Limit Sets Research Articles

Limit Sets of Discontinuous Vector Fields on Two-Dimensional Manifolds

In this paper, limit sets of trajectories of a discontinuous vector field Z defined on a two-dimensional manifold M are studied. As in the classical Poincaré–Bendixson theorem, trajectories of Z are supposed to be confined on some compact invariant set \(K\subset V\), where V is a coordinate neighborhood of M, and we require that Z and K fulfill some hypotheses analogous to the referred theorem. More precisely, M is split in an arbitrary number of regions by a set of smooth curves \(\Sigma \) so that Z is defined by pieces on those regions, being eventually discontinuous on \(\Sigma \). Moreover, it is assumed that K contains finite pseudo-equilibria of Z and at most two pieces of it, each piece having finite equilibria on \(K{\setminus }\Sigma \) and at most one tangency point on \(K\cap \Sigma \). We require no extra assumption on \(\Sigma \) but regularity, therefore the existence of the so-called sliding motion is allowed on \(\Sigma \) along with crossing and tangency points. The main results of the paper fully describe the limit sets of trajectories of Z under the previous hypotheses (see Theorem 1) and also state some features of a particular limit set presenting non-empty interior and nondeterministic chaos (see Theorem 2). They also generalize some previous results of the literature when trajectories either slide on \(\Sigma \) indefinitely or never slide again after some finite time (see the fundamental lemma). Some examples and classes of systems fitting the hypotheses of the main results are also provided in the paper along with an algorithm to apply Theorem 1 to robust discontinuous vector fields.

Discontinuous motions of limit sets

We characterise completely when limit sets, as parametrised by Cannon-Thurston maps, move discontinuously for a sequence of algebraically convergent quasi-Fuchsian groups.

On Limit Sets for Geodesics of Meromorphic Connections

Meromorphic connections on Riemann surfaces originate and are closely related to the classical theory of linear ordinary differential equations with meromorphic coefficients. Limiting behavior of geodesics of such connections has been studied by, e.g., Abate and Bianchi (Math Z 282:247–272, 2016) and Abate and Tovena (J Differ Equ 251(9):2612–2684, 2011) in relation with generalized Poincaré-Bendixson theorems. At present, it seems still to be unknown whether some of the theoretically possible asymptotic behaviors of such geodesics really exist. In order to fill the gap, we use the branched affine structure induced by a Fuchsian meromorphic connection to present several examples with geodesics having infinitely many self-intersections and quite peculiar ω-limit sets.

Kobayashi–Warren–Carter type systems with nonhomogeneous Dirichlet boundary data for crystalline orientation

In this paper we study the Dirichlet problem for the Kobayashi–Warren–Carter system. This system of parabolic PDE’s models the grain boundary motion in a polycrystal with a prescribed orientation at the boundary of the domain. We obtain global existence in time of energy-dissipative solutions. The regularity of the solutions as well as the energy-dissipation property permit us to derive the steady-state problem as the asymptotic in time limit of the system. We finally study the ω-limit set of the solutions; we completely characterize it in the one dimensional case, showing, in particular that orientations in the ω-limit set belong to the space of SBV functions. In the two dimensional case, we give sufficient conditions for existence of radial symmetric piecewise constant solutions.

On the dimension spectra of infinite conformal iterated function systems

The dimension spectrum of a conformal iterated function system (CIFS) is the set of all Hausdorff dimensions of its various subsystem limit sets. This brief note provides two constructions – (i) a compact perfect set that cannot be realized as the dimension spectrum of a CIFS; and (ii) a similarity IFS whose dimension spectrum has zero Hausdorff dimension, and thus is not uniformly perfect – which resolve questions posed by Chousionis, Leykekhman and Urbański, and goes on to provoke fresh conjectures and questions regarding the topological and metric properties of IFS dimension spectra.

When is the beginning the end? On full trajectories, limit sets and internal chain transitivity

Let f:X→X be a continuous map on a compact metric space X and let αf, ωf and ICTf denote the set of α-limit sets, ω-limit sets and nonempty closed internally chain transitive sets respectively. In this paper we characterise, by introducing novel variants of shadowing, maps for which every element of ICTf is equal to (resp. may be approximated by) the α-limit set and the ω-limit set of the same full trajectory. We construct examples highlighting the difference between these properties.

Spatial-Temporal Traffic Flow Control on Motorways Using Distributed Multi-Agent Reinforcement Learning

The prevailing variable speed limit (VSL) systems as an effective strategy for traffic control on motorways have the disadvantage that they only work with static VSL zones. Under changing traffic conditions, VSL systems with static VSL zones may perform suboptimally. Therefore, the adaptive design of VSL zones is required in traffic scenarios where congestion characteristics vary widely over space and time. To address this problem, we propose a novel distributed spatial-temporal multi-agent VSL (DWL-ST-VSL) approach capable of dynamically adjusting the length and position of VSL zones to complement the adjustment of speed limits in current VSL control systems. To model DWL-ST-VSL, distributed W-learning (DWL), a reinforcement learning (RL)-based algorithm for collaborative agent-based self-optimization toward multiple policies, is used. Each agent uses RL to learn local policies, thereby maximizing travel speed and eliminating congestion. In addition to local policies, through the concept of remote policies, agents learn how their actions affect their immediate neighbours and which policy or action is preferred in a given situation. To assess the impact of deploying additional agents in the control loop and the different cooperation levels on the control process, DWL-ST-VSL is evaluated in a four-agent configuration (DWL4-ST-VSL). This evaluation is done via SUMO microscopic simulations using collaborative agents controlling four segments upstream of the congestion in traffic scenarios with medium and high traffic loads. DWL also allows for heterogeneity in agents’ policies; cooperating agents in DWL4-ST-VSL implement two speed limit sets with different granularity. DWL4-ST-VSL outperforms all baselines (W-learning-based VSL and simple proportional speed control), which use static VSL zones. Finally, our experiments yield insights into the new concept of VSL control. This may trigger further research on using advanced learning-based technology to design a new generation of adaptive traffic control systems to meet the requirements of operating in a nonstationary environment and at the leading edge of emerging connected and autonomous vehicles in general.

Coexistence in a four-species food web model with general functional responses

The dynamics of an intraguild predation system is analyzed. Two shared resources, a middle predator and a top predator are considered in the model. Parameter conditions for coexistence in the intraguild system are given. The coexistence of species, through the existence of an equilibrium point or a stable limit cycle or an invariant torus is shown. Conditions to have Andronov-Hopf and Hopf-Hopf bifurcation are given. Then, limit sets and complex dynamical behaviors, can exhibit in the system. For a wide family of functional responses, these results are valid. Numerical simulations with varied functional responses are given and different limit sets are shown.

Stable limit cycles in an intraguild predation model with general functional responses

The conditions to have a stable limit set in the positive octant for an intraguild predation model are given, considering a general family of functional responses for the predators and a wide range of growth rate functions for the prey. The limit sets are obtained by Hopf or Bautin bifurcation. The results obtained in previous works about intraguild predation models, which consider logistic growth rate for the prey and Holling functional responses, are generalized.

Limit cycle oscillator in nonlinear systems with multiple time delays

This paper concerns with the problem of generation stable limit cycles in a class of nonlinear time-delay systems with multiple time delays. With the help of the Lyapunov–Krasovskii functional approach, a state-feedback controller is constructed based on the backstepping technique. According to the recursive procedure of the backstepping technique, at first, the desired limit cycle is generated in the second-order subsystem by utilizing the Lyapunov theorem analysis of the positive limit sets. Then, this method is expanded for higher-order systems. By introducing an appropriate Lyapunov-Krasovskii functional, the complexities raised by unknown time-delays are solved in the control design process. Based on this scheme, a delay-independent controller is explicitly designed which does not need the precise knowledge of time delays. The proposed method rigorously guarantees the practical stability of the closed-loop system and also ensures the convergence of the phase trajectories of the closed-loop system to the target limit cycle. Finally, to prove the theoretical achievement and also to illustrate the effective performance of the proposed approach, the simulation results are provided for several examples.

Redundancy of Triangle Groups in Spherical CR Representations

Falbel, Koseleff and Rouillier computed a large number of boundary unipotent CR representations of fundamental groups of non compact three-manifolds. Those representations are not always discrete. By experimentally computing their limit set, one can determine that those with fractal limit sets are discrete. Many of those discrete representations can be related to complex hyperbolic triangle groups. By exact computations, we verify the existence of those triangle representations, which have boundary unipotent holonomy. We also show that many representations are redundant: for n fixed, all the representations encountered are conjugate and only one among them is uniformizable.

On Limit Sets of Monotone Maps on Regular Curves

We investigate the structure of \(\omega \)-limit (resp. \(\alpha \)-limit) sets for a monotone map f on a regular curve X. We show that for any \(x\in X\) (resp. for any negative orbit \((x_{n})_{n\ge 0}\) of x), the \(\omega \)-limit set \(\omega _{f}(x)\) (resp. \(\alpha \)-limit set \(\alpha _{f}((x_{n})_{n\ge 0})\)) is a minimal set. This also holds for \(\alpha \)-limit set \(\alpha _{f}(x)\) whenever x is not a periodic point. These results extend those of Naghmouchi [24] established whenever f is a homeomorphism on a regular curve and those of Abdelli [1], whenever f is a monotone map on a local dendrite. Further results related to the basin of attraction of an infinite minimal set are also obtained.

On backward attractors of interval maps* *The research was supported by RVO, Czech Republic funding for IČ47813059. The first author was supported by the Marie Skłodowska-Curie Grant Agreement No. 883748 from the European Union’s Horizon 2020 research and innovation programme.

Special α-limit sets (sα-limit sets) combine together all accumulation points of all backward orbit branches of a point x under a noninvertible map. The most important question about them is whether or not they are closed. We challenge the notion of sα-limit sets as backward attractors for interval maps by showing that they need not be closed. This disproves a conjecture by Kolyada, Misiurewicz, and Snoha. We give a criterion in terms of Xiong’s attracting centre that completely characterizes which interval maps have all sα-limit sets closed, and we show that our criterion is satisfied in the piecewise monotone case. We apply Blokh’s models of solenoidal and basic ω-limit sets to solve four additional conjectures by Kolyada, Misiurewicz, and Snoha relating topological properties of sα-limit sets to the dynamics within them. For example, we show that the isolated points in a sα-limit set of an interval map are always periodic, the non-degenerate components are the union of one or two transitive cycles of intervals, and the rest of the sα-limit set is nowhere dense. Moreover, we show that sα-limit sets in the interval are always both F σ and G δ . Finally, since sα-limit sets need not be closed, we propose a new notion of β-limit sets to serve as backward attractors. The β-limit set of x is the smallest closed set to which all backward orbit branches of x converge, and it coincides with the closure of the sα-limit set. At the end of the paper we suggest several new problems about backward attractors.

On totally periodic ω-limit sets for monotone maps on regular curves

An ω-limit set of a continuous self-mapping of a compact metric space X is said to be totally periodic if all of its points are periodic. In [3] Askri and Naghmouchi proved that if f is a one-to-one continuous self mapping of a regular curve, then every totally periodic ω-limit set of f is finite. This also holds whenever f is a monotone map of a local dendrite by Abdelli in [1]. In this paper we generalize these results to monotone maps on regular curves. On the other hand, we give some remarks related to expansivity and totally periodic ω-limit sets for every continuous map on compact metric space.

On α-Limit Sets in Lorenz Maps

The aim of this paper is to show that -limit sets in Lorenz maps do not have to be completely invariant. This highlights unexpected dynamical behavior in these maps, showing gaps existing in the literature. Similar result is obtained for unimodal maps on . On the basis of provided examples, we also present how the performed study on the structure of -limit sets is closely connected with the calculation of the topological entropy.

Symmetry Properties of Sign-Changing Solutions to Nonlinear Parabolic Equations in Unbounded Domains

We study the asymptotic (in time) behavior of positive and sign-changing solutions to nonlinear parabolic problems in the whole space or in the exterior of a ball with Dirichlet boundary conditions. We show that, under suitable regularity and stability assumptions, solutions are asymptotically (in time) foliated Schwarz symmetric, i.e., all elements in the associated omega-limit set are axially symmetric with respect to a common axis passing through the origin and are nonincreasing in the polar angle. We also obtain symmetry results for solutions of Hénon-type problems, for equilibria (i.e. for solutions of the corresponding elliptic problem), and for time periodic solutions.

5D model of pancreatic cancer: Key features of ultimate dynamics

In this paper, we study various types of ultimate dynamics for the 5-dimensional pancreatic cancer model without treatment. This model has been created by Hu et al. and describes interactions between pancreatic cancer cells (PCCs); pancreatic stellate cells (PSCs); immune cells and two types of cytokines, tumour–promoting and tumour–suppressing. Using iteratively the localization theorem of compact invariant sets we derive ultimate upper bounds for all cell populations and establish the property of the existence of the attracting set. Next, we find several conditions under which our system demonstrates various types of convergence dynamics: to equilibrium points located in the intersection of PCC-free plane with the PSC-free plane; equilibrium points with positive values of PSC density located in PCC-free plane; ω-limit sets with positive values of PCC density. Further, we explore the dynamics within the PSC-free plane and find attraction conditions to ω-limit sets with positive values of PCC density. Our assertions are expressed by means of iterative sequence of algebraic inequalities respecting parameters of the model. Our theoretical studies are supplied by results of numerical simulation. Locations of attracting equilibrium points have been found with help of the iteration procedure containing from 2 steps up to 23 steps in the dependence of parameter conditions and the location with respect to the PSC-free plane.

Packing dimensions of basins generated by distributions on a finite alphabet

We consider a space of infinite signals composed of letters from a finite alphabet. Each signal generates a sequence of empirical measures on the alphabet and the limit set corresponding to this sequence. The space of signals is partitioned into narrow basins consisting of signals with identical limit sets for the sequence of empirical measures and for each narrow basin its packing dimension is computed. Furthermore, we compute packing dimensions for two other types of basins defined in terms of limit behaviour of the empirical measures.

Control of Power Converters With Hybrid Affine Models and Pulse-Width Modulated Inputs

In this paper, hybrid dynamical systems theory is applied to the analysis and control of switched converters with Pulse-Width Modulated (PWM) inputs. The system is described by a state-space model with continuous flows and discrete jumps, without averaged equations. The modulation effects are captured in full without using time-dependent signals, by enlarging the state vector to include the PWM waveform generation process. Furthermore, the sample-and-hold mechanism associated with the sampling frequency is also taken into account with this approach. A control law is proposed based on a Lyapunov function candidate. Furthermore, convergence sets and the steady state jitter, inherent to PWM-based controllers, are analyzed estimating limit sets for the augmented state. Consequently, output chattering can be bounded. By using hybrid dynamical system theory, the control designer gains a deeper understanding of the effect of modulation in the closed-loop dynamics, avoiding the problems associated with the use averaged models. Experimental results validate the proposed method.

A non-Borel special alpha-limit set in the square

AbstractWe consider the complexity of special $\alpha $ -limit sets, a kind of backward limit set for non-invertible dynamical systems. We show that these sets are always analytic, but not necessarily Borel, even in the case of a surjective map on the unit square. This answers a question posed by Kolyada, Misiurewicz, and Snoha.