Discrete-time Map Research Articles

Dynamic Analysis of a Simple Cournot Duopoly Model Based on a Computed Cost

The paper is organized to study some mathematical properties and dynamics of a simple Cournot duopoly game based on a computed quadratic cost. The time evolution of this game is described by a two-dimensional noninvertible discrete time map using the bounded rationality mechanism. For this map, some dynamic characteristics such as multistability and synchronization are investigated. Its equilibrium points are obtained for the asymmetric case, and their conditions of stability are obtained. Our results investigate that the Nash equilibrium point may be unstable due to flip bifurcation and under certain parameter values, and Neimark–Sacker bifurcation is born after the period-4 cycle. Through some restrictions, the coordinate axes of the map construct an invariant manifold, and therefore, their dynamics can be analyzed by using a one-dimensional map. In the symmetric case, both firms behave identically, and this implies that the diagonal set forms an invariant manifold, and hence the synchronization phenomena take place. Furthermore, the global bifurcation of the map is confirmed through contact between critical curves and the boundaries of infeasible domains.

Air traffic inefficiencies and predictability evaluation using route mapping—the Tokyo International Airport case

Air traffic inefficiencies lead to excess fuel burn, emissions and air traffic controller (ATCo) workload. Various stakeholders have developed metrics to assess the operation performance. Most metrics compare the actual trajectories to some benchmark ones to calculate excess time or distance. This research is inspired by cellular automata (CA) and develops a combined time-distance lateral inefficiency and predictability metric using discrete space and time mapping on published flight routes. The analysis is focused on Tokyo International Airport, but uses only track data and published routes, which makes it easily applicable to any other hub airport worldwide. The mapping and velocity analyses are used to investigate when and where ATCos are most likely to intervene to provide save separation. A metric which can be adjusted to evaluate both traffic flow predictability and efficiency is proposed. This metric can be applied to better understand current traffic and enable future improvements towards seamless air traffic flow management.

Influence of Heat Loss on the Chaotic Dynamics of Reaction Waves in the Model with Chain-Branching Reaction

In this paper, the complex dynamics of the pulsating regime of combustion waves propagation is numerically investigated within the Zel’dovich–Barenblatt–Dold model with two-step chain-branching reaction mechanism. It is shown that there exists a remarkable similarity in the dynamics of oscillations in this system and one-dimensional discrete-time maps such as the logistic map. In particular, both systems exhibit the period doubling route to chaos and the appearance of windows of stability as the bifurcation parameter is increased. For the first time for a model of combustion wave propagation the sequence of windows of stability is demonstrated and classified. Nevertheless, as the activation energy, which is taken as a bifurcation parameter, reaches a critical value corresponding to the dynamical quenching scenario the dynamics of the flame oscillations becomes multidimensional and therefore it cannot be described by a one-dimensional map anymore. The effect of the heat losses is also studied and it is demonstrated that the number of windows of stability decreases, the order of their appearance changes and the dynamical quenching appears for smaller values of the activation energy as the heat losses are increased.

Bifurcations in a new two-cell spiking map: a numerical and experimental study

AbstractIn this paper, a new nonlinear discrete-time map is presented. The map is based on a second-order dynamics that, despite the limited number of parameters, is able to produce a rich dynamical behavior, including the onset of spiking trends. This latter case will be particularly emphasized, since it allows to consider the introduced system as a novel discrete-time model for spiking neurons. The study is performed by using a numerical bifurcation approach. Moreover, the possibility to obtain a spiking behavior using noise is also shown. The implementation of the map using advanced microcontroller units and the obtained experimental results are discussed.

Lightweight encryption mechanism with discrete-time chaotic maps for Internet of Robotic Things

Chaotic systems have different characteristics that can be utilized for the purpose of cyber security. These features result from the erratic structure of images, making it difficult for systems to comprehend those images. Therefore, chaotic systems can serve as an effective means of securely storing and concealing images by implementing them in the IoRT for data encryption. In this study, a new lightweight encryption mechanism for IoRT based robots is proposed. A novel chaotic encryption mechanism is designed with discrete-time chaotic maps (Cubic Map and Ricker’s Population Model Map) for more efficiently encrypting large-sized data, such as images and videos, in robots. The test of the developed mechanism was conducted on the NVIDIA Jetson Orin development kit, which is utilized in artificial intelligence-supported robots and boasts high performance and energy efficiency. The study’s most important finding is that the established mechanism is suitable for parallel processing, making it possible to accomplish a high rate of image encryption per second utilizing the proposed method. In order to encrypt large files for IoRT, firstly, random number generation was performed and statistical tests were carried out. Then, encryption operations were performed with the developed encryption algorithm and security analyses were realized. The performances of the proposed mechanism will be compared with some studies in the literature. Based on the obtained image analysis and encryption performance, it has been shown that the developed mechanism can be easily used with high security for IoRT.

Modeling and Nonlinear Dynamic Analysis of Cascaded DC–DC Converter Systems Based on Simplified Discrete Mapping

As a basic structure of the distributed power systems, cascaded DC-DC converter systems have attracted a lot of attention. However, due to the interaction between the sub-converters, classic methods for modeling cascaded systems are either not accurate enough, such as state-space averaging model, or not simple enough, such as discrete-time mapping model, especially the systems with different switching frequencies. To overcome this drawback, a simplified discrete-time mapping modeling method for cascaded DC-DC converter systems is proposed in this paper. Based on the idea of state-space average, the original problem is simplified to modeling cascaded DC-DC converter systems with same switching frequencies. The proposed method is able to predict the dynamic properties of the system at all stages, such as slow-scale and fast-scale instabilities. Then, two-stage cascaded boost converter with different switching frequencies under peak current double loop control is taken as an example to present the simplified discrete-time mapping model. Finally, the effectiveness of the proposed method is verified by simulations and experiments.

Deep learning delay coordinate dynamics for chaotic attractors from partial observable data.

A common problem in time-series analysis is to predict dynamics with only scalar or partial observations of the underlying dynamical system. For data on a smooth compact manifold, Takens' theorem proves a time-delayed embedding of the partial state is diffeomorphic to the attractor, although for chaotic and highly nonlinear systems, learning these delay coordinate mappings is challenging. We utilize deep artificial neural networks (ANNs) to learn discrete time maps and continuous time flows of the partial state. Given training data for the full state, we also learn a reconstruction map. Thus, predictions of a time series can be made from the current state and several previous observations with embedding parameters determined from time-series analysis. The state space for time evolution is of comparable dimension to reduced order manifold models. These are advantages over recurrent neural network models, which require a high-dimensional internal state or additional memory terms and hyperparameters. We demonstrate the capacity of deep ANNs to predict chaotic behavior from a scalar observation on a manifold of dimension three via the Lorenz system. We also consider multivariate observations on the Kuramoto-Sivashinsky equation, where the observation dimension required for accurately reproducing dynamics increases with the manifold dimension via the spatial extent of the system.

Robust Chaos With Novel 4-Transistor Maps

This brief presents four discrete-time chaotic map circuits, including three new map circuits and one that has been previously reported. The designs are hardware efficient as they each contain only four Metal Oxide Semiconductor (MOS) transistors. They offer robust chaotic performance with wide chaotic space and uniform entropic properties. The wide chaotic region is essential for applications where parametric perturbation can push the system out of the desirable chaotic region. The chaotic performance is analyzed using a bifurcation plot, Lyapunov exponent, correlation coefficient, sample entropy, and Shannon entropy, and improvement over previous transistor-level designs is demonstrated. The proposed method can be useful in applications such as random number generation, reconfigurable and flexible computing, side-channel attack mitigation, logic obfuscation, and chaos-based cryptography. A chaotic oscillator design is proposed and its application in a chaos-based reconfigurable logic gate is presented, as well.

Stability Effect of Load Converter on Source Converter in a Cascaded Buck Converter

In a cascaded power converter, the input current ripple of load converter is fed forward to the output capacitor of source converter and results in reshaping the output capacitor current and output voltage ripples of source converter. To investigate the stability effect of feedforward current ripple of load converter on source converter, a cascaded power converter comprising two peak-voltage-ripple controlled buck converters is taken as an example. First, the stability effects of chosen circuit parameters of the source converter in cascaded operation and standalone operation are demonstrated and compared. Second, by describing three switching state sequences induced by the feedforward current ripple, a discrete-time map model is established for the cascaded power converter. Based on the map model, the instability mechanism of source converter with the variations of the circuit parameters is thereby expounded, and the stability boundaries are obtained in the circuit parameter planes. The results show that the stability effects of the circuit parameters of source converter with feedforward current ripple are essentially different from those in the standalone operation. The theoretical analyses are verified by the simulation results and experimental results.

A novel tacholess order tracking method for gearbox vibration signal based on extremums search of gearmesh harmonic

Computed order tracking (COT) is an effective vibration analysis method for mechanical equipment operating at time-varying speed. The principle is to resample the time-domain sampled vibration data under the guidance of angle information of reference shaft. The rotation angle obtained from numerical integration is generally limited by the deviation of integration on rotation speed. To address this problem, a novel acquisition method of rotation angle is proposed to improve the precision of order tracking analysis. Based on the equi-angular relationship between adjacent peaks of a single gearmesh harmonic, the discrete shaft phase vs time map is directly established by obtaining the peak time. The process error is thoroughly investigated, and a designed scheme is provided to eliminate the effects of discrete sampling and amplitude modulation. Theoretically, the rotation angle can be correctly depicted, which is useful for a highly precise order-tracking analysis. The accurate amplitude and high condensed spectral lines of order spectrum are achieved both in the simulation and experiment signals, which verified the effectiveness and robustness of the proposed method. It is helpful for the accurate reconstruction of speed-related vibration signal in the order spectrum, and enhances the capability of condition monitoring on gear systems.

Analysis of Fast-Scale Instability in Three-Level T-Type Single-Phase Inverter Feeding Diode-Bridge Rectifier With Inductive Load

In this article, the fast-scale instability in the three-level T-type single-phase inverter feeding diode-bridge rectifier with inductive load (3TSI-DR) is studied. Simulations suggest that such fast-scale instability on switching period scale can increase the harmonic content in the 3TSI-DR, which seriously affects its stable operation. To reveal the mechanism of this fast-scale instability, the state equation of the 3TSI-DR is derived, and state variables are solved based on quasi-static approximation principle. From state equation, the 3TSI-DR is periodic time-varying and piecewise smooth, belonging to Filippov system. Accordingly, the discrete-time mapping model of the 3TSI-DR is established, Filippov method is used for obtaining monodromy matrix, and Floquet theory is applied to explore instability mechanism. Theoretical results indicate that the fast-scale instability of the 3TSI-DR is caused by period-doubling bifurcation. Moreover, the Floquet multiplier sensitivities of different circuit parameters are calculated to identify key parameters; via comparing theoretical analyses with simulations, the unstable angle ranges of the fast-scale instability are given, and the stability boundaries in various parameter spaces are discussed. All these can provide design-oriented information for optimizing the 3TSI-DR to avoid instability due to period-doubling bifurcation. Finally, experimental results agreeing with simulations are presented to verify the correctness of theoretical analyses.

Metastable discrete time-crystal resonances in a dissipative central spin system

We consider the non-equilibrium behavior of a central spin system where the central spin is periodically reset to its ground state. The quantum mechanical evolution under this effectively dissipative dynamics is described by a discrete-time quantum map. Despite its simplicity this problem shows surprisingly complex dynamical features. In particular, we identify several metastable time-crystal resonances. Here the system does not relax rapidly to a stationary state but undergoes long-lived oscillations with a period that is an integer multiple of the reset period. At these resonances the evolution becomes restricted to a low-dimensional state space within which the system undergoes a periodic motion. Generalizing the theory of metastability in open quantum systems, we develop an effective description for the evolution within this long-lived metastable subspace and show that in the long-time limit a non-equilibrium stationary state is approached. Our study links to timely questions concerning emergent collective behavior in the 'prethermal' stage of a dissipative quantum many-body evolution and may establish an intriguing link to the phenomenon of quantum synchronization.

A Fully Distributed Hybrid Control Framework For Non-Differentiable Multi-Agent Optimization

This paper develops a fully distributed hybrid control framework for distributed constrained optimization problems. The individual cost functions are non-differentiable and convex. Based on hybrid dynamical systems, we present a distributed state-dependent hybrid design to improve the transient performance of distributed primal-dual first-order optimization methods. The proposed framework consists of a distributed constrained continuous-time mapping in the form of a differential inclusion and a distributed discrete-time mapping triggered by the satisfaction of local jump set. With the semistability theory of hybrid dynamical systems, the paper proves that the hybrid control algorithm converges to one optimal solution instead of oscillating among different solutions. Numerical simulations illustrate better transient performance of the proposed hybrid algorithm compared with the results of the existing continuous-time algorithms.

An Improved Gumowski–Mira Map with Symmetric Lyapunov Exponents

The symmetric Lyapunov exponents (LEs) are known to be an inherent property of continuous-time conservative systems. However, the research on this interesting phenomenon in a discrete-time chaotic map has not been reported. Thus, this paper presents an improved 2D chaotic map based on Gumowski–Mira (GM) transformation, which has a stable fixed point or an unstable fixed point depending on its control parameters. Furthermore, it can display symmetric LEs and an infinite number of coexisting attractors with different amplitudes and different shapes. To demonstrate the complex dynamics of the 2D chaotic map, this paper studies its control parameters related to dynamical behaviors employing numerical analysis methods. Then, the hardware implementation based on STM32 platform is established for illustrating the numerical simulation results. Next, the random performance of the 2D chaotic map is tested by NIST FIPS140-2 suite. Finally, an image encryption algorithm based on the 2D chaotic map is designed, and the results obtained reveal that the proposed chaotic map has excellent randomness and is more suitable for many chaos-based image encryptions.

Localization and delocalization properties in quasi-periodically-driven one-dimensional disordered systems.

Localization and delocalization of quantum diffusion in a time-continuous one-dimensional Anderson model perturbed by the quasiperiodic harmonic oscillations of M colors is investigated systematically, which has been partly reported by a preliminary Letter [H. S. Yamada and K. S. Ikeda, Phys. Rev. E 103, L040202 (2021)2470-004510.1103/PhysRevE.103.L040202]. We investigate in detail the localization-delocalization characteristics of the model with respect to three parameters: the disorder strength W, the perturbation strength ε, and the number of colors, M, which plays the similar role of spatial dimension. In particular, attention is focused on the presence of localization-delocalization transition (LDT) and its critical properties. For M≥3 the LDT exists and a normal diffusion is recovered above a critical strength ε, and the characteristics of diffusion dynamics mimic the diffusion process predicted for the stochastically perturbed Anderson model even though M is not large. These results are compared with the results of discrete-time quantum maps, i.e., the Anderson map and the standard map. Further, the features of delocalized dynamics are discussed in comparison with a limit model which has no static disordered part.

Global and Local Analysis for a Cournot Duopoly Game with Two Different Objective Functions

In this paper, a Cournot game with two competing firms is studied. The two competing firms seek the optimality of their quantities by maximizing two different objective functions. The first firm wants to maximize an average of social welfare and profit, while the second firm wants to maximize their relative profit only. We assume that both firms are rational, adopting a bounded rationality mechanism for updating their production outputs. A two-dimensional discrete time map is introduced to analyze the evolution of the game. The map has four equilibrium points and their stability conditions are investigated. We prove the Nash equilibrium point can be destabilized through flip bifurcation only. The obtained results show that the manifold of the game’s map can be analyzed through a one-dimensional map whose analytical form is similar to the well-known logistic map. The critical curves investigations show that the phase plane of game’s map is divided into three zones and, therefore, the map is not invertible. Finally, the contact bifurcation phenomena are discussed using simulation.

An approximate coupled cluster theory via nonlinear dynamics and synergetics: The adiabatic decoupling conditions.

The coupled cluster iteration scheme is analyzed as a multivariate discrete time map using nonlinear dynamics and synergetics. The nonlinearly coupled set of equations to determine the cluster amplitudes are driven by a fraction of the entire set of cluster amplitudes. These driver amplitudes enslave all other amplitudes through a synergistic inter-relationship, where the latter class of amplitudes behave as the auxiliary variables. The driver and the auxiliary variables exhibit vastly different time scales of relaxation during the iteration process to reach the fixed points. The fast varying auxiliary amplitudes are small in magnitude, while the driver amplitudes are large, and they have a much longer time scale of relaxation. Exploiting their difference in relaxation time scale, we employ an adiabatic decoupling approximation, where each of the fast relaxing auxiliary modes is expressed as a unique function of the principal amplitudes. This results in a tremendous reduction in the independent degrees of freedom. On the other hand, only the driver amplitudes are determined accurately via exact coupled cluster equations. We will demonstrate that the iteration scheme has an order of magnitude reduction in computational scaling than the conventional scheme. With a few pilot numerical examples, we would demonstrate that this scheme can achieve very high accuracy with significant savings in computational time.

Deep learning of conjugate mappings

Despite many of the most common chaotic dynamical systems being continuous in time, it is through discrete time mappings that much of the understanding of chaos is formed. Henri Poincaré first made this connection by tracking consecutive iterations of the continuous flow with a lower-dimensional, transverse subspace. The mapping that iterates the dynamics through consecutive intersections of the flow with the subspace is now referred to as a Poincaré map, and it is the primary method available for interpreting and classifying chaotic dynamics. Unfortunately, in all but the simplest systems, an explicit form for such a mapping remains outstanding. This work proposes a method for obtaining explicit Poincaré mappings by using deep learning to construct an invertible coordinate transformation into a conjugate representation where the dynamics are governed by a relatively simple chaotic mapping. The invertible change of variable is based on an autoencoder, which allows for dimensionality reduction, and has the advantage of classifying chaotic systems using the equivalence relation of topological conjugacies. Indeed, the enforcement of topological conjugacies is the critical neural network regularization for learning the coordinate and dynamics pairing. We provide expository applications of the method to low-dimensional systems such as the Rössler and Lorenz systems, while also demonstrating the utility of the method on infinite-dimensional systems, such as the Kuramoto–Sivashinsky equation.

Higher order sliding mode inspired nonlinear discrete-time observer

This work proposes a design scheme for arbitrary order discrete-time sliding mode observers for input-affine nonlinear systems. The dynamics of the estimation errors are represented in a pseudo-linear form, where the coefficients of the characteristic polynomial comprise the nonlinearities of the algorithm. The design process is reduced to a state-dependent eigenvalue placement procedure. Moreover, two different discrete-time eigenvalue mappings are proposed. As basis for the eigenvalue mappings serves a modified version of the continuous-time uniform robust exact differentiator. Due on the chosen eigenvalue mapping the proposed algorithm does not suffer from discretization chattering. Global asymptotic stability of the estimation errors for observers of order 2 and 3 is proven and the method to prove stability for higher order observers is demonstrated. The performance of a 3-rd order observer is illustrated in simulation. Simulation studies indicate that proposed discrete-time observer might possess an upper bound of its convergence time independent of the initial conditions.

Critical domain sizes of a discrete-map hybrid and reaction-diffusion model on hostile exterior domains

Abstract We study a hybrid impulsive reaction-diffusion equation composed with a discrete-time map in bounded domain $\varOmega $ in space dimension $n\in \mathbb N$. We assume that the exterior of domain is not lethal (not completely hostile) but hostile. We consider Robin boundary conditions which are used for mixed or reactive or semipermeable boundaries. Given geometry of the domain $\varOmega $, we establish critical domain sizes for the persistence and extinction of a species. Specifically, for habitats with the shape of $n$-hypercube and ball of fixed radius, we formulate the critical domain sizes depending on parameters of the model, including $h$, i.e. a measure of the hostility of the external (to $\varOmega $) environment. For a general habitat, called Lipschitz domains, we apply isoperimetric inequalities and variational methods to find the associated critical domain sizes. We also provide applications of the main results in marine reserve, terrestrial reserve and insect pest outbreaks.