Invariant Orbits Research Articles

Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups

We study the relation between two special classes of Riemannian Lie groups [Formula: see text] with a left-invariant metric [Formula: see text]: The Einstein Lie groups, defined by the condition [Formula: see text], and the geodesic orbit Lie groups, defined by the property that any geodesic is the integral curve of a Killing vector field. The main results imply that extensive classes of compact simple Einstein Lie groups [Formula: see text] are not geodesic orbit manifolds, thus providing large-scale answers to a relevant question of Nikonorov. Our approach involves studying and characterizing the [Formula: see text]-invariant geodesic orbit metrics on Lie groups [Formula: see text] for a wide class of subgroups [Formula: see text] that we call (weakly) regular. By-products of our work are structural and characterization results that are of independent interest for the classification problem of geodesic orbit manifolds.

Symplectic splitting of Hamiltonian structures and reduced monodromy matrices

In Hamiltonian systems monodromy and reduced monodromy matrices of periodic orbits are symplectic. A symmetry is a symplectic or anti-symplectic involution which leaves the Hamiltonian invariant. A natural class to study are periodic orbits which are invariant under such involutions. Depending on whether the orbit is invariant under a symplectic or anti-symplectic symmetry, its monodromy matrix satisfies special properties. In the symplectic case, the monodromy matrix admits a symplectic decomposition, depending on the symplectic involution's fixed point set. In the anti-symplectic case, the monodromy matrix is of special symmetric form, which was constructed by Frauenfelder–Moreno. Such monodromy matrices are very beneficial in applied problems. The goal of this paper is to design a general construction behind the reduced monodromy's symplectic splitting, by using the notion of Hamiltonian manifolds, and to give a general framework for monodromy and reduced monodromy matrices of invariant periodic orbits in Hamiltonian systems with symmetries. In addition, we show an application related to contact forms, especially to the spatial circular restricted three-body problem for energy level sets of contact type.

A Posteriori Validation of Generalized Polynomial Chaos Expansions

Generalized polynomial chaos (gPC) expansions are a powerful tool for studying differential equations with random coefficients, allowing, in particular, one to efficiently approximate random invariant sets associated with such equations. In this work, we use ideas from validated numerics in order to obtain rigorous a posteriori error estimates together with existence results about gPC expansions of random invariant sets. This approach also provides a new framework for conducting validated continuation, i.e., for rigorously computing isolated branches of solutions in parameter-dependent systems, which generalizes in a straightforward way to multiparameter continuation. We illustrate the proposed methodology by rigorously computing random invariant periodic orbits in the Lorenz system, as well as branches and 2 dimensional manifolds of steady states of the Swift–Hohenberg equation.

Horospherical two-orbit varieties as zero loci

We present geometric realizations of horospherical two-orbit varieties, by showing that their blow-up along the unique closed invariant orbit is the zero locus of a general section of a homogeneous vector bundle over some auxiliary variety. As an application, we compute the cohomology ring of the G 2 G_2 -variety, including its quantum version. We also consider the S p i n 7 Spin_7 -variety, which deserves a different treatment.

Orbit design and control for periodically revisiting multiple formation satellites

In the formation flying mission, the periodic revisit of multiple formation satellites can provide periodic surveillance of satellite status, e.g., on-orbit inspection. This paper studies the orbit design problem and the single-impulse control problem for periodically revisiting multiple formation satellites. Two periodic revisit conditions under the J2 perturbation are considered: the J2 invariant orbit and the same mean semi-major axis. By analyzing the numbers of free variables and constraint equations, the maximum numbers of revisiting satellites are obtained for both the coplanar and non-coplanar cases. A hybrid method is proposed by combining the state transition matrix and the second-order state transition tensor, which are used to solve the required velocity vector and to propagate the relative state vector, respectively. The original multi-dimensional nonlinear equation is reduced into a one- or two-dimensional equation, which is solved by numerical methods. Several numerical examples show that the revisit orbit drifts after 30 revolutions by the proposed method for the coplanar and non-coplanar cases are less than 0.3 km and 4.5 km, respectively.

Dynamical complexities and chaos control in a Ricker type predator-prey model with additive Allee effect

<abstract><p>This work investigates the dynamic complications of the Ricker type predator-prey model in the presence of the additive type Allee effect in the prey population. In the modeling of discrete-time models, Euler forward approximations and piecewise constant arguments are the most frequently used schemes. In Euler forward approximations, the model may undergo period-doubled orbits and invariant circle orbits, even while varying the step size. In this way, differential equations with piecewise constant arguments (Ricker-type models) are a better choice for the discretization of a continuous-time model because they do not involve any step size. First, the interaction between prey and predator in the form of the Holling-Ⅱ type is considered. The essential mathematical features are discussed in terms of local stability and the bifurcation phenomenon as well. Next, we apply the center manifold theorem and normal form theory to achieve the existence and directions of flip bifurcation and Neimark-Sacker bifurcation. Moreover, this paper demonstrates that the outbreak of chaos can stabilize in the considered model with a higher value of the Allee parameter. The existence of chaotic orbits is verified with the help of a one-parameter bifurcation diagram and the largest Lyapunov exponents, respectively. Furthermore, different control methods are applied to control the bifurcation and fluctuating phenomena, i.e., state feedback, the Ott-Grebogi-Yorke, and hybrid control methods. Finally, to ensure our analytical results, numerical simulations have been carried out using MATLAB software.</p></abstract>

Derived equivalence classification of Brauer graph algebras

We classify Brauer graph algebras up to derived equivalence by showing that the set of derived invariants introduced by Antipov is complete. These algebras first appeared in representation theory of finite groups and can be defined for any suitably decorated graph on an oriented surface. Motivated by the connection between Brauer graph algebras and gentle algebras we consider A∞-trivial extensions of partially wrapped Fukaya categories associated to surfaces with boundary. This construction naturally enlarges the class of Brauer graph algebras and provides a way to construct derived equivalences between Brauer graph algebras with the same derived invariants. As part of the proof we provide an interpretation of derived invariants of Brauer graph algebras as orbit invariants of line fields under the action of the mapping class group.

Detecting symmetries with neural networks

Identifying symmetries in data sets is generally difficult, but knowledge about them is crucial for efficient data handling. Here we present a method how neural networks can be used to identify symmetries. We make extensive use of the structure in the embedding layer of the neural network which allows us to identify whether a symmetry is present and to identify orbits of the symmetry in the input. To determine which continuous or discrete symmetry group is present we analyse the invariant orbits in the input. We present examples based on rotation groups SO(n) and the unitary group SU(2). Further we find that this method is useful for the classification of complete intersection Calabi-Yau manifolds where it is crucial to identify discrete symmetries on the input space. For this example we present a novel data representation in terms of graphs.

On a class of geodesic orbit spaces with abelian isotropy subgroup

Riemannian geodesic orbit spaces (G/H, g) are natural generalizations of symmetric spaces, defined by the property that their geodesics are orbits of one-parameter subgroups of G. We study the geodesic orbit spaces of the form (G/S, g), where G is a compact, connected, semisimple Lie group and S is abelian. We give a simple geometric characterization of those spaces, namely that they are naturally reductive. In turn, this yields the classification of the invariant geodesic orbit (and also the naturally reductive) metrics on any space of the form G/S. Our approach involves simplifying the intricate parameter space of geodesic orbit metrics on G/S by reducing their study to certain submanifolds and generalized flag manifolds, and by studying properties of root systems of simple Lie algebras associated to these manifolds.

Complete and computable orbit invariants in the geometry of the affine group over the integers

The subject matter of this paper is the geometry of the affine group over the integers, $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ . Turing-computable complete $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -orbit invariants are constructed for rational affine spaces, angles, segments, triangles and ellipses. In rational affine $${\mathsf {GL}}(n,{\mathbb {Q}})\ltimes {\mathbb {Q}}^n$$ -geometry, ellipses are classified by the Clifford–Hasse–Witt invariant, via the Hasse–Minkowski theorem. We classify ellipses in $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -geometry combining results by Apollonius of Perga and Pappus of Alexandria with the Hirzebruch–Jung continued fraction algorithm. We then consider rational polyhedra, i.e., finite unions of simplexes in $${\mathbb {R}}^n$$ with rational vertices. Markov’s unrecognizability theorem for combinatorial manifolds states the undecidability of the problem whether two rational polyhedra P and $$P'$$ are continuously $${\mathsf {GL}}(n,{\mathbb {Q}})\ltimes {\mathbb {Q}}^n$$ -equidissectable. The same problem for the continuous $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -equidissectability of P and $$P'$$ is open. We prove the decidability of the problem whether two rational polyhedra P and $$P'$$ in $${\mathbb {R}}^n$$ have the same $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -orbit.

Answering two open problems on Banks theorem for non-autonomous dynamical systems

ABSTRACTThis paper shows that (1) there exists a topologically transitive NADS having two disjoint invariant periodic orbits with dense periodic points, which is finitely generated but not periodic; (2) there exists a topologically transitive non-finitely generated NADS having two disjoint invariant periodic orbits with dense periodic points, which is not sensitive. This answers positively the Open Problems 4.1 and 4.2 posed in Yang and Li [A remark on Banks theorem, J. Difference Equ. Appl. 24 (2018), pp. 1777–1782].

Linking low- to high-energy dynamics of invariant manifolds, transit orbits, and singular collision orbits in the planar circular restricted three-body problem

The first part of the present paper reveals interplays among invariant manifolds emanating from planar Lyapunov orbits around the three collinear Lagrange points \(L_1, L_2\), and \(L_3\) for high energies. Once the energetically forbidden region vanishes, the invariant manifolds together form closed separatrices bounding transit orbits in the phase space, deviating from the low-energy picture of invariant manifold tubes. Though the qualitatively different behavior of invariant manifolds emerges for high energies, associated transit orbits possess a common feature generalized from that of low-energy transit orbits. The second part extends our previous proposal of using singular collision orbits associated with the secondary to find trajectories reaching the vicinity of the secondary to low energies. Statistical analyses indicate that singular collision orbits are useful to find such transfer trajectories except for the very low-energy regime. These results are numerically obtained in the Earth–Moon and Sun–Jupiter planar circular restricted three-body problems.

Bifurcations, Complex Behaviors, and Dynamic Transition in a Coupled Network of Discrete Predator-Prey System

The nonlinear dynamics of predator-prey systems coupled into network is an important issue in recent biological advances. In this research, we consider each node of the coupled network represents a discrete predator-prey system, and the network dynamics is investigated. By applying Jacobian matrix, center manifold theorem and bifurcation theorems, stability of fixed points, flip bifurcation and Neimark-Sacker bifurcation of the discrete predator-prey system are analyzed. Via the method of Lyapunov exponents, the nonchaos-chaos transition of the coupled network along the routes to chaos induced by bifurcations is determined. Numerical simulations are performed to demonstrate the bifurcations, various attractors and dynamic transitions of the coupled network. Via comparison, we find that the coupled network exhibits far richer and more complex behaviors than single predator-prey system, including period-doubling cascades in orbits of period-2, period-4, period-8, invariant closed curves, dynamic windows for periodic orbits and invariant curves, quasiperiodic orbits, tori, and chaotic sets. Moreover, the attractors of the coupled network show more diverse and complicated structures. These results may provide a new perspective on the predator-prey dynamics in complex networks.

Orbit closures and invariants

Let G be a reductive linear algebraic group, H a reductive subgroup of G and X an affine G-variety. Let X^H denote the set of fixed points of H in X, and N_G(H) the normalizer of H in G. In this paper we study the natural map of quotient varieties psi _{X,H}:X^H/N_G(H) rightarrow X/G induced by the inclusion X^H subseteq X. We show that, given G and H, psi _{X,H} is a finite morphism for all affine G-varieties X if and only if H is a G-completely reducible subgroup of G (in the sense defined by Serre); this was proved in characteristic 0 by Luna in the 1970s. We discuss some applications and give a criterion for psi _{X,H} to be an isomorphism. We show how to extend some other results in Luna’s paper to positive characteristic and also prove the following theorem. Let H and K be reductive subgroups of G; then the double coset HgK is closed for generic gin G if and only if Hcap gKg^{-1} is reductive for generic gin G.

Deformations, moduli stabilisation and gauge couplings at one-loop

We investigate deformations of {mathrm{mathbb{Z}}}_2 orbifold singularities on the toroidal orbifold {T}^6/left({mathrm{mathbb{Z}}}_2times {mathrm{mathbb{Z}}}_6right) with discrete torsion in the framework of Type IIA orientifold model building with intersecting D6-branes wrapping special Lagrangian cycles. To this aim, we employ the hypersurface formalism developed previously for the orbifold {T}^6/left({mathrm{mathbb{Z}}}_2times {mathrm{mathbb{Z}}}_6right) with discrete torsion and adapt it to the left({mathrm{mathbb{Z}}}_2times {mathrm{mathbb{Z}}}_6times Omega mathrm{mathcal{R}}right) point group by modding out the remaining {mathrm{mathbb{Z}}}_3 subsymmetry and the orientifold projection Omega mathrm{mathcal{R}} . We first study the local behaviour of the {mathrm{mathbb{Z}}}_3times Omega mathrm{mathcal{R}} invariant deformation orbits under non-zero deformation and then develop methods to assess the deformation effects on the fractional three-cycle volumes globally. We confirm that D6-branes supporting USp(2N) or SO(2N) gauge groups do not constrain any deformation, while deformation parameters associated to cycles wrapped by D6-branes with U(N) gauge groups are constrained by D-term supersymmetry breaking. These features are exposed in global prototype MSSM, Left-Right symmetric and Pati-Salam models first constructed in [1, 2], for which we here count the number of stabilised moduli and study flat directions changing the values of some gauge couplings.Finally, we confront the behaviour of tree-level gauge couplings under non-vanishing deformations along flat directions with the one-loop gauge threshold corrections at the orbifold point and discuss phenomenological implications, in particular on possible LARGE volume scenarios and the corresponding value of the string scale Mstring, for the same global D6-brane models.

On 3d bulk geometry of Virasoro coadjoint orbits: orbit invariant charges and Virasoro hair on locally AdS $$_3$$ 3 geometries

Expanding upon [arXiv:1404.4472, 1511.06079], we provide further detailed analysis of Ba\~nados geometries, the most general solutions to the AdS3 Einstein gravity with Brown-Henneaux boundary conditions. We analyze in some detail the causal, horizon and boundary structure, and geodesic motion on these geometries, as well as the two class of symplectic charges one can associate with these geometries: charges associated with the exact symmetries and the Virasoro charges. We elaborate further the one-to-one relation between the coadjoint orbits of two copies of Virasoro group and Ba\~nados geometries. We discuss that the information about the Ba\~nados goemetries fall into two categories: "orbit invariant" information and "Virasoro hairs". The former are geometric quantities while the latter are specified by the non-local surface integrals. We elaborate on multi-BTZ geometries which have some number of disconnected pieces at the horizon bifurcation curve. We study multi-BTZ black hole thermodynamics and discuss that the thermodynamic quantities are orbit invariants. We also comment on the implications of our analysis for a 2d CFT dual which could possibly be dual to AdS3 Einstein gravity.

Bifurcation analysis of a discrete-time ratio-dependent predator–prey model with Allee Effect

A discrete-time predator–prey model with Allee effect is investigated in this paper. We consider the strong and the weak Allee effect (the population growth rate is negative and positive at low population density, respectively). From the stability analysis and the bifurcation diagrams, we get that the model with Allee effect (strong or weak) growth function and the model with logistic growth function have somewhat similar bifurcation structures. If the predator growth rate is smaller than its death rate, two species cannot coexist due to having no interior fixed points. When the predator growth rate is greater than its death rate and other parameters are fixed, the model can have two interior fixed points. One is always unstable, and the stability of the other is determined by the integral step size, which decides the species coexistence or not in some extent. If we increase the value of the integral step size, then the bifurcated period doubled orbits or invariant circle orbits may arise. So the numbers of the prey and the predator deviate from one stable state and then circulate along the period orbits or quasi-period orbits. When the integral step size is increased to a critical value, chaotic orbits may appear with many uncertain period-windows, which means that the numbers of prey and predator will be chaotic. In terms of bifurcation diagrams and phase portraits, we know that the complexity degree of the model with strong Allee effect decreases, which is related to the fact that the persistence of species can be determined by the initial species densities.

On the existence of infinitely many invariant Reeb orbits

In this article we extend results of Grove and Tanaka (Bull Am Math Soc 82:497–498, 1976, Acta Math 140:33–48, 1978) and Tanaka (J Differ Geom 17:171–184, 1982) on the existence of isometry-invariant geodesics to the setting of Reeb flows and strict contactomorphisms. Specifically, we prove that if M is a closed connected manifold with the property that the Betti numbers of the free loop space $$ \Lambda (M)$$ are asymptotically unbounded then for every fibrewise star-shaped hypersurface $$\Sigma \subset T^*M$$ and every strict contactomorphism $$ \varphi :\Sigma \rightarrow \Sigma $$ which is contact-isotopic to the identity, there are infinitely many invariant Reeb orbits.

Deformations on tilted tori and moduli stabilisation at the orbifold point

We discuss deformations of orbifold singularities on tilted tori in the context of Type IIA orientifold model building with D6-branes on special Lagrangian cycles. Starting from $$ {T}^6/\left({\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2\right) $$ , we mod out an additional $$ {\mathrm{\mathbb{Z}}}_3 $$ symmetry to describe phenomenologically appealing backgrounds and reduce to $$ {\mathrm{\mathbb{Z}}}_3 $$ and $$ \Omega \mathrm{\mathcal{R}} $$ invariant orbits of deformations. While D6-branes carrying SO(2N) or USp(2N) gauge groups do not constrain deformations, D6-branes with U(N) gauge groups develop non-vanishing D-terms if they couple to previously singular, now deformed cycles. We present examples for both types of D6-branes, and in a three-generation Pati-Salam model on $$ {T}^6/\left({\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_6^{\prime}\right) $$ we find that ten out of 15 twisted complex structure moduli are indeed stabilised at the orbifold point by the existence of the brane stacks.

Invariant measures and orbit equivalence for generalized Toeplitz subshifts

We show that for every metrizable Choquet simplex K and for every group G which is innite, countable, amenable and residually nite, there exists a Toeplitz G -subshift whose set of shift-invariant probability measures is anely homeomorphic to K . Furthermore, we get that for every integer d > 1 and every Toeplitz flow (X, T), there exists a Toeplitz \mathbb Z^d -subshift which is topologically orbit equivalent to (X, T)$.