of different parameters including nonlocal small‐scale parameter, foundation coefficients, biaxial load ratio, geometric aspect ratio on the critical buckling load and ratio of nonlocal to local natural frequency have been studied. The critical threshold surface for the synchronous and asynchronous manifolds to predict transversely condition for the chaotic motion are discussed on the parametric space.

Curve intersection based on cubic hybrid clipping

This study presents a novel approach to computing all intersections between two Bézier curves using cubic hybrid clipping. Each intersection is represented by two strip intervals that contain an intersection. In each step, one curve is bounded by two fat lines, and the other is bounded by two cubic Bézier curves, clipping away the domain that does not contain the intersections. By selecting the moving control points of the cubic hybrid curves, better cubic polynomial bounds are obtained to make the proposed method more efficient. It was proved that the two strip intervals have second- and fourth-order convergence rates for transversal intersections. Experimental results show that the new algorithm is the most efficient among all existing curve/curve intersection approaches.

Characterizing chaos in systems subjected to parameter drift.

To characterize chaos in systems subjected to parameter drift, where a number of traditional methods do not apply, we propose viable alternative approaches, both in the qualitative and quantitative sense. Qualitatively, following stable and unstable foliations is shown to be efficient, which are easy to approximate numerically, without relying on the need for the existence of an analog of hyperbolic periodic orbits. Chaos originates from a Smale horseshoe-like pattern of the foliations, the transverse intersections of which indicate a chaotic set changing in time. In dissipative cases, the unstable foliation is found to be part of the so-called snapshot attractor, but the chaotic set is not dense on it if regular time-dependent attractors also exist. In Hamiltonian cases stable and unstable foliations turn out to be not equivalent due to the lack of time-reversal symmetry. It is the unstable foliation, which is found to correlate with the so-called snapshot chaotic sea. The chaotic set appears to be locally dense in this sea, while tori with originally quasiperiodic character might break up, their motion becoming chaotic as time goes on. A quantity called ensemble-averaged pairwise distance evaluated in relation to unstable foliations is shown to be an appropriate tool to provide the instantaneous strength of time-dependent chaos.

W-translated Schubert divisors and transversal intersections

We study the toric degeneration of Weyl group translated Schubert divisors of a partial flag variety \(F{\ell _{{n_1}, \ldots,{n_k};n}}\) via Gelfand-Cetlin polytopes. We propose a conjecture that Schubert varieties of appropriate dimensions intersect transversally up to translation by Weyl group elements, and verify it in various cases, including the complex Grassmannian Gr(2, n) and the complete flag variety Fℓ1,2,3,4.

Computer assisted proof of drift orbits along normally hyperbolic manifolds II: Application to the restricted three body problem

We present a computer assisted proof of diffusion in the Planar Elliptic Restricted Three Body Problem. We treat the elliptic problem as a perturbation of the circular problem, where the perturbation parameter is the eccentricity of the primaries. The unperturbed system preserves energy, and we show that for sufficiently small perturbations we have orbits with explicit energy changes, independent from the size of the perturbation. The result is based on shadowing of orbits along transversal intersections of stable/unstable manifolds of a normally hyperbolic cylinder.

Kac-Rice formula for transverse intersections

We prove a generalized Kac-Rice formula that, in a well defined regular setting, computes the expected cardinality of the preimage of a submanifold via a random map by expressing it as the integral of a density. Our proof starts from scratch and although it follows the guidelines of the standard proofs of Kac-Rice formula, it contains some new ideas coming from the point of view of measure theory. Generalizing further, we extend this formula to any other type of counting measure, such as the intersection degree. We discuss in depth the specialization to smooth Gaussian random sections of a vector bundle. Here, the formula computes the expected number of points where the section meets a given submanifold of the total space, it holds under natural non-degeneracy conditions and can be simplified by using appropriate connections. Moreover, we point out a class of submanifolds, that we call sub-Gaussian, for which the formula is locally finite and depends continuously with respect to the covariance of the first jet. In particular, this applies to any notion of singularity of sections that can be defined as the set of points where the jet prolongation meets a given semialgebraic submanifold of the jet space. Various examples of applications and special cases are discussed. In particular, we report a new proof of the Poincaré kinematic formula for homogeneous spaces and we observe how the formula simplifies for isotropic Gaussian fields on the sphere.

Reaction forces of singular pendulum

Various behavior types of reaction forces and Lagrange multipliers for the case of mechanical systems with configuration space singularities are studied in this paper. The motion of a one-dimensional double pendulum (or a singular pendulum) with a transversal singular point or a first order tangency singular point is considered. Properties of the configuration space of singular pendulum depends on the constraint line which the free vertex of the double pendulum moves along. Configuration space of singular pendulum could be represented by two smooth curves on a torus without common points, two transversely intersecting smooth curves or two curves with first-order tangency. In order to study the pendulum motion, the reaction forces on a two-dimensional torus are found. The expressions for the reaction forces are obtained analytically in angular coordinates. In the case of a transverse intersection it is proved that the reaction forces must be zero at the singular point. In the case of a first-order tangency singularity, the reaction forces are nonzero at the singular point. The Lagrange multiplier which depends on the motion along the ellipse becomes unlimited near to the singular point. Two mechanisms with a different type of singular points in the configuration space are described: a nonsmooth singular pendulum and a broken singular pendulum. There are no smooth regular curves passing through a singular point in the configuration spaces of these mechanical systems. In the case of nonsmooth singular pendulum, the Lagrange multiplier which depends on the motion along the ellipse becomes undefined when the singular point is passed. In the case of broken singular pendulum, the Lagrange multiplier makes a jump from a finite value to an infinite one.

Differential geometry properties by using the perturbation methods

In this paper, we present a new method to evaluate the moving Frenet frame along the intersection curves of two parametric surfaces in the tangential intersection situations. To resolve such situation, we have combined a perturbation method with a classical method that works in transversal intersection situation. Unlike the existent methods, our method works even if the order contact of the point is more than one.

Families of similar simplices inscribed in most smoothly embedded spheres

Abstract Let $\Delta $ denote a nondegenerate k-simplex in $\mathbb {R}^k$ . The set $\operatorname {\mathrm {Sim}}(\Delta )$ of simplices in $\mathbb {R}^k$ similar to $\Delta $ is diffeomorphic to $\operatorname {O}(k)\times [0,\infty )\times \mathbb {R}^k$ , where the factor in $\operatorname {O}(k)$ is a matrix called the pose. Among $(k-1)$ -spheres smoothly embedded in $\mathbb {R}^k$ and isotopic to the identity, there is a dense family of spheres, for which the subset of $\operatorname {\mathrm {Sim}}(\Delta )$ of simplices inscribed in each embedded sphere contains a similar simplex of every pose $U\in \operatorname {O}(k)$ . Further, the intersection of $\operatorname {\mathrm {Sim}}(\Delta )$ with the configuration space of $k+1$ distinct points on an embedded sphere is a manifold whose top homology class maps to the top class in $\operatorname {O}(k)$ via the pose map. This gives a high-dimensional generalisation of classical results on inscribing families of triangles in plane curves. We use techniques established in our previous paper on the square-peg problem where we viewed inscribed simplices in spheres as transverse intersections of submanifolds of compactified configuration spaces.

Rank-one strange attractors <i>versus</i> heteroclinic tangles

<p style='text-indent:20px;'>We present two different mechanisms for the emergence of strange attractors (observable chaos) in a two-parameter periodically-perturbed family of differential equations on the plane. The two parameters are independent and act on different ways in the invariant manifolds of consecutive saddles in the cycle. When both parameters are zero, the flow exhibits an attracting heteroclinic cycle associated to two equilibria. The first parameter makes the two-dimensional invariant manifolds of consecutive saddles in the cycle to pull apart; the second forces transverse intersection. The unfolding of each parameter is associated to the emergence of different dynamical scenarios (rank-one attractors and heteroclinic tangles). These relative positions may be determined using the Melnikov method.</p><p style='text-indent:20px;'>Extending the previous theory on the field, we prove the existence of many complicated dynamical objects in the two-parameter family, ranging from "lar-ge" strange attractors supporting SRB (Sinai-Ruelle-Bowen) measures to superstable sinks and non-uniformly hyperbolic attractors. We draw a plausible bifurcation diagram associated to the problem under consideration and we show that the occurrence of heteroclinic tangles is a <i>prevalent</i> phenomenon.</p>

Chains in 3D Filippov systems: A chaotic phenomenon

This work is devoted to the study of global connections between typical generic singularities, named T-singularities, in piecewise smooth dynamical systems. Such a singularity presents the so-called nonsmooth diabolo, which consists on a pair of invariant cones emanating from it.We analyze global features arising from the communication between the branches of a nonsmooth diabolo of a T-singularity and we prove that, under generic conditions, such communication leads to a chaotic behavior of the system. More specifically, we relate crossing orbits of a Filippov system presenting certain crossing self-connections to a T-singularity, with a Smale horseshoe of a first return map associated to the system. The techniques used in this work rely on the detection of transverse intersections between invariant manifolds of a hyperbolic fixed point of saddle type of such a first return map and the analysis of the Smale horseshoe associated to it.From the specific case discussed in our approach, we present a robust chaotic phenomenon for which its counterpart in the smooth case seems to happen only for highly degenerate systems.

Intersection curve of two parametric surfaces in Euclidean n-space

The aim of this paper is to study the differential geometric properties of the intersection curve of two parametric surfaces in Euclidean n-space. For this aim, we first present the mth order derivative formula of a curve lying on a parametric surface. Then, we obtain curvatures and Frenet vectors of the transversal intersection curve of two parametric surfaces in Euclidean n-space. We also provide computer code produced in MATLAB to simplify determining the coefficients relative to Frenet frame of higher order derivatives of a curve.

Control of chaotic behavior in the dynamics of generalized Bonhoeffer-van der Pol system: Effect of asymmetric parameter

The effect of the asymmetric parameter in the dynamics, both of the forced Duffing van der Pol (DVdP) system and of the forced generalized Bonhoeffer-van der Pol (BVdP) system is investigated, namely the possibility for these systems to execute dynamics chaos even for large values of this asymmetric parameter. We have shown that the dynamics of the associated non dissipative and unforced system can be interpreted by means of an effective energy potential which may exhibit a two-hump or a two-well configuration according to the magnitude of the parameters of the system. In the two-well configuration, a pair of pulses of different amplitudes describing homoclinic orbits useful in the prediction of irregular dynamics for the associated forced system are analytically derived as well as the expression of the single pulse describing the single homoclinic orbit in the case of two-hump configuration and which leads to a kink-antikink pair (heteroclinic orbits) in the absence of the asymmetric parameter. By means of the Melnikov theory, the conditions for the existence of the transverse intersection of stable and unstable orbits or dynamics chaos are also derived. In the particular case for which the system can exhibit a pair of homoclinic orbits, the presence of the asymmetric parameter induces two types of domains among which a more favorable domain for the existence of chaotic behavior. In both cases, the asymmetric parameter reduces the domain range in which the system can execute regular dynamics and consequently increases its domain of a possible dynamics chaos. The accuracy of these analytical results is checked through the bifurcation diagrams and the corresponding Lyapunov exponents resulting from numerical simulations. It appears that the background properties of the unforced system, namely its capacity to exhibit autonomous oscillations and the stability of its equilibrium points which are closely connected to the magnitude of the asymmetric parameter, influence the accuracy of the analytical predictions.