Discrete Dynamical Systems Research Articles

Vanilla Feedforward Neural Networks as a Discretization of Dynamical Systems

Vanilla Feedforward Neural Networks as a Discretization of Dynamical Systems

Huygens synchronization of three aligned clocks

AbstractThis study examines the synchronization of three identical oscillators arranged in an array and coupled by small impacts, wherein each oscillator interacts solely with its nearest neighbour. The synchronized state, which is asymptotically stable, is characterized by phase opposition among alternating oscillators. We analyse the system using a non-linear discrete dynamical system based on a difference equation derived from the iteration of a plane diffeomorphism. We illustrate these results with the application to a system of three aligned Andronov clocks, showcasing their applicability to a broad range of oscillator systems.

Universal dynamical function behind all genetic codes: [formula omitted]-adic attractor dynamical model

The genetic code is a map which gives the correspondence between codons in DNA and amino acids. In the attractor dynamical model (ADM), genetic codes can be described as the sets of the cyclic attractors of discrete dynamical systems - the iterations of functions acting in the ring of 2-adic integers Z2. This ring arises from representation of nucleotides by binary vectors and hence codons by triples of binary vectors. We construct a Universal Function B such that the dynamical functions for all known genetic codes can be obtained from B by simple transformations on the set of codon cycles - the “Addition” and “Division” operations. ADM can be employed for study of phylogenetic dynamics of genetic codes. One can speculate that the “common ancestor genetic code” was caused by B. We remark that this function has 24 cyclic attractors which distribution coincides with the distribution for the hypothetical pre-LUCA code. This coupling of the Universal Function with the pre-LUCA code assigns the genetic codes evolution perspective to ADM. All genetic codes are generated from B through the special chains of the “Addition” and “Division” operations. The challenging problem is to assign the biological meaning to these mathematical operations.

Non-standard Lagrangians and a model of branched Hamiltonians

Take up some examples of multi-valued structure of the governing Hamiltonians namely, the branched Hamiltonians are explored, as recently advocated by Shapere and Wilczek.These are in fact cases of switchback potential, in the continuous interpolation of discrete time dynamical systems that exhibit chaotic behaviour enabling incorporation of a canonical quasi-Hamiltonian formalism

A dynamical system analysis of non-interacting cold dark matter and dark energy at perturbative level

This work deals with a cosmological model consisting of cold dark matter and dark energy without any interaction. The study has been done both at the background level and at the perturbative level for the homogeneous and isotropic FLRW spacetime. By suitable transformation of the variables, the cosmic evolution equations are converted into an autonomous system. A discrete dynamical system analysis has been employed for cosmic inferences.

New cluster algebras from old: integrability beyond Zamolodchikov periodicity

Abstract We consider discrete dynamical systems obtained as deformations of mutations in cluster algebras associated with finite-dimensional simple Lie algebras. The original (undeformed) dynamical systems provide the simplest examples of Zamolodchikov periodicity: they are affine birational maps for which every orbit is periodic with the same period. Following on from preliminary work by one of us with Kouloukas, here we present integrable maps obtained from deformations of cluster mutations related to the simple root systems 
$A_3$, $B_2$, $B_3$ and $D_4$. We further show how new cluster algebras arise, by considering Laurentification, that is, a lifting to a higher-dimensional map expressed in a set of new variables (tau functions), for which the dynamics exhibits the Laurent property. For the integrable map obtained by deformation of type $A_3$, which already appeared in our previous work, we show that there is a commuting map of Quispel-Roberts-Thompson (QRT) type which is built from a composition of mutations and a permutation applied to 
the same cluster algebra of rank 6, with an additional 2 frozen variables. Furthermore, both the deformed $A_3$ map and the QRT map correspond to translation by a generator in the Mordell-Weil group of a rational elliptic surface of rank two, and the underlying cluster algebra comes from a quiver that is mutation equivalent to the $q$-Painlev\'e III quiver found by Okubo. 
The deformed integrable maps of types $B_2$, $B_3$ and $D_4$ are also related to elliptic surfaces. 

From a dynamical systems viewpoint, the message of the paper is that special families of birational maps with completely periodic dynamics under iteration admit natural deformations that are aperiodic yet completely integrable.

A discrete fractional order cournot duopoly game model with relative profit delegation: Stability, bifurcation, chaos, 0-1 testing and control

Due to the memory effect, fractional order dynamical systems provide more realistic results compared with ordinary counterparts. In this study, we consider a Cournot-duopoly game model with relative profit delegation in the sense of Caputo fractional derivative. To describe richer dynamical behavior such as chaos in the model, a discrete dynamical system is needed. As a result of the discretization method based on the use of piecewise constant arguments, we obtain a two dimensional system of difference equations. The stability conditions of all equilibrium points of the discrete dynamical system are given comprehensively. The existence of the flip bifurcation in the system has been demonstrated theoretically. Lyapunov exponents and 0–1 test chaos imply that chaotic structures are formed as a result of this bifurcation. In addition, we present the chaos control technique such as Pyragas method to eliminate chaos in the model. All theoretical results dealing with the stability, bifurcation and chaos in the model are stimulated by numerical simulations.

Interpersonal strategy for controlling unpredictable opponents in soft tennis

Competition in sports, unlike cooperation in everyday life, does not involve a single solution because individuals aim to behave unpredictably, thereby preventing others from predicting their actions. This study determined how individuals in court-based sports attempted to control others’ unpredictable behaviors, addressing the gap in previous studies regarding the lack of clarity around strategies employed by individuals in competitive situations. We achieved this by applying a switching hybrid dynamics model, considering external inputs to analyze individual behaviors. Consequently, the study indicates that skilled individuals, in contrast to intermediate players, exhibit greater consistency in their behaviors. These skilled individuals lead others to anticipate their consistency and subsequently employ strategies to disrupt these expectations. This strategy exploits the principles of active human inference, implying that competition involves cooperation. This was revealed by an analysis of both human decision-making and behavior in actual matches as discrete and continuous dynamical systems. This interpersonal strategy could assist policymakers in the field of everyday life to enhance competitiveness. This strategy enables policymakers to adopt new policies that promote cooperation with competitors, ultimately increasing competitiveness in various aspects of our daily lives.

Stabilization of Nonlinear Continuous–Discrete Dynamic Systems with a Constant Sampling Step

Stabilization of Nonlinear Continuous–Discrete Dynamic Systems with a Constant Sampling Step

Stability and Convergence Analysis of the Discrete Dynamical System for Simulating a Moving Bed

The efficiency of controlling the simulated moving bed (SMB) has long been a critical issue in the chemical engineering industry. Most existing research relies on finite element methods, which often result in lower control efficiency and are unable to achieve online control. To enhance control over the SMB process, this paper employs the Crank–Nicolson method to develop a discrete dynamical model. This approach allows for the investigation of system stability and convergence, fundamentally addressing the sources of error. During the discretization of partial differential equations (PDEs), two main types of errors arise: intrinsic errors from the method itself and truncation errors due to derivative approximations and the discretization process. Research indicates that for the former, the iterative process remains convergent as long as the time and spatial steps are sufficiently small. Regarding truncation errors, studies have demonstrated that they exhibit second-order behavior relative to time and spatial steps. The theoretical validation shows that the iteration works effectively, and simulations confirm that the finite difference method is stable and performs well with varying SMB system parameters and controller processes. This provides a solid theoretical foundation for practical, real-time online control.

Analyzing Stability and Controlling Chaos in a Unique Host Parasitoid Discrete Dynamical System with Applications in Artificial Intelligence

Analyzing Stability and Controlling Chaos in a Unique Host Parasitoid Discrete Dynamical System with Applications in Artificial Intelligence

Furstenberg Family and Chaos for Time-Varying Discrete Dynamical Systems.

Assume that (Y,ρ) is a nontrivial complete metric space, and that (Y,g1,∞) is a time-varying discrete dynamical system (T-VDDS), which is given by sequences (gl)l=1∞ of continuous selfmaps gl:Y→Y. In this paper, for a given Furstenberg family G and a given T-VDDS (Y,g1,∞), G-scrambled pairs of points of the system (Y,g1,∞) (which contains the well-known scrambled pairs) are provided. Some properties of the set of G-scrambled pairs of a given T-VDDS (Y,g1,∞) are studied. Moreover, the generically G-chaotic T-VDDS and the generically strongly G-chaotic T-VDDS are defined. A sufficient condition for a given T-VDDS to be generically strongly G-chaotic is also presented.

Research on Stability and Bifurcation for Two-Dimensional Two-Parameter Squared Discrete Dynamical Systems

This study investigates a class of two-dimensional, two-parameter squared discrete dynamical systems. It determines the conditions for local stability at the fixed points for these proposed systems. Theoretical and numerical analyses are conducted to examine the bifurcation behavior of the proposed systems. Conditions for the existence of Naimark–Sacker bifurcation, transcritical bifurcation, and flip bifurcation are derived using center manifold theorem and bifurcation theory. Results of the theoretical analyses are validated by numerical simulation studies. Numerical simulations also reveal the complex bifurcation behaviors exhibited by the proposed systems and their advantage in image encryption.

Mapping Petri Nets onto a Calculus of Context-Aware Ambients

Petri nets are a graphical notation for describing a class of discrete event dynamic systems whose behaviours are characterised by concurrency, synchronisation, mutual exclusion and conflict. They have been used over the years for the modelling of various distributed systems applications. With the advent of pervasive systems and the Internet of Things, the Calculus of Context-aware Ambients (CCA) has emerged as a suitable formal notation for analysing the behaviours of these systems. In this paper, we are interested in comparing the expressive power of Petri nets to that of CCA. That is, can the class of systems represented by Petri nets be modelled in CCA? To answer this question, an algorithm is proposed that maps any Petri net onto a CCA process. We prove that a Petri net and its corresponding CCA process are behavioural equivalent. It follows that CCA is at least as expressive as Petri nets, i.e., any system that can be specified in Petri nets can also be specified in CCA. Moreover, tools developed for CCA can also be used to analyse the behaviours of Petri nets.

Decoding of MDP Convolutional Codes over the Erasure Channel under Linear Systems Point of View

This paper attempts to highlight the decoding capabilities of MDP convolutional codes over the erasure channel by defining them as discrete linear dynamical systems, with which the controllability property and the observability characteristics of linear system theory can be applied, in particular those of output observability, easily described using matrix language. Those are viewed against the decoding capabilities of MDS block codes over the same channel. Not only is the time complexity better but the decoding capabilities are also increased with this approach because convolutional codes are more flexible in handling variable-length data streams than block codes, where they are fixed-length and less adaptable to varying data lengths without padding or other adjustments.

Stability analysis of discrete dynamical system using khan-iteration technique

Stability analysis of discrete dynamical system using khan-iteration technique

Bifurcation and Chaos in a Fractional-Order Cournot Duopoly Game Model on Scale-Free Networks

In this study, a Cournot duopoly model describing Caputo fractional-order differential equations with piecewise constant arguments is discussed. We have obtained two-dimensional discrete dynamical system as a result of applying the discretization process to the model. By using the center manifold theory and the bifurcation theory, it is shown that the discrete dynamical system undergoes flip bifurcation about the Nash equilibrium point. Phase portraits, bifurcation diagrams, and Lyapunov exponents show the existence of many complex dynamical behaviors in the model such as the stable equilibrium point, period-2 orbit, period-4 orbit, period-8 orbit, period-16 orbit, and chaos according to changing the speed of the adjustment parameter [Formula: see text]. The discrete Cournot duopoly game model is also considered on two scale-free networks with different numbers of nodes. It is observed that the complex dynamical networks exhibit similar dynamical behaviors such as the stable equilibrium point, flip bifurcation, and chaos depending on changing the coupling strength parameter [Formula: see text]. Moreover, flip bifurcation and transition chaos take place earlier in more heterogeneous networks. Calculating the largest Lyapunov exponents guarantees the transition from nonchaotic to chaotic states in complex dynamical networks.

Simulating Bluetooth virus propagation on the real map via infectious attenuation algorithm and discrete dynamical system

Simulating Bluetooth virus propagation on the real map via infectious attenuation algorithm and discrete dynamical system

Elementary Catastrophe’s Chaos in One-Dimensional Discrete Systems Based on Nonlinear Connections and Deviation Curvature Statistics

This study shows, by means of numerical analysis, that the characteristics of discrete dynamical systems, in which chaos and catastrophe coexist, are closely related to the geometric statistics in Finsler geometry. The two geometric statistics introduced are nonlinear connections information, denoted as [Formula: see text], and the mean deviation curvature, denoted as [Formula: see text]. The quantity [Formula: see text] can be used to determine the occurrence of chaos in terms of nonequilibrium stability. The resulting chaos is characterized by [Formula: see text] in terms of the trajectory’s robustness, which is related to the localization or globalization of chaos. The characteristics of catastrophe-induced chaos are clearly visualized through the contour topography of [Formula: see text], in which an abrupt change is represented by cliff topography (i.e. a line of critical points); initial dependence is reflected in the reversibility of topographic patterns. On overlaying the contour topography with the singularity pattern, it is evident that chaos does not arise around the singular point. Furthermore, the extensive development of cusp and butterfly chaos demands information on the nonlinear connections within the singularity pattern. The asymmetry in swallowtail chaos is less distinguishable in an equilibrated state, but becomes more evident when the system is in a state of nonequilibrium. In many analyses, chaos and catastrophe are examined separately. However, these results demonstrate that when both are present, the two have a complex relationship constrained by the singularity.

Synchronous Dynamical Systems on Directed Acyclic Graphs: Complexity and Algorithms

Discrete dynamical systems serve as useful formal models to study diffusion phenomena in social networks. Several recent articles have studied the algorithmic and complexity aspects of some decision problems on synchronous Boolean networks, which are discrete dynamical systems whose underlying graphs are directed, and may contain directed cycles. Such problems can be regarded as reachability problems in the phase space of the corresponding dynamical system. Previous work has shown that some of these decision problems become efficiently solvable for systems on directed acyclic graphs (DAGs). Motivated by this line of work, we investigate a number of decision problems for dynamical systems whose underlying graphs are DAGs. We show that computational intractability (i.e., PSPACE -completeness) results for reachability problems hold even for dynamical systems on DAGs. We also identify some restricted versions of dynamical systems on DAGs for which reachability problem can be solved efficiently. In addition, we show that a decision problem (namely, Convergence), which is efficiently solvable for dynamical systems on DAGs, becomes PSPACE -complete for Quasi-DAGs (i.e., graphs that become DAGs by the removal of a single edge). In the process of establishing the above results, we also develop several structural properties of the phase spaces of dynamical systems on DAGs.