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This paper introduces self-similarity inherent in planar Milich-Jennings centered flip graphs derived from the Narayana sequence. We show that self-similarity found in a Narayana sequence yields a connected spanning subgraph with a centered flip. This paper has several main results (1) Every Narayana sequence constructs a flip graph, (2) Every Narayana sequence is self-similar and (3) Every Pascal 3-triangle has a free group presentation.

Complexity may be one of the most important measurements for analysing time series data; it covers or is at least closely related to different data characteristics within nonlinear system theory. This paper provides a comprehensive literature review examining the complexity testing techniques for time series data. According to different features, the complexity measurements for time series data can be divided into three primary groups, i.e., fractality (mono- or multi-fractality) for self-similarity (or system memorability or long-term persistence), methods derived from nonlinear dynamics (via attractor invariants or diagram descriptions) for attractor properties in phase-space, and entropy (structural or dynamical entropy) for the disorder state of a nonlinear system. These estimations analyse time series dynamics from different perspectives but are closely related to or even dependent on each other at the same time. In particular, a weaker self-similarity, a more complex structure of attractor, and a higher-level disorder state of a system consistently indicate that the observed time series data are at a higher level of complexity. Accordingly, this paper presents a historical tour of the important measures and works for each group, as well as ground-breaking and recent applications and future research directions.

This paper studies the problem of finding optimal parameters for a Poincare section used for determining the type of behavior of a time series: a deterministic or stochastic one. To reach that goal optimization algorithms are coupled with the Poincare & Higuchi (P&H) method, which calculates the Higuchi dimension using points obtained by performing a Poincare section of a certain attractor. The P&H method generates distinctive patterns that can be used for determining if a given attractor is produced by a deterministic or a stochastic system, but this method is sensitive to the parameters of the Poincare section. Patterns generated by the P&H method can be characterized using numerical measures which in turn can be used for finding such parameters for the Poincare section for which the patterns produced by the P&H method are the most prominent. This paper studies several approaches to parameterization of the Poincare section. Proposed approaches are tested on twelve time series, six produced by deterministic chaotic systems and six generated randomly. The obtained results show, that finding good parameters of the Poincare section is important for determining the type of behavior of a time series. Among the tested methods the evolutionary algorithm was able to find the best Poincare sections for use with the P&H method.

The subject of this paper is to give the description, up to topological conjugacy, of possible minimal sets of triangular maps of the square of type 2 ∞ . In [4] , we give a general method allowing to embed any zero-dimensional almost 1–1 extension of the dyadic odometer (in particular any dyadic Toeplitz system) as a minimal set of a triangular map of this type. In this paper we present a method (a combination of that described in [4] with one introduced in [1] ) of similarly embedding a special class of zero-dimensional almost 2–1 extensions of the odometer. We conjecture that these two embedding theorems exhaust all possibilities for nonperiodic minimal sets. The paper was inspired by the last unsolved problem in the Sharkovski classification program of triangular maps: does there exist a triangular map with positive entropy attained on the set of uniformly recurrent points but with entropy zero on the set of regularly recurrent points. The paper answers this question positively, concluding the program.

In an earlier paper (arxiv:1108.4292) we introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. For a compact metric space $K$ let $\dim_{H}K$ and $\dim_{tH} K$ denote its Hausdorff and topological Hausdorff dimension, respectively. We proved that this new dimension describes the Hausdorff dimension of the level sets of the generic continuous function on $K$, namely $\sup{\dim_{H}f^{-1}(y) : y \in \mathbb{R}} = \dim_{tH} K - 1$ for the generic $f \in C(K)$, provided that $K$ is not totally disconnected, otherwise every non-empty level set is a singleton. We also proved that if $K$ is not totally disconnected and sufficiently homogeneous then $\dim_{H}f^{-1}(y) = \dim_{tH} K - 1$ for the generic $f \in C(K)$ and the generic $y \in f(K)$. The most important goal of this paper is to make these theorems more precise. As for the first result, we prove that the supremum is actually attained on the left hand side of the first equation above, and also show that there may only be a unique level set of maximal Hausdorff dimension. As for the second result, we characterize those compact metric spaces for which for the generic $f\in C(K)$ and the generic $y\in f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$. We also generalize a result of B. Kirchheim by showing that if $K$ is self-similar then for the generic $f\in C(K)$ for every $y\in \inter f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$. Finally, we prove that the graph of the generic $f\in C(K)$ has the same Hausdorff and topological Hausdorff dimension as $K$., 20 pages

Remarkably benign looking logistic transformations x n +1 = r x n (1 − x n ) for choosing x 0 between 0 and 1 and 0 r ⩽ 4 have found a celebrated place in chaos, fractals and discrete dynamics. The strong physical meaning of Mandelbrot and Julia sets is broadly accepted and nicely connected by Christian Beck [Beck C. Physical meaning for Mandelbrot and Julia sets. Physica D 1999;125(3–4):171–182. Zbl0988.37060] to the complex logistic maps, in the former case, and to the inverse complex logistic map, in the latter case. The purpose of this paper is to study the bounded behavior of the complex logistic map using superior iterates and generate fractals from the same. The analysis in this paper shows that many beautiful properties of the logistic map are extendable for a larger value of r .

In this paper, we study the following Neumann–Robin boundary value problem - ( ϕ p ( u ′ ( x ) ) ) ′ = λ f ( u ( x ) ) , x ∈ ( 0 , 1 ) , u ′ ( 0 ) = 0 , u ′ ( 1 ) + α u ( 1 ) = 0 , whereα ∈ R, λ > 0 are parameters and p > 1, and p ′ = p p - 1 is the conjugate exponent of p and ϕp(x): = ∣x∣p−2x for all x ∈ R where (ϕp(u′))′ is the one dimensional p-Laplacian and f ∈ C2[0, ∞) such that f(0) 0, and also f is increasing and concave up. We shall investigate the existence and multiplicity of nonnegative solutions. Note that in this paper, we shall establish our existence results by using the quadrature method.

Motivated by Mandelbrot's [The Fractal Geometry of Nature, Freeman, San Francisco, 1983] idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, we study the properties of these translation sets and show how they can be used for a classification purpose. This first paper of a series of two will be devoted to set up the fundamental properties of Hausdorff measures of those intersection sets. Using the triadic expansion of the shifting number, we determine the fractal structure of intersection of triadic Cantor sets with their translates. We found that the Hausdorff measure of these sets forms a discrete spectrum whose non-zero values come only from those shifting numbers with a finite triadic expansion. We characterize this set of shifting numbers by giving a partition expression of it and the steps of its construction from a fundamental root set. Finally, we prove that intersection of Cantor sets with their translates verify a measure-conservation law with scales. The second paper will take advantage of the properties exposed here in order to utilize them in a classification context. Mainly, it will deal with the use of the discrete spectrum of measures to distinguish two Cantor-like sets of the same fractal dimension.

Some definitions such as m-chain recurrent set, weakly gradient flow and generalized Morse decomposition for a flow defined on a topological space are introduced in this paper. Some conclusions, include the chain recurrent set contain the m-chain recurrent set; the m-chain recurrent set contain the non-wandering set, are proved. In some flows the non-wandering set is proper subset of the m-chain recurrent set; in the meantime the m-chain recurrent set is proper subset of the chain recurrent set. Moreover some criterions for the existence of trajectories joining singular points and a necessary and sufficient condition of the weakly gradient flow are also given here. At last the generalized Morse decomposition of the invariant set are discussed in the paper.

In this paper, dual-complex k-Fibonacci numbers are defined. Also, some algebraic properties of dual-complex k-Fibonacci numbers which are connected with dual-complex numbers and k-Fibonacci numbers are investigated. Furthermore, the Honsberger identity, the d’Ocagne’s identity, Binet’s formula, Cassini’s identity, Catalan’s identity for these numbers are given.

This paper presents detailed procedure for determining the formula for calculation Tribonacci sequence numbers with arbitrary initial numbers Ta,b,c,(n). Initial solution is based on the concept of damped oscillations of Lucas type series with initial numbers T3,1,3(n). Afterwards coefficient θ3 has been determined which reduces Lucas type Tribonacci series to Tribonacci sequence T0,0,1(n). Determined relation had to be corrected with a phase shift ω3. With known relations of unitary series T0,0,1(n) with remaining two equations of Tribonacci series sequence T1,0,0(n) and T0,1,0(n), Binet type equation of Tribonacci sequence that has initial numbers Ta,b,c(n) is obtained.

An Apollonian packing is one of the most beautiful circle packings based on an old theorem of Apollonius of Perga. Ford circles are important objects for studying the geometry of numbers and the hyperbolic geometry. In this paper we pursue a research on the Ford sphere packing, which is not only the three dimensional extension of Ford circle packing, but also a degenerated case of the Apollonian sphere packing. We focus on two interesting sequences in Ford sphere packings. One sequence converges slowly to an infinitesimal sphere touching the origin of the horizontal plane. The other sequence converges at fastest rate to an infinitesimal sphere in a particular position on the plane. All these sequences have their counterparts in Ford circle packings and keep similar features. For example, our finding shows that the x-coordinate of one Ford circle sequence converges to the golden ratio gracefully. We define a Ford sphere group to interpret the Ford sphere packing and its sequences finally.

In this paper, we define the co-complex-type k -Fibonacci numbers and then give the relationships between the k -step Fibonacci numbers and the co-complex-type k -Fibonacci numbers. Also, we produce various properties of the co-complex-type k -Fibonacci numbers such as the generating matrices, the Binet formulas, the combinatorial, permanental and determinantal representations, and the finite sums by matrix methods. In addition, we study the co-complex-type k -Fibonacci sequence modulo m and then we give some results concerning the periods and the ranks of the co-complex-type k -Fibonacci sequences for any k and m . Furthermore, we extend the co-complex-type k -Fibonacci sequences to groups. Finally, we obtain the periods of the co-complex-type 2-Fibonacci sequences in the semidihedral group S D 2 m , ( m ≥ 4 ) with respect to the generating pair ( x , y ) .

Mutualistic networks are used to study the structure and processes inherent to mutualistic relationships. In this paper, we introduce a random matrix ensemble (RME) representing the adjacency matrices of mutualistic networks composed by two vertex sets of sizes n and m − n . Our RME depends on three parameters: the network size n , the size of the smaller set m , and the connectivity between the two sets α , where α is the ratio of current adjacent pairs over the total number of possible adjacent pairs between the sets. We focus on the spectral, eigenvector and topological properties of the RME by computing, respectively, the ratio of consecutive eigenvalue spacings r , the Shannon entropy of the eigenvectors S , and the Randic index R . First, within a random matrix theory approach (i.e. a statistical approach), we identify a parameter ξ ≡ ξ ( n , m , α ) that scales the average normalized measures X ¯ > (with X representing r , S and R ). Specifically, we show that (i) ξ ∝ α n with a weak dependence on m , and (ii) for ξ 1 / 10 most vertices in the mutualistic network are isolated, while for ξ > 10 the network acquires the properties of a complete network, i.e., the transition from isolated vertices to a complete-like behavior occurs in the interval 1 / 10 ξ 10 . Then, we demonstrate that our statistical approach predicts reasonably well the properties of real-world mutualistic networks; that is, the universal curves X ¯ > vs. ξ show good correspondence with the properties of real-world networks.

The spatial distributions of cities fall into two groups: one is the simple distribution with characteristic scale (e.g. exponential distribution), and the other is the complex distribution without characteristic scale (e.g. power-law distribution). The latter belongs to scale-free distributions, which can be modeled with fractal geometry. However, fractal dimension is not suitable for the former distribution. In contrast, spatial entropy can be used to measure any types of urban distributions. This paper is devoted to generalizing multifractal parameters by means of dual relation between Euclidean and fractal geometries. The main method is mathematical derivation and empirical analysis, and the theoretical foundation is the discovery that the normalized fractal dimension is equal to the normalized entropy. Based on this finding, a set of useful spatial indexes termed dummy multifractal parameters are defined for geographical analysis. These indexes can be employed to describe both the simple distributions and complex distributions. The dummy multifractal indexes are applied to the population density distribution of Hangzhou city, China. The calculation results reveal the feature of spatio-temporal evolution of Hangzhou's urban morphology. This study indicates that fractal dimension and spatial entropy can be combined to produce a new methodology for spatial analysis of city development., Comment: 23 pages, 3 figures, 5 tables

In this paper we apply the chaos game to n -sided regular polygons to generate fractals that are similar to the Sierpinski gasket. We show that for each n -gon, there is an exact ratio that will yield a perfect gasket. We then find a formula for this ratio that depends only on the angle π / n .

This paper deals with developing a new class of quaternions, octonions and sedenions called higher order Fibonacci 2 m -ions (or-higher order Fibonacci hyper complex numbers) whose components are higher order Fibonacci numbers. We give recurrence relation, Binet formula, generating function and exponential generating function of higher order Fibonacci 2 m -ions. We also derive some identities such as Vajda’s identity, Catalan’s identity, Cassini’s identity, and d’Ocagne’s identity with the aid of the Binet formula. Finally, we develop some matrix identities involving higher order Fibonacci 2 m -ions which allow us to obtain some properties of these higher order hyper complex numbers.

The main aim of this paper is to introduce new concept of bivariate Mersenne Lucas polynomials { m n ( x , y ) } n = 0 ∞ , we first give the recurrence relation of them. We then obtain Binet’s formula, generating function, Catalan’s identity and Cassini’s identity for this type of polynomials. After that, we give the symmetric function, explicit formula and d’Ocagne’s identity of bivariate Mersenne and bivariate Mersenne Lucas polynomials. By using the Binet’s formula we obtain some well-known identities of these bivariate polynomials. Also, some summation formulas of bivariate Mersenne and bivariate Mersenne Lucas polynomials are investigated.

In this paper, we introduced dual complex Fibonacci p -numbers and some properties of dual complex Fibonacci p -numbers which are related to complex Fibonacci numbers and complex Fibonacci p -numbers.

In this paper, we introduce h(x) – Fibonacci polynomials in an arbitrary finite-dimensional unitary algebra over a field K ( K = R , C ) . These polynomials generalize h(x) – Fibonacci quaternion polynomials andh(x) – Fibonacci octonion polynomials. For h(x) – Fibonacci polynomials in an arbitrary algebra, we provide generating function, Binet-style formula, Catalan-style identity, and d’Ocagne-type identity.

Let m, r, k be three positive integers. Let G be a graph with vertex set V(G) and edge set E(G), and let f: V(G) → N be a function such that f ( x ) ≥ ( k + 2 ) r − 1 for any x ∈ V(G). Let H1, H2, … , Hk be k vertex disjoint mr-subgraphs of a graph G. In this paper, we prove that every ( 0 , m f − ( m − 1 ) r ) -graph admits a (0, f)-factorization randomly r-orthogonal to each Hi ( i = 1 , 2 , … , k ).

In theoretical chemistry, the eccentric connectivity index ξ(G) of a molecular graph G was introduced as ξ ( G ) = ∑ v ∈ V ( G ) d ( v ) ɛ ( v ) where d(v) expresses the degree of vertex v and ɛ(v) is the largest distance between v and any other vertex of G. The corresponding eccentric connectivity polynomial is denoted by ξ ( G , x ) = ∑ v ∈ V ( G ) d ( v ) x ɛ ( v ) . In this paper, we present the exact expressions of eccentric connectivity polynomial for V-phenylenic nanotubes and Zig-Zag polyhex nanotubes.

A novel model is proposed to overcome the shortages of the classical hypothesis of the two-dimensional discrete hidden Markov model. In the proposed model, the state transition probability depends on not only immediate horizontal and vertical states but also on immediate diagonal state, and the observation symbol probability depends on not only current state but also on immediate horizontal, vertical and diagonal states. This paper defines the structure of the model, and studies the three basic problems of the model, including probability calculation, path backtracking and parameters estimation. By exploiting the idea that the sequences of states on rows or columns of the model can be seen as states of a one-dimensional discrete 1 × 2 order hidden Markov model, several algorithms solving the three questions are theoretically derived. Simulation results further demonstrate the performance of the algorithms. Compared with the two-dimensional discrete hidden Markov model, there are more statistical characteristics in the structure of the proposed model, therefore the proposed model theoretically can more accurately describe some practical problems.

The purpose of this paper is to study and analyze an iterative method for finding a common element of the solution set Ω of the split feasibility problem and the set F ( T ) of fixed points of a right Bregman strongly nonexpansive mapping T in the setting of p -uniformly convex Banach spaces which are also uniformly smooth. By combining Mann's iterative method and the Halpern's approximation method, we propose an iterative algorithm for finding an element of the set F( T )⋂ Ω; moreover, we derive the strong convergence of the proposed algorithm under appropriate conditions and give numerical results to verify the efficiency and implementation of our method. Our results extend and complement many known related results in the literature.

Let E , F ⊂ R be two given closed intervals, and let τ: E → F and θ: F → E be continuous maps. In this paper, we consider Koto’s chaos, sensitivity and accessibility of a given system Ψ ( u , v ) = ( θ ( v ) , τ ( u ) ) on a given product space E × F where u ∈ E and v ∈ F. In particular, it is proved that for any Cournot map Ψ ( u , v ) = ( θ ( v ) , τ ( u ) ) on the product space E × F, the following hold: (1) If Ψ satisfies Kato’s definition of chaos then at least one of Ψ 2 | Q 1 and Ψ 2 | Q 2 does, where Q 1 = { ( θ ( v ) , v ) : v ∈ F } and Q 2 = { ( u , τ ( u ) ) : u ∈ E } . (2) Suppose that Ψ 2 | Q 1 and Ψ 2 | Q 2 satisfy Kato’s definition of chaos, and that the maps θ and τ satisfy that for any e > 0, if ∣ ( τ ∘ θ ) n ( v 1 ) − ( τ ∘ θ ) n ( v 2 ) ∣ ɛ and ∣ ( θ ∘ τ ) m ( u 1 ) − ( θ ∘ τ ) m ( u 2 ) ∣ ɛ for some integers n, m > 0, then there is an integer l(n, m, e) > 0 with ∣ ( τ ∘ θ ) l ( n , m , ɛ ) ( v 1 ) − ( τ ∘ θ ) l ( n , m , ɛ ) ( v 2 ) ∣ ɛ and ∣ ( θ ∘ τ ) l ( n , m , ɛ ) ( u 1 ) − ( θ ∘ τ ) l ( n , m , ɛ ) ( u 2 ) ∣ ɛ . Then Ψ satisfies Kato’s definition of chaos.

Let (Gm)m ≥ 0 be an integer linear recurrence sequence (under some weak technical conditions) and let x ≥ 1 be an integer. In this paper, we are interested in the problem of finding combinations of the form x G n + G n − 1 which belongs to (Gm)m ≥ 0 for infinitely many positive integers n. In this case, we shall make explicit an upper bound for x which only depends on the roots of the characteristic polynomial of this recurrence. As application, we shall study the k-nacci case.

The Hofstadter Q-sequence is perhaps the most known example of meta-Fibonacci sequence. Many authors have been interested in meta-Fibonacci sequences related to this sequence. For example, recently, it was studied by A. Alkan, N. Fox, and O. Aybar the connection between the Q-sequence and the Hofstadter-Conway $ 10,000 sequence. Here, in the similar spirit as their paper, we study the interplay between the Hofstadter Q-sequence and one of the more important multiplicative arithmetic function, namely, the Mobius function.

In this paper, we introduce h ( x ) − Fibonacci quaternion polynomials that generalize the k − Fibonacci quaternion numbers, which in their turn are a generalization of the Fibonacci quaternion numbers. We also present a Binet-style formula, ordinary generating function and some basic identities for the h ( x ) − Fibonacci quaternion polynomial sequences.

Alternate Julia sets have been studied in Picard iterative procedures. The purpose of this paper is to study the quadratic and cubic maps using superior iterates to obtain Julia sets with different alternate structures. Analytically, graphically and computationally it has been shown that alternate superior Julia sets can be connected, disconnected and totally disconnected, and also fattier than the corresponding alternate Julia sets. A few examples have been studied by applying different type of alternate structures.

It is known that the whole space can be a Li–Yorke scrambled set in a compact dynamical system, but this does not hold for distributional chaos. In this paper we construct a noncompact weekly mixing dynamical system, and prove that the whole space is a transitive extremal distributionally scrambled set in this system.

Let ( T , d ) be a dendrite and f be a continuous map from T to T . Denote by ω ( x , f ) the ω -limit set of x under f . Write Ω ( x , f ) = { y | there exist a sequence of points x k ∈ T and a sequence of positive integers n 1 n 2 ⋯ such that lim k ⟶ ∞ x k = x and lim k ⟶ ∞ f n k ( x k ) = y } . In this paper, we show that if the cardinal number of the set of endpoints of T is less than the cardinal number c of the continuum, then f is equicontinuous if and only if Ω ( x , f n ) = ω ( x , f n ) for any x ∈ T and any positive integer n .

In this paper, we study the nonperiodic second order Hamiltonian systems u ¨ ( t ) - λ L ( t ) u ( t ) + ∇ W ( t , u ( t ) ) = 0 , ∀ t ∈ R , where λ ⩾ 1 is a parameter, the matrix L ( t ) is not necessarily positive definite for all t ∈ R nor coercive. Replacing the Ambrosetti–Rabinowitz condition by general superquadratic assumptions, we establish the existence and multiplicity results for the above system when λ > 1 large. We also consider the situation where W is a combination of subquadratic and superquadratic terms, and obtain infinitely many homoclinic solutions.

The present paper formulates and solves a problem of dividing coins. The basic form of the problem seeks the set of the possible ways of dividing coins of face values 1,2,4,8,... between three people. We show that this set possesses a nested structure like the Sierpinski-gasket fractal. For a set of coins with face values power of r, the number of layer of the gasket becomes r. A higher-dimensional Sierpinski gasket is obtained if the number of people is more than three. In addition to Sierpinski-type fractals, the Cantor set is also obtained in dividing an incomplete coin set between two people., 12 pages, 7 figures

A weighted hierarchical network model is introduced in this paper. We study the trapping problem for weighted-dependent walks taking place on a hierarchical weighted network at a given trap. We concentrate on the average trapping time (ATT) for three cases, i.e., the immobile trap located at the root node, the external nodes and a neighbor of the root with a single connectivity, respectively. The closed-form formulae for the ATT for the three cases are obtained. In different range of the weight factor r, the leading term of ATT grows differently, i.e., superlinearly, linearly and sublinearly with the network size. For all the three cases of trapping problems, the leading scaling of ATT can reach the minimum scaling.

This paper investigates the evolution of reactive strategies ( p , q ) on the homogeneous regular and random networks with different network densities. Here p and q mean the probabilities to cooperate after a cooperative and defective opponent. Through the prisoner’s dilemma model, we show that the intermediate number of neighbors of both regular and random networks can promote the emergence of generous tit-for-tat (GTFT) strategy and improve the individuals’ gains. The sparse network inhibits the diffusion of the GTFT-like strategy, while the dense network promotes the spread of the defective strategy. Moreover, our investigation shows that compared with the regular ring, the individuals on the random network with proper number of neighbors can obtain higher gains, whereas the cooperative behavior is inhibited for the denser random network.

Let X be a compact metric space and f : X → X be a continuous map. In this paper, we prove that if f has the asymptotic average shadowing property (Abbrev. AASP) and an invariant Borel probability measure with full support or the positive upper Banach density recurrent points of f are dense in X , then for all n ⩾ 1, f × f × ⋯ × f ( n times) and f n are totally strongly ergodic. Moreover, we also give some sufficient conditions for an interval map having the AASP to be Li-Yorke chaotic.

This paper concerns a special class of Moran sets, the λ-Moran sets, and obtains the necessary and sufficient conditions to describe the thickness and thinness of λ-Moran sets.

In this paper we show how the notions of conductance and cutoff can be used to determine the length of the random walks in some clustering algorithms. We consider graphs which are globally sparse but locally dense. They present a community structure: there exists a partition of the set of vertices into subsets which display strong internal connections but few links between each other. Using a distance between nodes built on random walks we consider a hierarchical clustering algorithm which provides a most appropriate partition. The length of these random walks has to be chosen in advance and has to be appropriate. Finally, we introduce an extension of this clustering algorithm to dynamical sequences of graphs on the same set of vertices.

This paper is concerned with a class of second order Hamiltonian systems with superlinear and sublinear nonlinearity (P) u ¨ ( t ) + b ( t ) | u ( t ) | μ - 2 u ( t ) + ∇ H ( t , u ( t ) ) = 0 , a . e . t ∈ [ 0 , T ] ; u ( 0 ) - u ( T ) = u ˙ ( 0 ) - u ˙ ( T ) = 0 , where b(t) is a real function defined on [0, T], μ > 2 and H : [0, T] × RN → R is a Caratheodory function. Some new multiplicity results of periodic orbits for the problem (P) are obtained via some critical point theorems.

In this paper, we investigate the bifurcations of limit cycles in a class of planar quadratic reversible system of genus one x ˙ = y + 4 x 2 , y ˙ = - x 1 - 8 3 y under quadratic perturbations. It is proved that the cyclicity of the period annulus is equal to two.

The limit set of zeros of partition function of the Potts model on diamond-like hierarchical lattices is studied. It is shown that the limit set is the Julia set of a family rational maps, it is shown in a mathematically exact way that the Julia set tends to a geometrical circle and its Hausdorff dimension tends to 1 when the parameter ∣λ∣ → +∞, which gives a true answer that Bambi Hu and Bin Lin proposed in 1989, furthermore, in this paper, it give a perfect description about this relations. Also the continuity of level diameter of Aλ(1) of this physical model about λ is discussed.

Let (X, d) be a compact metric space and fn : X → X a sequence of continuous maps such that (fn) converges uniformly to a map f. The purpose of this paper is to study the Devaney’s chaos on the uniform limit f. On the one hand, we show that f is not necessarily transitive even if all fn mixing, and the sensitive dependence on initial conditions may not been inherited to f even if the iterates of the sequence have some uniform convergence, which correct two wrong claims in [1] . On the other hand, we give some equivalence conditions for the uniform limit f to be transitive and to have sensitive dependence on initial conditions. Moreover, we present an example to show that a non-transitive sequence may converge uniformly to a transitive map.

In this paper, we shall study the multifractal decomposition behavior for a family of sets E known as Moran fractals. For each value of the parameter α ∈ (αmin, αmax), we define “multifractal components” Eα of E, and show that they are non-regularity fractals (in the sense of Taylor). By obtaining the new sufficient conditions for the valid multifractal formalisms of non-regularity Moran measures, we give explicit formula for the Hausdorff dimension and Packing dimension of Eα respectively. In particular, we describe a large class of non-regularity Moran measure satisfying the explicit formula.

Very often traditional approaches studying dynamics of self-similarity processes are not able to give their quantitative characteristics at infinity and, as a consequence, use limits to overcome this difficulty. For example, it is well known that the limit area of Sierpinski’s carpet and volume of Menger’s sponge are equal to zero. It is shown in this paper that recently introduced infinite and infinitesimal numbers allow us to use exact expressions instead of limits and to calculate exact infinitesimal values of areas and volumes at various points at infinity even if the chosen moment of the observation is infinitely faraway on the time axis from the starting point. It is interesting that traditional results that can be obtained without the usage of infinite and infinitesimal numbers can be produced just as finite approximations of the new ones. The importance of the possibility to have this kind of quantitative characteristics for E-Infinity theory is emphasized.

In this paper, the Hausdorff dimension of the range for a Markov process of Ornstein–Uhlenbeck type { X ( t ) , t ∈ R + } on R is given. We also investigate the Hausdorff dimension of the process in R d case.

In this paper, we defined order-m generalized Fibonacci k -numbers by matrix representation. Using this matrix representation we obtained sums, some identities and the generalized Binet formula of generalized order-m Fibonacci k-numbers.

In this paper we intend to generalize the L q -spectrum relatively to two measures and to study its behavior under orthogonal projections. A uniform result is obtained in the case of “Frostman like measures”.

Let G be a connected planar graph with n vertices and m edges. It is known that the fundamental group of G has 1 −(n − m) generators. In this paper, we show that if G is a self-dual graph, then its fundamental group has (n − 1) generators. We indicate that these results are relevant to quantum space time.

In this paper, we introduce a class of fractal sets, which can be recursively constructed by two sequences {nk}k⩾1 and {ck}k⩾1. We obtain the exact Hausdorff dimensions of these types of fractal sets using the continued fraction map. Connection of continued fraction with El Naschie’s fractal spacetime is also illustrated. Furthermore, we restrict one sequence {ck}k⩾1 to make every irrational number α ∈ (0, 1) correspond to only one of the above fractal sets called α-Cantor sets, and we found that almost all α-Cantor sets possess a common Hausdorff dimension of 0.618, which is also the Hausdorff dimension of the one-dimensional random recursive Cantor set and it is the foundation stone of E-infinity fractal spacetime theory.

We have considered a class of square Fibonacci matrix of order (p + 1) whose elements are based on the Fibonacci p numbers with determinant equal to +1 or −1. There is a relation between Fibonacci numbers with initial terms which is known as cassini formula. Fibonacci series and the golden mean plays a very important role in the construction of a relatively new space–time theory, which is known as E-infinity theory. An original Fibonacci coding/decoding method follows from the Fibonacci matrices. There already exists a relation between the code matrix elements for the case p = 1 [Stakhov AP. Fibonacci matrices, a generalization of the cassini formula and a new coding theory. Chaos, Solitons and Fractals 2006;30:56–66.]. In this paper, we have established generalized relations among the code matrix elements for all values of p. For p = 2, the correct ability of the method is 99.80%. In general, correct ability of the method increases as p increases.