Manifolds Of Different Dimensions Research Articles

Cascades of heterodimensional cycles via period doubling

A heterodimensional cycle is formed by the intersection of stable and unstable manifolds of two saddle periodic orbits that have unstable manifolds of different dimensions: connecting orbits exist from one periodic orbit to the other, and vice versa. The difference in dimensions of the invariant manifolds can only be achieved in vector fields of dimension at least four. At least one of the connecting orbits of the heterodimensional cycle will necessarily be structurally unstable, meaning that is does not persist under small perturbations. Nevertheless, the theory states that the existence of a heterodimensional cycle is generally a robust phenomenon: any sufficiently close vector field (in the C1-topology) also has a heterodimensional cycle.We investigate a particular four-dimensional vector field that is known to have a heterodimensional cycle. We continue this cycle as a codimension-one invariant set in a two-parameter plane. Our investigations make extensive use of advanced numerical methods that prove to be an important tool for uncovering the dynamics and providing insight into the underlying geometric structure. We study changes in the family of connecting orbits as two parameters vary and Floquet multipliers of the periodic orbits in the heterodimensional cycle change. In particular the Floquet multipliers of one of the periodic orbits change from real positive to real negative prior to a period-doubling bifurcation. We then focus on the transitions that occur near this period-doubling bifurcation and find that it generates new families of heterodimensional cycles with different geometric properties. Our careful numerical study suggests that further two-parameter continuation of the ‘period-doubled heterodimensional cycles’ gives rise to an abundance of heterodimensional cycles of different types in the limit of a period-doubling cascade.Our results for this particular example vector field make a contribution to the emerging bifurcation theory of heterodimensional cycles. In particular, the bifurcation scenario we present can be viewed as a specific mechanism behind so-called stabilisation of a heterodimensional cycle via the embedding of one of its constituent periodic orbits into a more complex invariant set.

On the evolution of the hierarchy of shock waves in a two-dimensional isobaric medium

In the proposed paper, the process of propagation of shock waves in two-dimensional media without its own pressure drop is studied. The model of such media is a system of equations of gas dynamics, where formally the pressure is assumed to be zero. From the point of view of the theory of systems of conservation laws, the system of equations under consideration is in some sense degenerate, and, consequently, the corresponding generalized solutions have strong singularities (evolving shock waves with density in the form of delta functions on manifolds of different dimensions). We will denote this property as the evolution of the hierarchy of strong singularities or the evolution of the hierarchy of shock waves. In the paper, in the two-dimensional case, the existence of such an interaction of strong singularities with density delta function along curves in the space $\mathbb{R}^2$ is proved, at which a density concentration occurs at a point, that is, a hierarchy of shock waves arises. The properties of such dynamics of strong singularities are described. The results obtained provide a starting point for moving on to a much more interesting multidimensional case in the future.

Unstable dimension variability and heterodimensional cycles in the border-collision normal form.

Chaotic attractors commonly contain periodic solutions with unstable manifolds of different dimensions. This allows for a zoo of dynamical phenomena not possible for hyperbolic attractors. The purpose of this Letter is to emphasize the existence of these phenomena in the border-collision normal form. This is a continuous, piecewise-linear family of maps that is physically relevant as it captures the dynamics created in border-collision bifurcations in diverse applications. Since the maps are piecewise linear, they are relatively amenable to an exact analysis. We explicitly identify parameter values for heterodimensional cycles and argue that the existence of heterodimensional cycles between two given saddles can be dense in parameter space. We numerically identify key bifurcations associated with unstable dimension variability by studying a one-parameter subfamily that transitions continuously from where periodic solutions are all saddles to where they are all repellers. This is facilitated by fast and accurate computations of periodic solutions; indeed the piecewise-linear form should provide a useful testbed for further study.

On Model Two-Dimensional Pressureless Gas Flows: Variational Description and Numerical Algorithm Based on Adhesion Dynamics

Weak solutions of the system of pressureless gas dynamics equations in two dimensions are studied. Theoretical issues are considered, namely, the general mathematical theory of conservation laws for the system is addressed. Emphasis is placed on an important distinctive feature of this system: the emergence of strong density singularities along manifolds of different dimensions. This property is characterized as the formation of a hierarchy of singularities. In earlier application-oriented works (e.g., by A.N. Krayko, et al., including more complicated cases of 3D two-phase flows), this property was studied at the physical level of rigor. In this paper, the formation of a hierarchy of singularities is examined mathematically, since, for example, the existence of a solution with a strong singularity at a point (in the 2D case) is rather difficult to prove rigorously. Accordingly, a special numerical algorithm is used to develop mathematical hypotheses concerning solution behavior. Approaches to the construction of a variational principle for weak solutions are considered theoretically. A numerical algorithm based on approximate adhesion dynamics in the multidimensional case is implemented. The algorithm is tested on several examples (2D Riemann problem) in terms of internal convergence and is compared with mathematical results, including those obtained by other authors.

Matching geometric and expansion characteristics of wild chaotic attractors

Wild chaotic attractors exhibit chaotic dynamics with a robustness property that cannot be destroyed with small perturbations. We consider a discrete-time system with the smallest possible dimension, namely, defined by a non-invertible map on the complex plane. For this map, wild chaos has been proven to exist in a small parameter region. Recently, it was conjectured to exist in a much larger region of parameter space, past a so-called backward critical tangency, at which a sequence of pre-images of a critical point converges to a saddle fixed point. Geometrically, a backward critical tangency leads to an abundance of homoclinic and heteroclinic tangencies between invariant manifolds of different dimensions, generating precisely what are believed to be the necessary ingredients for wild chaos. In this paper, we present corroborating evidence for this conjecture by computing Lyapunov exponents associated with the attractor. When the sum of the two (largest) Lyapunov exponents is positive, the dynamics is wild chaotic for this non-invertible map. We find that the zero-sum locus matches the locus of backward critical tangency, confirming its role as a boundary of existence of wild chaos.

Determining the global manifold structure of a continuous-time heterodimensional cycle

<p style='text-indent:20px;'>A heterodimensional cycle consists of two saddle periodic orbits with unstable manifolds of different dimensions and a pair of connecting orbits between them. Recent theoretical work on chaotic dynamics beyond the uniformly hyperbolic setting has shown that heterodimensional cycles may occur robustly in diffeomorphisms of dimension at least three. We consider the first explicit example of a heterodimensional cycle in the continuous-time setting, which has been identified by Zhang, Krauskopf and Kirk [<i>Discr. Contin. Dynam. Syst. A</i> <b>32</b>(8) 2825-2851 (2012)] in a four-dimensional vector-field model of intracellular calcium dynamics.</p><p style='text-indent:20px;'>We show here how a boundary-value problem set-up can be employed to determine the organization of the dynamics in a neighborhood in phase space of this heterodimensional cycle, which consists of a single connecting orbit of codimension one and an entire cylinder of structurally stable connecting orbits between two saddle periodic orbits. More specifically, we compute the relevant stable and unstable manifolds, which we visualize in different projections of phase space and as intersection sets with a suitable three-dimensional Poincaré section. In this way, we show that, locally near the intersection set of the heterodimensional cycle, the manifolds interact as described by the theory for three-dimensional diffeomorphisms. On the other hand, their global structure is more intricate, which is due to the fact that it is not possible to find a Poincaré section that is transverse to the flow everywhere. More generally, our results show that advanced numerical continuation techniques enable one to investigate how abstract concepts â€" such as that of a heterodimensional cycle of a diffeomorphism â€" arise and manifest themselves in explicit continuous-time systems from applications.</p>

Simulating sticky particles: A Monte Carlo method to sample a stratification.

Many problems in materials science and biology involve particles interacting with strong, short-ranged bonds that can break and form on experimental timescales. Treating such bonds as constraints can significantly speed up sampling their equilibrium distribution, and there are several methods to sample probability distributions subject to fixed constraints. We introduce a Monte Carlo method to handle the case when constraints can break and form. More generally, the method samples a probability distribution on a stratification: a collection of manifolds of different dimensions, where the lower-dimensional manifolds lie on the boundaries of the higher-dimensional manifolds. We show several applications of the method in polymer physics, self-assembly of colloids, and volume calculation in high dimensions.

Perturbative-Iterative Computation of Inertial Manifolds of Systems of Delay-Differential Equations with Small Delays

Delay-differential equations belong to the class of infinite-dimensional dynamical systems. However, it is often observed that the solutions are rapidly attracted to smooth manifolds embedded in the finite-dimensional state space, called inertial manifolds. The computation of an inertial manifold yields an ordinary differential equation (ODE) model representing the long-term dynamics of the system. Note in particular that any attractors must be embedded in the inertial manifold when one exists, therefore reducing the study of these attractors to the ODE context, for which methods of analysis are well developed. This contribution presents a study of a previously developed method for constructing inertial manifolds based on an expansion of the delayed term in small powers of the delay, and subsequent solution of the invariance equation by the Fraser functional iteration method. The combined perturbative-iterative method is applied to several variations of a model for the expression of an inducible enzyme, where the delay represents the time required to transcribe messenger RNA and to translate that RNA into the protein. It is shown that inertial manifolds of different dimensions can be computed. Qualitatively correct inertial manifolds are obtained. Among other things, the dynamics confined to computed inertial manifolds display Andronov–Hopf bifurcations at similar parameter values as the original DDE model.

Functional analysis and exterior calculus on mixed-dimensional geometries

We are interested in differential forms on mixed-dimensional geometries, in the sense of a domain containing sets of d-dimensional manifolds, structured hierarchically so that each d-dimensional manifold is contained in the boundary of one or more d + 1-dimensional manifolds. On any given d-dimensional manifold, we then consider differential operators tangent to the manifold as well as discrete differential operators (jumps) normal to the manifold. The combined action of these operators leads to the notion of a semi-discrete differential operator coupling manifolds of different dimensions. We refer to the resulting systems of equations as mixed-dimensional, which have become a popular modeling technique for physical applications including fractured and composite materials. We establish analytical tools in the mixed-dimensional setting, including suitable inner products, differential and codifferential operators, Poincaré lemma, and Poincaré–Friedrichs inequality. The manuscript is concluded by defining the mixed-dimensional minimization problem corresponding to the Hodge Laplacian, and we show that this minimization problem is well-posed.

Rolling Manifolds of Different Dimensions

If (M,g) and $(\hat{M},\hat{g})$ are two smooth connected complete oriented Riemannian manifolds of dimensions n and $\hat{n}$ respectively, we model the rolling of (M,g) onto $(\hat{M},\hat{g})$ as a driftless control affine systems describing two possible constraints of motion: the first rolling motion (Σ)NS captures the no-spinning condition only and the second rolling motion (Σ)R corresponds to rolling without spinning nor slipping. Two distributions of dimensions $(n + \hat{n})$ and n are then associated to the rolling motions (Σ)NS and (Σ)R respectively. This generalizes the rolling problems considered in Chitour and Kokkonen (Rolling manifolds and controllability: the 3D case, 2012) where both manifolds had the same dimension. The controllability issue is then addressed for both (Σ)NS and (Σ)R and completely solved for (Σ)NS. As regards to (Σ)R, basic properties for the reachable sets are provided as well as the complete study of the case $(n,\hat{n})=(3,2)$ and some sufficient conditions for non-controllability.

Semi-Classical Wavefront Set and Fourier Integral Operators

AbstractHere we define and prove some properties of the semi-classical wavefront set. We also define and study semi-classical Fourier integral operators and prove a generalization of Egorov’s theorem to manifolds of different dimensions.

Elastic Principal Graphs and Manifolds and their Practical Applications

Principal manifolds serve as useful tool for many practical applications. These manifolds are defined as lines or surfaces passing through "the middle" of data distribution. We propose an algorithm for fast construction of grid approximations of principal manifolds with given topology. It is based on analogy of principal manifold and elastic membrane. The first advantage of this method is a form of the functional to be minimized which becomes quadratic at the step of the vertices position refinement. This makes the algorithm very effective, especially for parallel implementations. Another advantage is that the same algorithmic kernel is applied to construct principal manifolds of different dimensions and topologies. We demonstrate how flexibility of the approach allows numerous adaptive strategies like principal graph constructing, etc. The algorithm is implemented as a C++ package elmap and as a part of stand-alone data visualization tool VidaExpert, available on the web. We describe the approach and provide several examples of its application with speed performance characteristics.

Second-Order Elliptic Equations on a Stratified Set

A series of problems in mathematical physics connected with complex systems consisting of elements having different dimensionalities or different physical characteristics are conveniently modeled by boundary-value problems on stratified sets Ω. The approaches to equations on graphs described in other articles in this issue also allow development in this case. The geometry of a stratified set is notably more complex than the geometry of a graph (Ω is now a connected union of a finite number of manifolds of different dimensions), but, on the whole, the methodological principles developed for equations on graphs can be realized. Among these principles, the interpretation of all differential relations associated with the constituent elements of the set as a single equation on Ω of the divergent type is fundamental. Divergence, as in the classical case, is the density of the current of a vector field on Ω with respect to a special “stratified” measure. But to obtain substantial results, it is necessary to isolate a special class of so-called solid stratified sets. The nature of the results obtained is determined by the type of solidity of Ω. Two types of solidity of Ω are described in this paper. In connection with this, it is divided into two parts.

Removing coincidences of maps between manifolds of different dimensions

We consider sufficient conditions of local removability of coincidences of maps $f,g\colon N\rightarrow M$, where $M$, $N$ are manifolds with dimensions $\dim N\geq\dim M$. The coincidence index is the only obstruction to the removability for maps with fibers either acyclic or homeomorphic to spheres of certain dimensions. We also address the normalization property of the index and coincidence-producing maps.

Sobolev problem in the complete scale of Banach Spaces

In a bounded domainG ⊂ ℝ n , whose boundary is the union of manifolds of different dimensions, we study the Sobolev problem for a properly elliptic expression of order 2m. The boundary conditions are given by linear differential expressions on manifolds of different dimensions. We study the Sobolev problem in the complete scale of Banach spaces. For this problem, we prove the theorem on a complete set of isomorphisms and indicate its applications.

The dirichlet problem in a domain whose boundary is partly degenerated

In this paper we bring a formulation of the Dirichlet problem for strongly elliptic equations in domains vhose boundaries may include manifolds of different dimensions. It is shown that, under certain regularity conditions, this problem is equivalent to the generalized Dirichlet problem, with respect to existence and uniquennes of solutions.