Invariant Sets Research Articles

Stability of the twisted states in a ring of oscillators interacting with distance-dependent delays

We consider a ring of phase oscillators interacting with distance-dependent time delays in general forms. It is found that the distance-dependent interaction delay is an essential driving mechanism for the twisted state to occur as a generic state. We carry out rigorous stability analysis for the existence and stability of the twisted states along the Ott–Antonsen invariant manifold and derive the exact characteristic equations for the stability conditions for arbitrarily chosen distance-dependent delay function. The complete stability diagrams are illustrated for two types of distance-dependent delay schemes, the propagational and step-wise interaction delays. Our theoretical results are verified using the direct numerical simulations of the model system.

Rapid Approximation of Low-Thrust Spacecraft Reachable Sets within Complex Two-Body and Cislunar Dynamics

The reachable set of controlled dynamical systems is the set of all reachable states from an initial condition over a certain time horizon, subject to operational constraints and exogenous disturbances. In astrodynamics, rapid approximation of reachable sets is invaluable for trajectory planning, collision avoidance, and ensuring safe and optimal performance in complex dynamics. Leveraging the connection between minimum-time trajectories and the boundary of reachable sets, we propose a sampling-based method for rapid and efficient approximation of reachable sets for finite- and low-thrust spacecraft. The proposed method combines a minimum-time multi-stage indirect formulation with the celebrated primer vector theory. Reachable sets are generated under two-body and circular restricted three-body (CR3B) dynamics. For the two-body dynamics, reachable sets are generated for (1) the heliocentric phase of a benchmark Earth-to-Mars problem, (2) two scenarios with uncertainties in the initial position and velocity of the spacecraft at the time of departure from Earth, and (3) a scenario with a bounded single impulse at the time of departure from Earth. For the CR3B dynamics, several cislunar applications are considered, including L1 Halo orbit, L2 Halo orbit, and Lunar Gateway 9:2 NRHO. The results indicate that low-thrust spacecraft reachable sets coincide with invariant manifolds existing in multi-body dynamical environments. The proposed method serves as a valuable tool for qualitatively analyzing the evolution of reachable sets under complex dynamics, which would otherwise be either incoherent with existing grid-based reachability approaches or computationally intractable with a complete Hamilton–Jacobi–Bellman method.

Boundary analysis and energy feedback control of fractional-order extended Malkus–Robbins dynamo system

In this paper, the fractional-order extended Malkus-Robbins dynamo system (FOEMRD) is introduced. Additionally, the stability, complexity diagrams, Lyapunov dimension, Lyapunov exponents, and bifurcation diagrams of the system are studied. Also, we estimate the global Mittag-Leffler positive invariant sets and global Mittag-Leffler attractive sets of the proposed system. In addition, the Hamilton energy function of the FOEMRD system is computed by the Helmholtz theorem. Next, we design an energy feedback controller to control the chaos. Due to the relationship of Hamilton energy with the occurrence of chaos, the energy control method is very efficient and powerful. The applicability of the presented theories was proven with numerical simulations and related diagrams.

Theoretical and experimental study of a stable state adjustable nonlinear energy sink

Due to high sensitivity to the changes in external excitation intensity, the implementation of nonlinear energy sinks (NESs) have been greatly limited in practical applications. This study presents a stable state adjustable NES (SA-NES) structure. By adjusting the structural geometric relationship, the SA-NES structure can easily switch its stable state among monostable, bistable, and tristable. The governing equation of the coupled system is obtained by Hamilton's principle. The approximate analytical solution is obtained by the harmonic balance method, and the slow invariant manifold is obtained complexification-averaging method, which are validated through numerical analysis. Through theoretical analysis and experimental validation, the dynamic response of the SA-NES and vibration reduction performance are studied under different levels of excitations. The obtained results show that the designed SA-NES can effectively suppress the vibration under different level excitation intensities. It is also shown that the SA-NES structure can be adjusted to the monostable state, and the vibration of the linear oscillator can be effectively suppressed by the strong modulation response under a certain excitation intensity. When the excitation intensity is weak or strong, the SA-NES can be adjusted to the bistable state and the tristable state, respectively, thus effectively suppressing the vibration through chaotic inter-well oscillation. This study reveals the vibration suppression mechanism of the SA-NES, and introduces a new NES configuration to suppress different levels of engineering excitations.

Accurate neural-network-based fitting of full-dimensional N2 –Ar and N2–CH4 two-body potential energy surfaces aimed at spectral simulations

We describe the development of machine-learned (ML) potentials for flexible, weakly interacting monomers. A recently suggested permutationally invariant polynomial neural network (PIP-NN) approach is utilised to represent the full-dimensional two-body component of the molecular pair energy. To ensure the asymptotic zero-interaction limit, a tailored subset of the full invariant polynomial basis set is utilised, and their variables are modified to achieve a better fit of the correct asymptotic behaviour at a long range. This new technique is used to build full-dimensional potentials for the two-body N 2 –Ar and N 2 –CH 4 interactions by fitting databases of ab initio energies calculated at the coupled-cluster level of theory. The second virial coefficient, fully accounting for molecular flexibility, is then calculated within the classical framework using the obtained PIP-NN potential surfaces. A trajectory-based simulation of the N 2 –Ar collision-induced absorption is conducted, covering both the far- and mid-infrared ranges.

Switching positive system event triggering T-S fuzzy control

This paper presents a saturation control problem with a positive switching T-S fuzzy system based on event-triggering mechanism under continuous time. Firstly, a 1-norm event triggering mechanism is designed. Then, the multilinear co-positive Lyapunov function is applied to the closed-loop positive switching system with a T-S fuzzy control strategy under continuous time. Next, we consider constructing a cone invariant set to transform the actuator saturation problem into an interval form. The designed controller with a linear programming method guarantees both the positivity and stability of the closed-loop system. Finally, an example of a communication data network proves the effectiveness of the design. The main contents are (i) The event-triggering strategy under continuous time is used in the T-S switching positive system. (ii) A controller with input saturation is introduced into controllable subsystems. (iii) When determining the attractor domain of the system state, a cone domain is considered as a potential restriction.

On the lifting and reconstruction of nonlinear systems with multiple invariant sets

On the lifting and reconstruction of nonlinear systems with multiple invariant sets

A Note on Coregular Elements in Certain Semigroups of Transformations

For a non-empty set X, let P(X) represent the partial transformation semigroup on X. For a non-empty subset Y of X, define as the semigroup . Then is a generalization of P(X) , encompassing all partial transformations on that maintain Y as an invariant set. This paper delves into exploring the characterization of coregular elements within and applies the results to obtain a similar characterization in relevant semigroups.

Tube-based model predictive control for linear systems with bounded disturbances and input delay

This paper proposes a tube-based model predictive control strategy for linear systems with bounded disturbances and input delay to ensure input-to-state stability. Firstly, the actual disturbed system is decomposed into a nominal system without disturbances and an error system. For the nominal system, solving an optimization problem, where the delayed control input is set as an optimization variable, yields a nominal control law that enables the nominal state signal to approach to zero. Then, for the error system, the Razumikhin approach is used to identify a robust control invariant set. Using the set invariance theorem, an ancillary control law is developed to confine the error state signal in the invariant set. Combining the two results, we obtain a control law that enables the state signal to remain within a robustly invariant tube. Finally, the effectiveness of the developed control strategy is validated by simulations.

Analysis of long transients and detection of early warning signals of extinction in a class of predator-prey models exhibiting bistable behavior.

In this paper, we develop a method of analyzing long transient dynamics in a class of predator-prey models with two species of predators competing explicitly for their common prey, where the prey evolves on a faster timescale than the predators. In a parameter regime near a singular zero-Hopf bifurcation of the coexistence equilibrium state, we assume that the system under study exhibits bistability between a periodic attractor that bifurcates from the singular Hopf point and another attractor, which could be a periodic attractor or a point attractor, such that the invariant manifolds of the coexistence equilibrium point play central roles in organizing the dynamics. To find whether a solution that starts in a vicinity of the coexistence equilibrium approaches the periodic attractor or the other attractor, we reduce the equations to a suitable normal form, and examine the basin boundary near the singular Hopf point. A key component of our study includes an analysis of the long transient dynamics, characterized by their rapid oscillations with a slow variation in amplitude, by applying a moving average technique. We obtain a set of necessary and sufficient conditions on the initial values of a solution near the coexistence equilibrium to determine whether it lies in the basin of attraction of the periodic attractor. As a result of our analysis, we devise a method of identifying early warning signals, significantly in advance, of a future crisis that could lead to extinction of one of the predators. The analysis is applied to the predator-prey model considered in Sadhu (Discrete Contin Dyn Syst B 26:5251-5279, 2021) and we find that our theory is in good agreement with the numerical simulations carried out for this model.

Dynamics of traveling waves for predator-prey systems with Allee effect and time delay

This article aims to establish the existence of traveling waves for a predator-prey system with Beddington-DeAngelis functional response, reproductive Allee effect, and time delay. We investigate the existence of solutions for a system with two special delay kernels by geometric singular perturbation theory, invariant manifold theory, and Fredholm orthogonality theory. In addition, we discuss the asymptotic behaviors of traveling waves with the aid of the asymptotic theory. For more information see https://ejde.math.txstate.edu/Volumes/2024/33/abstr.html

Lyapunov-like functions for almost global convergence in discrete-time systems

This paper presents new results on discrete-time almost global convergence to an invariant set. First, we present new sufficient conditions for almost global convergence using density functions. In contrast to existing density-function results, the new results in this paper do not rely on a local convergence assumption, and they do not require knowledge of the inverse map of the difference equation. Next, we present new sufficient conditions for almost global convergence using Lyapunov-like functions rather than density functions. These Lyapunov-like results can be useful because constructing and analyzing density functions is often difficult in comparison to Lyapunov-like analysis. This paper also presents a variety of simple examples to illustrate these new methods for almost global convergence analysis.

Full Classification of Finite Singleton Local Rings

The main objective of this article is to classify all finite singleton local rings, which are associative rings characterized by a unique maximal ideal and a distinguished basis consisting of a single element. These rings are associated with four positive integer invariants p,n,s, and t, where p is a prime number. In particular, we aim to classify these rings and count them up to isomorphism while maintaining the same set of invariants. We have found interesting cases of finite singleton local rings with orders of p6 and p7 that hold substantial importance in the field of coding theory.

Bifurcation of the travelling wave solutions in a perturbed (1 + 1)-dimensional dispersive long wave equation via a geometric approach

Choosing ${\kappa }$ (horizontal ordinate of the saddle point associated to the homoclinic orbit) as bifurcation parameter, bifurcations of the travelling wave solutions is studied in a perturbed $(1 + 1)$ -dimensional dispersive long wave equation. The solitary wave solution exists at a suitable wave speed $c$ for the bifurcation parameter ${\kappa }\in \left (0,1-\frac {\sqrt 3}{3}\right )\cup \left (1+\frac {\sqrt 3}{3},2\right )$ , while the kink and anti-kink wave solutions exist at a unique wave speed $c^*=\sqrt {15}/3$ for $\kappa =0$ or $\kappa =2$ . The methods are based on the geometric singular perturbation (GSP, for short) approach, Melnikov method and invariant manifolds theory. Interestingly, not only the explicit analytical expression of the complicated homoclinic Melnikov integral is directly obtained for the perturbed long wave equation, but also the explicit analytical expression of the limit wave speed is directly given. Numerical simulations are utilized to verify our mathematical results.

Constrained State Regulation Problem of Descriptor Fractional-Order Linear Continuous-Time Systems

This paper deals with the constrained state regulation problem (CSRP) of descriptor fractional-order linear continuous-time systems (DFOLCS) with order 0<α<1. The objective is to establish the existence of conditions for a linear feedback control law within state constraints and to propose a method for solving the CSRP of DFOLCS. First, based on the decomposition and separation method and coordinate transformation, the DFOLCS can be transformed into an equivalent fractional-order reduced system; hence, the CSRP of the DFOLCS is equivalent to the CSRP of the reduced system. By means of positive invariant sets theory, Lyapunov stability theory, and some mathematical techniques, necessary and sufficient conditions for the polyhedral positive invariant set of the equivalent reduced system are presented. Models and corresponding algorithms for solving the CSRP of a linear feedback controller are also presented by the obtained conditions. Under the condition that the resulting closed system is positive, the given model of the CSRP in this paper for the DFOLCS is formulated as nonlinear programming with a linear objective function and quadratic mixed constraints. Two numerical examples illustrate the proposed method.

Blow-up at infinity of solutions to a class of pseudo-parabolic equations with logarithmic nonlinearity and singular potential

We study the initial–boundary value problem for a class of pseudo-parabolic equations in a bounded domain Ω with smooth boundary ∂Ω and 0∈Ω⊂BR(0)⊂Rn(n≥1), where BR(0) is a ball centered at 0 with radius R. First, we simplify the scheme of potential well method, which helps us obtain the invariant set in an easier way. Combining logarithmic Sobolev inequality and Hardy inequality, we prove that the solutions to the pseudo-parabolic equations with logarithmic nonlinearity ulog|u| blow up at infinity when R≤1. However, there is no restriction on R if the singular potential is 1. Second, when the singular potential is 1 and the pseudo-parabolic term vanishes, the equations model the epitaxial growth of thin films. We also show that the solutions have better growth rate under weaker conditions, which improves the result in Han et al. (2019).

A customizable cam-typed bistable nonlinear energy sink

A novel cam-typed bistable nonlinear energy sink (CBNES) is first proposed in this paper. The CBNES achieves a customized design of the ideal configuration of BNES by introducing a cam-roller-spring negative stiffness mechanism (CRS-NSM). The design criteria for continuous and smooth bistable cam profiles are studied, and the extreme values of dimensionless stiffness and pre-compression are predicted. Furthermore, the periodic and quasi-periodic dynamics of the system are studied by using the complexification-averaging (CA-X) method. By combining Slow Invariant Manifold (SIM) and phase trajectory, the energy dissipations of different response types of the CBNES are predicted. The SMR and chaotic response have the best and lowest targeted energy transfer (TET) efficiency, respectively. The saddle-node (S-N) and Holf bifurcation characteristics of the periodic fixed points are analyzed to predict the possibility of SMR occurrence. The analysis of the amplitude frequency curve with the variation of system parameters indicates that weak negative stiffness and intermediate cubic stiffness are more beneficial to vibration suppression. The highlight is that the excitation thresholds of SMR are predicted by using analytical and numerical methods, and the influences of system parameters on the thresholds are analyzed. The deviation between the analytical and numerical results will decrease with increasing damping. Finally, the correctness of the analytical results is verified through a series of fixed-frequency experiments. The proposed CBNES provides a valuable potential solution for designing efficient and robust mechanical BNES physical implementations.

Developing a continuous SIR epidemic model and its discrete version using Euler method: Analyzing dynamics with analytical and numerical methods

In this paper, we develop a continuous SIR epidemic model and its discrete version using the Euler method. The continuous SIR epidemic model is a widely used mathematical tool for understanding the dynamics of infectious diseases. We derive the system of differential equations that describe the rates of change in the number of individuals in each compartment and discuss the key parameters that influence the dynamics of the model. Next, we use the Euler method to discretize the continuous SIR epidemic model and obtain a discrete‐time version. We analyze the dynamics of the discrete SIR epidemic model using both analytical and numerical methods. We derive the positively invariant set of the model and discuss the initial conditions of the model and how they relate to the total population. We compare the behavior of the continuous and discrete SIR epidemic models and show that the discretized model captures the essential features of the continuous model. We also investigate the impact of key parameters such as the transmission rate, recovery rate, and death rate on the dynamics of the model. Our findings suggest that the discrete SIR epidemic model is a powerful mathematical tool for understanding the dynamics of infectious diseases and developing effective strategies for their control. By studying the behavior of the model under different scenarios, researchers can gain insights into the transmission and prevalence of infectious diseases and develop interventions that reduce their impact on public health.

Additive and geometric transversality of fractal sets in the integers

AbstractBy juxtaposing ideas from fractal geometry and dynamical systems, Furstenberg proposed a series of conjectures in the late 1960's that explore the relationship between digit expansions with respect to multiplicatively independent bases. In this work, we introduce and study — in the discrete context of the integers — analogs of some of the notions and results surrounding Furstenberg's work. In particular, we define a new class of fractal sets of integers that parallels the notion of ‐invariant sets on the 1‐torus and investigate the additive and geometric independence between two such fractal sets when they are structured with respect to multiplicatively independent bases. Our main results in this direction parallel the works of Furstenberg, Hochman–Shmerkin, Shmerkin, Wu, and Lindenstrauss–Meiri–Peres and include: a classification of all subsets of the positive integers that are simultaneously ‐ and ‐invariant; integer analogs of two of Furstenberg's transversality conjectures pertaining to the dimensions of the intersection and the sumset of ‐ and ‐invariant sets and when and are multiplicatively independent; and a description of the dimension of iterated sumsets for any ‐invariant set . We achieve these results by combining ideas from fractal geometry and ergodic theory to build a bridge between the continuous and discrete regimes. For the transversality results, we rely heavily on quantitative bounds on the ‐dimensions of projections of restricted digit Cantor measures obtained recently by Shmerkin. We end by outlining a number of open questions and directions regarding fractal subsets of the integers.

О методах исследования динамических систем

Wide application of dynamical systems resulted in designing many various methods for investigations, both analytical and computer-oriented ones. In this paper we consider methods based on ideas of symbolic dynamics, and discuss the application of dynamical systems for solving identification and prognosis problems. The representation of the system dynamics by an oriented graph constructed in accordance with the system and a finite partition of the phase space gives a possibility to matchtrajectories paths on the graph. By this method one may construct approximation to invariant sets and invariant measures, and the Morse spectrum as well. Using dynamical systems in identification problems is consideredon examples of probability chainsfor the modeling social and economic resource distribution, and the method of nonlinear dynamics for reconstruction of an attractor on a given time series.