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In this paper, we investigate the impact of functional shifts in a time-discrete cross-catalytic system. We use the hypercycle model considering that one of the species shifts from a cooperator to a degrader. At the bifurcation caused by this functional shift, an invariant curve collapses to a point P while, simultaneously, two fixed points collide with P in a transcritical bifurcation. Moreover, all points of a line containing P become fixed points at the bifurcation and only at the bifurcation in a degenerate scenario. We provide a complete analytical description of this degenerate bifurcation. As a result of our study, we prove the existence of the invariant curve arising from the transition to cooperation. [ABSTRACT FROM AUTHOR]

The problem of finding the optimal open-loop control for discrete-time stochastic dynamical systems is considered. It is assumed that the initial conditions and external influences are random. The average value of the Bolza functional defined on individual trajectories is minimized. It is proposed to solve the problem by means of classical and modified migrating optimization algorithms. The modification of the migrating algorithm consists of cloning the members of the initial population and choosing different strategies of migratory behavior for the main population and for populations formed by clones. At the final stage of the search for an extremum, an intensively clarifying migration cycle is implemented with the participation of three leaders of the populations participating in the search process. Problems of optimal control of bundles of trajectories of deterministic discrete dynamical systems, as well as individual trajectories, are considered as special cases. Seven model examples illustrating the performance of the proposed approach are solved. [ABSTRACT FROM AUTHOR]

Methods for the algorithmic construction of stability indicators of nonnegative matrices is described, and the application of these indicators to problems of modern mathematical biology and epidemiology is discussed. Specific features of such indicators when they are applied to problems about the parametric loss of stability of trivial equilibrium states of discrete dynamical systems are pointed out. Estimates of the efficiency of algorithms based on the proposed methods for the case of systems determined by sparse matrices are given. Examples of using the constructed algorithms for such systems are discussed. [ABSTRACT FROM AUTHOR]

Boolean Networks can be converted to discrete linear dynamical systems on finite spaces by a semi-tensor-product approach. This approach has been used by many to study the dynamics and control of Boolean systems. However, the process of getting the linear representation using the semi-tensor-product method is complicated even for a simple three-node network and requires the help of a computer program. In this work, we show that we can skip the semi-tensor process and obtain the same linear representation with a straightforward mapping. Moreover, our approach produces a large number of isomorphic representations which provides a flexible framework. Importantly, it could simplify the analytical study of networks with unspecified number of nodes that have some structure. [ABSTRACT FROM AUTHOR]

We present two types of systems of differential equations that can be derived from a set of discrete integrable systems which we call the closed geometric crystal chains. One is an extended Lotkaâ€"Volterra type system, and the other seems to be generally new but reduces to a previously known system in a special case. Both equations have Lax representations associated with what are known as the loop elementary symmetric functions, which were originally introduced to describe products of affine type A geometric crystals for symmetric tensor representations. Examples of the derivations of the continuous time Lax equations from a discrete time one are described in detail, where a novel method of taking a continuum limit by assuming asymptotic behaviors of the eigenvalues of the Lax matrix in Puiseux series expansions is used. [ABSTRACT FROM AUTHOR]

A new bio-inspired method for optimizing the objective function on a parallelepiped set of admissible solutions is proposed. It uses a model of the behavior of tomtits during the search for food. This algorithm combines some techniques for finding the extremum of the objective function, such as the memory matrix and the Levy flight from the cuckoo algorithm. The trajectories of tomtits are described by the jump-diffusion processes. The algorithm is applied to the classic and nonseparable optimal control problems for deterministic discrete dynamical systems. This type of control problem can often be solved using the discrete maximum principle or more general necessary optimality conditions, and the Bellman's equation, but sometimes it is extremely difficult or even impossible. For this reason, there is a need to create new methods to solve these problems. The new metaheuristic algorithm makes it possible to obtain solutions of acceptable quality in an acceptable time. The efficiency and analysis of this method are demonstrated by solving a number of optimal deterministic discrete open-loop control problems: nonlinear nonseparable problems (Luus–Tassone and Li–Haimes) and separable problems for linear control dynamical systems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give a complete description of the input–output methods for uniform exponential stability of discrete dynamical systems. We present a new study from four perspectives, in each case providing a deep analysis of the input–output criteria and of the axiomatic structure of the admissible pairs. The first stage is devoted to global conditions of uniform type for stability of discrete variational systems, in the most general case. Next, the results are applied to characterize the stability of discrete nonautonomous systems, without any assumptions on their coefficients. In both cases, the optimality of the methods is motivated by examples, showing that the hypotheses regarding the input and output spaces cannot be removed. The third method is focused on ergodic theory approaches, providing criteria of nonuniform type for stability of discrete variational systems. We characterize the uniform exponential stability by means of nonuniform input–output stabilities relative to ergodic measures, using some very general classes of sequence spaces. Thus, we extend our recent stability results obtained in Dragičević et al. (J Differ Equ 268:4786–4829, 2020). At the fourth stage, we prove even more, that in certain conditions, in the variational case, the exponential stability can be characterized in terms of stabilities along periodic orbits. [ABSTRACT FROM AUTHOR]

A new hybrid metaheuristic method for optimizing the objective function on a parallelepiped set of admissible solutions is proposed. It mimics the behavior of a school of river perch when looking for food. The algorithm uses the ideas of several methods: a frog-leaping method, migration algorithms, a cuckoo algorithm and a path-relinking procedure. As an application, a wide class of problems of finding the optimal control of deterministic discrete dynamical systems with a nonseparable performance criterion is chosen. For this class of optimization problems, it is difficult to apply the discrete maximum principle and its generalizations as a necessary optimality condition and the Bellman equation as a sufficient optimality condition. The desire to extend the class of problems to be solved to control problems of trajectory bundles and stochastic problems leads to the need to use not only classical adaptive random search procedures, but also new approaches combining the ideas of migration algorithms and swarm intelligence methods. The efficiency of this method is demonstrated and an analysis is performed by solving several optimal deterministic discrete control problems: two nonseparable problems (Luus–Tassone and LiHaimes) and five classic linear systems control problems with known exact solutions. [ABSTRACT FROM AUTHOR]

One of the applications of dynamical systems with chaotic behavior is data encryption. Chaos-based cryptography uses chaotic dynamical systems as the basis for creating algorithms. The present article discusses a new dynamical system called M-map with its analysis: fixed points, bifurcation diagram, Lyapunov exponent, and invariant density. The obtained bifurcation diagram and the plot of the Lyapunov exponent (with a minimum value of ln 2 and a maximum value of ln 4 ) suggest that the so-called robust chaos characterizes this map. Moreover, the obtained results are compared with other dynamical systems used in cryptography. Additionally, the article proposes a new image encryption algorithm. It uses, among others, cyclically shifted S-box or saving encrypted pixels on the first or last free space in the cipher-image. The conducted analysis shows that the cipher-images are characterized by an entropy value close to 8, a correlation of adjacent pixels value close to 0, or values of Number of Pixel of Change Rate ( N P C R ) and Unified Average Changing Intensity ( U A C I ) measures close to 100% and 33%, respectively. [ABSTRACT FROM AUTHOR]

A set of nonlinear feedback control strategies were designed to realize the bifurcation solutions of codimensional bifurcations in discrete dynamical systems with 1∶2 resonance from the perspective of bifurcation anti-controlling. Firstly, aimed at the limitation of traditional bifurcation criteria for determination of high codimensional bifurcation points, a new explicit criterion for codimension-2 bifurcation in 1∶2 resonance was proposed. Based on this explicit criterion, the linear control gain was designed to ensure the existence of such codimension-2 bifurcation. Then, the central manifold of 1∶2 resonance was derived. Based on the normal form method, the types and stability of codimension-2 bifurcation solutions in 1∶2 resonance were analyzed through design of nonlinear control gain. Finally, an Arneodo-Coullet-Tresser mapping was taken as an example, and various bifurcation solutions with 1∶2 resonance bifurcation properties were realized by control at the specified parameter points, to further validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

In the applied research of nonlinear system, the low degree of chaos in the dynamical system leads to the limitation of using the chaos method to solve some practical problems. In this paper, we use the product trigonometric function and ternary polynomial to build a dynamical system, which has strong chaotic characteristics. The dynamical system is constructed by two product trigonometric functions and a ternary linear equation, and its chaotic properties are verified by bifurcation diagrams, Lyapunov exponents, fractal dimensions, etc. The system has many parameters and large parameter intervals and is not prone to cycles. The conditions for the non-divergence of this system are given by mathematical derivation, and it is found that the linear part of the system can be replaced by an arbitrary ternary polynomial system and still not diverge, and the bifurcation diagram is drawn to verify it. Finally, the chaotic sequence is distributed more uniformly in the value domain space by adding the modulo operation. Then, the bit matrix of multiple images is directly permuted by the above system, and the experiment confirms that the histogram, information entropy, and pixel correlation of its encrypted images are satisfactory, as well as a very large key space. [ABSTRACT FROM AUTHOR]

Lawvere's open problem on quotient toposes has been solved for boolean Grothendieck toposes but not for non-boolean toposes. As a simple and non-trivial example of a non-boolean topos, this paper provides a complete classification of the quotient toposes of the topos of discrete dynamical systems, which, in this context, are sets equipped with an endofunction. This paper also offers an order-theoretic framework to address the open problem, particularly useful for locally connected toposes. Our result is deeply related to monoid epimorphisms. At the end of this paper, utilizing the theory of lax epimorphisms in the 2-category Cat, we explain how (non-surjective) monoid epimorphisms from N correspond to (non-periodic) behaviors in discrete dynamical systems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a way to associate "relevant" reductions to subgroupoids G of X×Z×X with G(0) finite. In order to obtain such a reduction we use singular value decomposition of a matrix associated with the groupoid. [ABSTRACT FROM AUTHOR]

A dynamical system describes the consequence of the current state of an event or particle in future. The models expressed by functions in the dynamical systems are more often deterministic, but these functions might also be stochastic in some cases. The prediction of the system's behavior in future is studied with the analytical solution of the implicit relations (Differential, Difference equations) and simulations. A discrete-time first order system of equations with quadratic nonlinearity is considered for study in this work. Classical approach of stability analysis using Jury's condition is employed to analyze the system's stability. The chaotic nature of the dynamical system is illustrated by the bifurcation theory. The enhancement of chaos is performed using Cosine Chaotification Technique (CCT). Simulations are carried out for different parameter values. [ABSTRACT FROM AUTHOR]

An effort to study one-dimensional nonuniform elementary number conserving cellular automata (NCCA) rules from an exponential order rule space of cellular automata is an excellent computational task. To perform this task effectively, a mathematical heritage under the number of conserving functions over binary strings of length n has been highlighted along with their number conserving cellular automata rules (either uniform or nonuniform). A basic approach for the construction of some feasible nonuniform NCCA rules of any finite configuration with the assistance of nine uniform elementary CA rules has been investigated. From our construction procedure, recurrence equations are formulated as suitably solved to ascertain the actual range of NCCA rules. The state transition diagrams (STDs) of NCCA rules are analyzed. While classifying the binary strings through STDs, we found a fascinating optical insight that equal weight strings from a class whose cardinality is the same as the binomial coefficient C (n , k) where n is the length and k is the weight of the binary string. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study some basic properties of the orbits of elements in dynamical systems defined by groups. The relation between orbit of an element and the orbit of its inverse element has been established. The nature of the orbit of the identity element in the dynamical system has been studied and a singleton set containing the identity element itself has been obtained. It is observed that the set of all fixed points in the dynamical system is strongly invariant (Sinvariant) and is a closed subgroup of the group. By using the topological conjugacy between dynamical systems defined by groups, a relation between the orbit of an element and the orbit of its image element in the other dynamical system has been obtained. Finally, we obtained that if an element is in the kernel of the homomorphism, then it must be eventually fixed at the identity element of the group. [ABSTRACT FROM AUTHOR]

Decreasing sandpiles model the dynamics of configurations where each position i ∈ N contains a finite number of stacked grains h i , such that h i ≥ h i + 1 (decrease property). Grains move according to a decreasing local rule R = (r 1 , r 2 , ... , r p) such that r j ≥ r j + 1 , meaning that r j grains move from columns i to i + j for all 1 ≤ j ≤ p , if it does not contradict the decrease property. We are interested in the fixed point reached starting from a finite number of grains on a unique column. In [21] , we proved the emergence of wave patterns periodically covering fixed points, for rules of the form (1 , ... , 1) (Kadanoff sandpile models). The present work is a significative extension: for large classes of decreasing sandpile model instances, we prove the emergence of patterns of various shapes periodically covering fixed points. We introduce new automata to analyze their asymptotic structure, and use the least action principle. The difficulty of understanding the behavior of sandpile models, despite the simplicity of the rules, is what makes the problem challenging. [ABSTRACT FROM AUTHOR]

Let X be a compact metric countable space, let f : X → X be a homeomorphism and let E(X, f) be its Ellis semigroup. Among other results we show that the following statements are equivalent: (1) (X, f) is equicontinuous, (2) (X, f) is distal and (3) every point is periodic. We use this result to give a direct proof of a theorem of Ellis saying that (X, f) is distal if, and only if, E(X, f) is a group. [ABSTRACT FROM AUTHOR]

We study numerically the α - and ω -limits of the Newton maps of quadratic polynomial transformations of the plane into itself. Our results confirm the conjectures posed in a recent work about the general dynamics of real Newton maps on the plane. [ABSTRACT FROM AUTHOR]

Many problems in biology and medicine have a control component. Often, the goal might be to modify intracellular networks, such as gene regulatory networks or signaling networks, in order for cells to achieve a certain phenotype, what happens in cancer. If the network is represented by a mathematical model for which mathematical control approaches are available, such as systems of ordinary differential equations, then this problem might be solved systematically. Such approaches are available for some other model types, such as Boolean networks, where structure-based approaches have been developed, as well as stable motif techniques. However, increasingly many published discrete models are mixed-state or multistate, that is, some or all variables have more than two states, and thus the development of control strategies for multistate networks is needed. This paper presents a control approach broadly applicable to general multistate models based on encoding them as polynomial dynamical systems over a finite algebraic state set, and using computational algebra for finding appropriate intervention strategies. To demonstrate the feasibility and applicability of this method, we apply it to a recently developed multistate intracellular model of E2F-mediated bladder cancerous growth and to a model linking intracellular iron metabolism and oncogenic pathways. The control strategies identified for these published models are novel in some cases and represent new hypotheses, or are supported by the literature in others as potential drug targets. Our Macaulay2 scripts to find control strategies are publicly available through GitHub at https://github.com/luissv7/multistatepdscontrol. [ABSTRACT FROM AUTHOR]

This paper tries to investigate the complex characteristics behind quantity competition using generalized relative profit functions. For this purpose, we study the dynamics of a nonlinear Cournot - type duopoly game with differentiated goods and generalized relative profit maximization. We suppose a linear demand and asymmetric cost functions. The game is modelled with a system of two difference equations. Results concerning the equilibria of the economic model and their stability are presented and the occurrence of bifurcations is stated. We show that the model gives more complex, chaotic and unpredictable trajectories. Numerical experiments are presented. [ABSTRACT FROM AUTHOR]

We present properties of equivalence classes of the codivergency relation defined for a Brouwer homeomorphism for which there exists a family of invariant pairwise disjoint lines covering the plane. In particular, using the codivergency relation we describe the sets of regular and irregular points for such Brouwer homeomorphisms. Moreover, we show that under this assumption the interior of each equivalence class of this relation is invariant and simply connected. [ABSTRACT FROM AUTHOR]

We introduce a new class of locally compact groups, namely the strongly compactly covered groups, which are the Hausdorff topological groups G such that every element of G is contained in a compact open normal subgroup of G. For continuous endomorphisms ϕ : G → G of these groups we compute the algebraic entropy and study its properties. Also an Addition Theorem is available under suitable conditions. [ABSTRACT FROM AUTHOR]

Given a discrete dynamical system (X, f), we consider the function wf -limit set from X to 2X as wf (x) = {y ∊ X : there exists a sequence of positive integers n1 < n2 < ... such that lim k→∞ fnk (x) = y}, for each x ∊ X. In the article [1], A. M. Bruckner and J. Ceder es- tablished several conditions which are equivalent to the continuity of the function wf where f : [0, 1] → [0, 1] is continuous surjection. It is natural to ask whether or not some results of [1] can be extended to finite graphs. In this direction, we study the function wf when the phase space is a n-od simple T. We prove that if wf is a continuous map, then Fix(f²) and Fix(f³) are connected sets. We will provide examples to show that the inverse implication fails when the phase space is a simple triod. However, we will prove that. [ABSTRACT FROM AUTHOR]

We obtain new characterizations for nonuniform and uniform asymptotic behaviors of variational systems in infinite dimensional spaces, following the line of studies recently developed in Dragičević et al. (J Differ Equ 268:4786–4829, 2020, J Dyn Differ Equ 34:1107–1137, 2022, Appl Math Comput 414:1–22, 2022, J Math Anal Appl 515:1–37, 2022). First, we give complete descriptions for both nonuniform and uniform stability by means of some nonuniform conditions of convergence for series of suitable nonlinear trajectories. We apply our criteria to present a novel method of exploring the robustness of the nonuniform exponential stability under additive and multiplicative perturbations and we also deduce some consequences for the robustness of the uniform exponential stability. After that, we obtain characterizations for nonuniform and uniform instability in terms of certain nonuniform conditions imposed to the sums of some well-chosen series of nonlinear trajectories. We apply our results to provide a new technique of exploring the robustness of the nonuniform exponential instability under additive and multiplicative perturbations and discuss the consequences for uniform exponential instability. Hence, for both stability and instability we generalize the previous Zabczyk type results and extend their applicability. Next, as another class of applications, we provide nonuniform criteria of Rolewicz type for stability and instability of skew-product semiflows. Thus, we extend the Zabczyk–Rolewicz type methods in two directions: by giving local conditions for global behaviors and by employing specific nonuniform boundedness conditions. Our criteria can be applied to broad classes of discrete and continuous variational dynamical systems. [ABSTRACT FROM AUTHOR]

Abstract Results and tools on discrete interaction networks are often concerned with Boolean variables, whereas considering more than two levels is sometimes useful. Multivalued networks can be converted to partial Boolean maps, in a way that preserves the asynchronous dynamics. We investigate the problem of extending these maps to non-admissible states, i.e. states that do not have a multivalued counterpart. We observe that attractors are preserved if a stepwise version of the original function is considered for conversion. Different extensions of the Boolean conversion affect the structure of the interaction graphs in different ways. A particular technique for extending the partial Boolean conversion is identified, that ensures that feedback cycles are preserved. This property, combined with the conservation of the asymptotic behaviour, can prove useful for the application of results and analyses defined in the Boolean setting to multivalued networks, and vice versa. As a first application, by considering the conversion of a known example for the discrete multivalued case, we create a Boolean map showing that the existence of a cyclic attractor and the absence of fixed points are compatible with the absence of local negative cycles. We then state a multivalued version of a result connecting mirror states and local feedback cycles. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present several dichotomy properties of nonautonomous systems in Banach spaces. We discuss the connections between discrete-time and continuous-time dichotomic behavior on the whole line. The key points of the method are the discrete asymptotic properties and the input-output techniques. As consequences, we present several applications and criteria for exponential dichotomy of evolution families on the whole line, which extend the previous results in this framework. [ABSTRACT FROM AUTHOR]

Let E(X, f) be the Ellis semigroup of a dynamical system (X, f) where X is a compact metric space. We analyze the cardinality of E(X, f) for a compact countable metric space X. A characterization when E(X, f) and E(X,f)∗=E(X,f)\{fn:n∈N} are both finite is given. We show that if the collection of all periods of the periodic points of (X, f) is infinite, then E(X, f) has size 2ℵ0. It is also proved that if (X, f) has a point with a dense orbit and all elements of E(X, f) are continuous, then |E(X,f)|≤|X|. For dynamical systems of the form (ω2+1,f), we show that if there is a point with a dense orbit, then all elements of E(ω2+1,f) are continuous functions. We present several examples of dynamical systems which have a point with a dense orbit. Such systems provide examples where E(ω2+1,f) and ω2+1 are homeomorphic but not algebraically homeomorphic, where ω2+1 is taken with the usual ordinal addition as semigroup operation. [ABSTRACT FROM AUTHOR]

Markov map is one example of interval maps where it is a piecewise expanding map and obeys the Markov property. One well-known example of Markov map is the doubling map, a map which has two subintervals with equal partitions. In this paper, we are interested to investigate another type of Markov map, the so-called skewed doubling map. This map is a more generalized map than the doubling map. Thus, the aims of this paper are to find the fixed points as well as the periodic points for the skewed doubling map and to investigate the sensitive dependence on initial conditions of this map. The method considered here is the cobweb diagram. Numerical results suggest that there exist dense of periodic orbits for this map. The sensitivity of this map to initial conditions is also verified where small differences in initial conditions give different behaviour of the orbits in the map. [ABSTRACT FROM AUTHOR]

We present properties of sets of invariant lines for Brouwer homeomorphisms which are not necessarily embeddable in a flow. Using such lines we describe the structure of equivalence classes of the codivergency relation. We also obtain a result concerning the set of regular points. [ABSTRACT FROM AUTHOR]

Many discrete models of biological networks rely exclusively on Boolean variables and many tools and theorems are available for analysis of strictly Boolean models. However, multilevel variables are often required to account for threshold effects, in which knowledge of the Boolean case does not generalise straightforwardly. This motivated the development of conversion methods for multilevel to Boolean models. In particular, Van Ham’s method has been shown to yield a one-to-one, neighbour and regulation preserving dynamics, making it the de facto standard approach to the problem. However, Van Ham’s method has several drawbacks: most notably, it introduces vast regions of “non-admissible” states that have no counterpart in the multilevel, original model. This raises special difficulties for the analysis of interaction between variables and circuit functionality, which is believed to be central to the understanding of dynamic properties of logical models. Here, we propose a new multilevel to Boolean conversion method, with software implementation. Contrary to Van Ham’s, our method doesn’t yield a one-to-one transposition of multilevel trajectories; however, it maps each and every Boolean state to a specific multilevel state, thus getting rid of the non-admissible regions and, at the expense of (apparently) more complicated, “parallel” trajectories. One of the prominent features of our method is that it preserves dynamics and interaction of variables in a certain manner. As a demonstration of the usability of our method, we apply it to construct a new Boolean counter-example to the well-known conjecture that a local negative circuit is necessary to generate sustained oscillations. This result illustrates the general relevance of our method for the study of multilevel logical models. [ABSTRACT FROM AUTHOR]

In this paper, I continue the study of the mathematical models presented in [J. C. Larsen, Models of cancer growth, J. Appl. Math. Comput. 53(1-2) (2015) 613-645] and [J. C. Larsen, The bistability theorem in a model of metastatic cancer, to appear in Appl. Math.]. I shall prove the bistability theorem for the ODE model from [Larsen, 2015]. It is a mass action kinetic system in the variables cancer, GF growth factor and GI growth inhibitor. This theorem says that for some values of the parameters, there exist two positive singular points , of the vector field. Here and is stable and is unstable, see Sec. 2. There is also a discrete model in [Larsen, 2015], it is a linear map () on three-dimensional Euclidean vector space with variables where these variables have the same meaning as in the ODE model above. In [Larsen, 2015], I showed that one can sometimes find affine vector fields on three-dimensional Euclidean vector space whose time one map is . I shall also show this in the present paper in a more general setting than in [Larsen, 2015]. This enables me to find an expression for the rate of change of cancer growth on the coordinate hyperplane in Euclidean vector space. I also present an ODE model of cancer metastasis with variables where is primary cancer and is metastatic cancer and GF, GI are growth factors and growth inhibitors, respectively. [ABSTRACT FROM AUTHOR]

Consider the difference equation →xn+1 = f(n,→xn,…,→xn-k), n = 0,1,.. where kϵ{0; 1…} and the initial conditions are real vectors. We investigate the asymptotic behavior of the solutions of the considered equation. We give some effective conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation. Our results are based on application of the linearizations technique. We illustrate our results with many examples that include some equations from mathematical biology. [ABSTRACT FROM AUTHOR]

We introduce theEllis semigroupof a nonautonomous discrete dynamical system (X,f1,∞) whenXis a metric compact space. The underlying set of this semigroup is the pointwise closure ofin the spaceXX. By using the convergence of a sequence of points with respect to an ultrafilter it is possible to give a precise description of the semigroup and its operation. This notion extends the classical Ellis semigroup of a discrete dynamical system. We show several properties that connect this semigroup and the topological properties of the nonautonomous discrete dynamical system. [ABSTRACT FROM AUTHOR]

Suppose that a closed surface S ⊆ R 3 is an attractor, not necessarily global, for a discrete dynamical system. Assuming that its set of wild points W is totally disconnected, we prove that (up to an ambient homeomorphism) it has to be contained in a straight line. As a corollary we show that there exist uncountably many different 2-spheres in R 3 none of which can be realized as an attractor for a homeomorphism. Our techniques hinge on a quantity r ( K ) that can be defined for any compact set K ⊆ R 3 and is related to “how wildly” it sits in R 3 . We establish the topological results that (i) r ( W ) ≤ r ( S ) and (ii) any totally disconnected set having a finite r must be contained in a straight line (up to an ambient homeomorphism). The main result follows from these and the fact that attractors have a finite r . [ABSTRACT FROM AUTHOR]

nConsider the difference equation x→n+1 = f(n,x→n,...,x→n-k), n=0,1,..., where k ∊ {0,1,...} and the initial conditions are real vectors. We investigate the asymptotic behavior of the solutions of the considered equation. We give some effective conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation. Our results are based on application of the linearizations technique. We illustrate our results with many examples that include some equations from mathematical biology. [ABSTRACT FROM AUTHOR]

We obtain sufficient conditions for total topological transitivity (transitivity of all iterations) for a class of C skew products defined on cells in ℝ, n ≥ 2. [ABSTRACT FROM AUTHOR]

This paper provides a collection of mathematical and computational tools for the study of robustness in nonlinear gene regulatory networks, represented by time- and state-discrete dynamical systems taking on multiple states. The focus is on networks governed by nested canalizing functions (NCFs), first introduced in the Boolean context by S. Kauffman. After giving a general definition of NCFs we analyze the class of such functions. We derive a formula for the normalized average c -sensitivities of multistate NCFs, which enables the calculation of the Derrida plot, a popular measure of network stability. We also provide a unique canonical parametrized polynomial form of NCFs. This form has several consequences. We can easily generate NCFs for varying parameter choices, and derive a closed form formula for the number of such functions in a given number of variables, as well as an asymptotic formula. Finally, we compute the number of equivalence classes of NCFs under permutation of variables. Together, the results of the paper represent a useful mathematical framework for the study of NCFs and their dynamic networks. [ABSTRACT FROM AUTHOR]

In this study we investigate the dynamics of a nonlinear Cournot- type duopoly game with differentiated goods, linear demand and a cost function that includes emission costs. The game is modeled with a system of two difference equations. Existence and stability of equilibria of this system are studied. We show that the model gives more complex chaotic and unpredictable trajectories as a consequence of change in the parameter of horizontal product differentiation and a higher (lower) degree of product differentiation (weaker or fiercer competition) destabilize (stabilize) the economy. The chaotic features are justified numerically via computing Lyapunov numbers and sensitive dependence on initial conditions. Also, we show that in this case there are stable trajectories and a higher (lower) degree of product differentiation does not tend to destabilize the economy. [ABSTRACT FROM AUTHOR]

A fractal structure is a countable family of coverings which displays accurate information about the irregularities that a set presents when being explored with enough level of detail. It is worth noting that fractal structures become especially appropriate to provide new definitions of fractal dimension, which constitutes a valuablemeasure to test for chaos in dynamical systems. In this paper, we explore several approaches to calculate the fractal dimension of a subset with respect to a fractal structure. These models generalize the classical box dimension in the context of Euclidean subspaces from a discrete viewpoint. To illustrate the flexibility of the new models, we calculate the fractal dimension of a family of self-affine sets associated with certain discrete dynamical systems. [ABSTRACT FROM AUTHOR]

Immune memories of antigens are formed as limit behavior of antibody dynamics, which is a simplified model of cyclic idiotypic immune networks. Rapid responses of immune systems to secondary antigens are due to the memory recalls of the stored antigenic attributes. The immune-memory mechanism is studied by combining network structure and dynamical systems. Moreover, associative memory can be explored by network dynamics of a cross-correlation matrix determined by an antibody chain. Analysis of cross-reactive immune responses can be accomplished by introducing the overlapping difference function ρ of network dynamics. Classification of four types of memory recalls was performed here. It explains the phenomena of associative memory and provides methods for control of vaccinations via tuning of suitable parameters. Other issues with immune memory and cyclic idiotypic immune networks are also discussed. [ABSTRACT FROM AUTHOR]

A model is considered for a spatially distributed population of male and female individuals that mate and reproduce only once in their life during a very short reproductive season. Between birth and mating, females and males move by diffusion on a bounded domain $$\Omega $$ under Dirichlet boundary conditions. Mating and reproduction are described by a (positively) homogeneous function (of degree one). We identify a basic reproduction number $${\mathcal {R}}_0$$ that acts as a threshold between extinction and persistence. If $${\mathcal {R}}_0 <1$$ , the population dies out while it persists (uniformly weakly) if $${\mathcal {R}}_0 > 1$$ . $${\mathcal {R}}_0$$ is the cone spectral radius of a bounded homogeneous map. [ABSTRACT FROM AUTHOR]

We introduce the algebraic entropy for endomorphisms of arbitrary abelian groups, appropriately modifying existing notions of entropy. The basic properties of the algebraic entropy are given, as well as various examples. The main result of this paper is the Addition Theorem showing that the algebraic entropy is additive in appropriate sense with respect to invariant subgroups. We give several applications of the Addition Theorem, among them the Uniqueness Theorem for the algebraic entropy in the category of all abelian groups and their endomorphisms. Furthermore, we point out the delicate connection of the algebraic entropy with the Mahler measure and Lehmer Problem in Number Theory. [ABSTRACT FROM AUTHOR]

Can the interaction graph of a finite dynamical system force this system to have a “complex” dynamics? In other words, given a finite interval of integers A , which are the signed digraphs G such that every finite dynamical system f : A n → A n with G as interaction graph has a “complex” dynamics? If | A | ≥ 3 we prove that no such signed digraph exists. More precisely, we prove that for every signed digraph G there exists a system f : A n → A n with G as interaction graph that converges toward a unique fixed point in at most ⌊ log 2 ⁡ n ⌋ + 2 steps. The boolean case | A | = 2 is more difficult, and we provide partial answers instead. We exhibit large classes of unsigned digraphs which admit boolean dynamical systems which converge toward a unique fixed point in polynomial, linear or constant time. [ABSTRACT FROM AUTHOR]

In this paper, a discrete dynamical system (DDS) is derived from the generalized Navier-Stokes equations for incompressible flow in porous media via a Galerkin procedure. The main difference from the previously studied poor man's Navier-Stokes equations is the addition of forcing terms accounting for linear and nonlinear drag forces of the medium - Darcy and Forchheimer terms. A detailed numerical investigation focusing on the bifurcation parameters due to these additional terms is provided in the form of regime maps, time series, power spectra, phase portraits and basins of attraction, which indicate system behaviors in agreement with expected physical fluid flow through porous media. As concluded from the previous studies, this DDS can be employed in subgrid-scale models of synthetic-velocity form for large-eddy simulation of turbulent flow through porous media. [ABSTRACT FROM AUTHOR]

We obtain some results on the dynamics of maps F ( x 1 , x 2 , … , x n ) = ( f σ ( 1 ) ( x σ ( 1 ) ) , f σ ( 2 ) ( x σ ( 2 ) ) , … , f σ ( n ) ( x σ ( n ) ) ) (we call them cyclically permuted direct product maps), defined from the Cartesian product X 1 × X 2 × ⋯ × X n into itself, where X 1 , X 2 , … , X n are general topological spaces, each map f σ ( i ) : X σ ( i ) → X i is continuous, i = 1 , … , n , and σ is a cyclic permutation of { 1 , 2 , … , n } , n ≥ 2 . We study the topics of (totally) topological transitivity and (weakly) topological mixing for cyclically permuted direct product maps from the following point of view: we analyze the relationship between the dynamics of F and that of the compositions f σ ( i ) ∘ … ∘ f σ n ( i ) , i ∈ { 1 , … , n } . [ABSTRACT FROM AUTHOR]

This article reports an algebraic criterion of the eigenvalue assignment, transversality condition and non-resonance condition for fold-N–S bifurcations. By means of symbolic computation methods, we propose an algorithmic approach for systematically analyzing codimension 2 bifurcations for high-dimensional discrete systems. The effectiveness of the proposed symbolic approach is verified by experiments. In particular, the flip- and fold-N–S bifurcations of a five-dimensional discrete dynamical system with a washout-filter feedback controller are analyzed. [ABSTRACT FROM AUTHOR]

We consider two new granular dynamical systems obtained from the Sand Pile Model SPM(n) by adding a smoothness condition. First, we define the Smooth Sand Pile Model SmSPM(n) and we provide a characterization of the reachable states, together with some interesting properties of the resulting lattice. Then we extend it to SmSPM(n), a related dynamical system with a more complex lattice structure. [ABSTRACT FROM AUTHOR]