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This paper develops the analytical conditions for the existence of periodic solutions of a second-order,time-delayed, discontinuous dynamical system. A sample model consists of two linear delayed sub-systems with a switching boundary. The defined G-functions for the delayed, discontinuous systems are introduced, and sufficient and necessary conditions for a flow crossing, sliding and grazing along the switch boundary are developed for such a delayed, discontinuous system. Furthermore, nine (9) regular basic mappings in phase plane and thirty-three (33) delay-related mappings for the second-order, time-delayed, discontinuous systems are classified. Constraint equations are predicted analytically for two periodic orbits with initial functions provided posteriorly. Finally, three numerical examples are illustrated to verify the existence of generalized slowly oscillating periodic orbits without and with sliding portions. This paper improves and extends motion switchability conditions at the boundary in discontinuous dynamical systems without delay.

The qualitative and quantitative combined nonlinear dynamics model proposed in this paper fill the gap in nonlinear dynamics model in terms of qualitative and quantitative combined methods, allowing the qualitative model and quantitative model to perfectly combine and overcome their weaknesses by learning from each other. These two types of models use their strengths to make up for the other’s deficiencies. The qualitative and quantitative combined models can surmount the weakness that the qualitative model cannot be applied and verified in a quantitative manner, and the high costs and long time of multiple construction as well as verification of the quantitative model. The combined model is more practical and efficient, which is of great significance for nonlinear dynamics. The qualitative and quantitative combined modeling and model analytical method raised in this paper is not only applied to nonlinear dynamics, but can be adopted and drawn on in the modeling and model analysis of other fields. Additionally, the analytical method of qualitative and quantitative combined nonlinear dynamics model proposed in this paper can satisfactorily resolve the problems with the price system’s existing nonlinear dynamics model analytical method. The three-dimensional dynamics model of price, supply–demand ratio and selling rate established in this paper make estimates about the best commodity prices using the model results, thereby providing a theoretical basis for the government’s macro-control of price. Meanwhile, this model also offer theoretical guidance to how to enhance people’s purchasing power and consumption levels through price regulation and hence to improve people’s living standards.

Stability in distribution, implying the existence of the invariant probability measure, is an important measure of stochastic hybrid system. However, the effect of Levy jumps on the stability in distribution is still unclear. In this paper, we consider a n-species competitive Lotka–Volterra model with Levy jumps under regime-switching. First, we prove the existence of the global positive solution, obtain the upper and lower boundedness. Then, asymptotic stability in distribution as the main result of our paper is derived under some sufficient conditions. Finally, numerical simulations are carried out to support our theoretical results and a brief discussion is given.

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.

In this paper we obtain different sufficient conditions for a non-autonomous discrete system to be multi-sensitivite. We study properties of a multi-sensitive non-autonomous system in detail. It is proved that on a compact metric space every finitely generated non-autonomous system which is topologically transitive having dense set of periodic points is thickly syndetically sensitive. We introduce and study the notion of totally sensitive non-autonomous systems. We also provide counter examples to support our results.

In this paper, we are interested in the study of a stochastic viral infection model with immune impairment driven by Levy noise. First we prove the existence of a unique global solution to the model. By means of the Lyapunov method we study the stability of the equilibria. We present sufficient conditions for the extinction and persistence in mean. Furthermore, we present some numerical results to support the theoretical work.

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.

In this paper, we use fast oscillating integrated Ornstein–Uhlenbeck (abbreviated as O-U) processes to pathwisely approximate Wiener processes. In physics, such approximation process is known as a colored noise approximation, and is suitable for dealing with stochastic flow problems. Our first result shows that if the drift term of a stochastic differential equation (abbreviated as SDE) satisfies usual Lipschitz constrains and a linear growth condition, then the solution of the SDE can be almost surely approximated with a polynomial rate. Next, we explore the O-U process approximation on the random manifold of stochastic evolution equations with linear multiplicative noise. Our second result shows that if the stochastic evolution equation further satisfies a uniformly hyperbolic condition, then the corresponding random manifold approximation also converges almost surely, with a polynomial rate.

In this paper, we study the bifurcation of limit cycles in two special near-Hamiltonian polynomial planer systems which their corresponding Hamiltonian systems have a heteroclinic loop connecting a hyperbolic saddle and a cusp of order two. In these systems, we will compute the asymptotic expansions of corresponding first order Melnikov functions near the loop and the center to analyze the number of limit cycles. Moreover, in the first system, by using the Chebychev criterion, we study the Poincare bifurcation.

Many problems on computer science, chemistry, physics and network theory are related to factors, factorizations and orthogonal factorizations in graphs. For example, the telephone network design problems can be converted into maximum matchings of graphs; perfect matchings or 1-factors in graphs correspond to Kekule structures in chemistry; the file transfer problems in computer networks can be modelled as (0, f )-factorizations in graphs; the designs of Latin squares and Room squares are related to orthogonal factorizations in graphs; the orthogonal ( g, f )-colorings of graphs are related to orthogonal ( g, f )-factorizations of graphs. In this paper, the orthogonal factorizations in graphs are discussed and we show that every bipartite ( 0 , m f − ( m − 1 ) r ) -graph G has a (0, f )-factorization randomly r -orthogonal to n vertex disjoint mr -subgraphs of G in certain conditions.

This paper is concerned with the fractional differential equations of Sobolev type with boundary conditions in a Banach space. With the help of properties of Hilfer fractional calculus, the theory of propagation family as well as the theory of the measure of noncompactness and the fixed point methods, we obtain the existence results of mild solutions for Sobolev type fractional evolution differential equations involving Hilfer fractional derivative. Finally, two examples are presented to illustrate the main result.

In this paper, we deal with Abel equations: d x d y = A ( x ) y 2 + B ( x ) y 3 , where A(x) and B(x) are real polynomials. If a solution y = φ ( x ) of the above equations satisfies that φ ( 0 ) = φ ( 1 ) , then we say that it is a periodic solution. If a periodic solution is isolated, then we call it a limit cycle. If a limit cycle y = φ ( x ) is a rational function but not a polynomial, then we call it a nontrivial rational limit cycle. Firstly, we study the existence of nontrivial rational limit cycles. We prove that there exist Abel equations, which have at least two nontrivial rational limit cycles, and there also exists other Abel equations, which have at least one nontrivial rational limit cycle and one non-rational limit cycle. Secondly, we discuss the relation between the existence of nontrivial rational limit cycle and the degrees of A(x) and B(x). Finally we show that the multiplicity of a nontrivial rational limit cycle can be unbounded.

Models with uncertain values of their parameters are of a practical significance in different fields of science and technology. Uncertainty in the competitive models can greatly affect their dynamical behavior and, therefore, the analysis of the effects of uncertain terms in such models is very important in both theory and application. In this paper, we extend the existing N-species impulsive competitive models to the uncertainty case. The existence of a unique strictly positive and robustly exponentially stable almost periodic solution is investigated for the model. The main results are obtained by using Lyapunov-type functions and a comparison principle.

In this paper, the dynamical analysis of a delayed reaction-diffusion virus infection model with logistic growth and chemotaxis for the uninfected target cells and humoral immune impairment is studied. By analyzing corresponding characteristic equations, the local stability of the infection-free equilibrium is established. The stability properties and Turing instability of the antibody-free equilibrium and antibody-present infection equilibrium have been extensively discussed. The existence of Hopf bifurcation with antibody response delay as a bifurcation parameter at the antibody-present infection equilibrium is established. The numerical simulations are carried out in order to illustrate the dynamical behavior of the model.

This paper introduces an efficient fractional differential transform that is called “conformable fractional partial differential transform (CFPDT)” and its properties for solving linear and nonlinear conformable space-time fractional partial differential equations (CSTFPDEs). Moreover, a CFPDT is more practical and helpful for solving abroad CSTFPDEs. Analytical solutions to linear Navier–Stokes equation and nonlinear gas dynamic equations in sense of conformable space-time fractional partial derivatives are successfully obtained to confirm the accuracy and efficiency of the proposed transform.

In this paper, we study a stochastic tuberculosis model with antibiotic resistance. By constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Moreover, we obtain sufficient conditions for extinction of the disease. The existence of a stationary distribution implies stochastic weak stability.

In this paper we consider Hausdorff dimension of the sets of Li-Yorke pairs for some chaotic dynamical systems including A-coupled expanding systems. We prove that Li-Yorke pairs of A-coupled-expanding system under some conditions have full Hausdorff dimension in the invariant set. Moreover we give a generalization of the result of [5] which is on the Hausdorff dimension of Li-Yorke pairs of dynamical systems topologically conjugate to the full shift and have a self-similar invariant set, to the case of the dynamical systems topologically semi-conjugate to some kinds of subshifts. Furthermore Hausdorff dimension of “chaotic invariant set” for some kinds of A-coupled-expanding maps is shown.

In this paper, we present a single species model with cannibalism and nonlocal effect. The existence of traveling wave fronts connecting the equilibrium 0 to the equilibrium K r r + K h is proved when the wave speed c ≥ 2 r .

In this paper, we identify the unknown space-dependent source term in a time-fractional diffusion equation with variable coefficients in a bounded domain where additional data are consider at a fixed time. Using the generalized and revised generalized Tikhonov regularization methods, we construct regularized solutions. Convergence estimates for both methods under an a-priori and a-posteriori regularization parameter choice rules are given, respectively. Numerical example shows that the proposed methods are effective and stable.

In this paper we present applications of special numbers obtained from a difference equation of degree three. As a particular case of this difference equation of degree three, we obtain the generalized Pell-Fibonacci-Lucas numbers, which were extended to the generalized quaternion algebras. Using properties of these quaternion elements, we can define a set with an interesting algebraic structure, namely, an order on a generalized rational quaternion algebra. Another presented application is in the Coding Theory, since some of these numbers can be used to built cyclic codes with good properties (MDS codes).

This paper is concerned with the Diophantine properties of the orbits of real numbers in continued fraction system under the doubling metric. More precisely, let φ be a positive function defined on N . We determine the Lebesgue measure and Hausdorff dimension of the set E ( φ ) = { ( x , y ) ∈ [ 0 , 1 ) × [ 0 , 1 ) : | T n x − y | φ ( n ) for i.m. n } , where T is the Gauss map and “i.m.” stands for “infinitely many”.

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 study the bifurcation and stability of a ratio-dependent predator-prey model with nonconstant predator harvesting rate. The analysis is carried out both analytically and numerically. We determine stability and dynamical behaviours of the equilibria of this system and characterize codimension 1 and codimension 2 bifurcations of the system analytically. Our bifurcation analysis indicates that the system exhibits numerous types of bifurcation phenomena, including Fold, Hopf, Cusp, and Bogdanov–Takens bifurcations. We use the numerical software MATCONT, to compute curves of equilibria and to compute several bifurcation curves. We especially approximate a family of limit cycles emanating from a Hopf point. Our results generalize and improve some known results and show that the model has more rich dynamics than the ratio-dependent predator-prey model without harvesting rate.

A stochastic three species food chain model with general L e ´ vy jumps is investigated in the paper. The criterion on the global stability in the mean with probability one for each species is established. The results show that the dynamics of the model can be significantly changed by L e ´ vy jumps, and strongly indicate that the general jump N(dt, du) is more complicated than compensated jump N ˜ ( dt , du ) .

In this paper, bicomplex Fibonacci quaternions are defined. Also, some algebraic properties of bicomplex Fibonacci quaternions which are connected with bicomplex numbers and Fibonacci numbers are investigated. Furthermore, Binet’s formula, Cassini’s identity, Catalan’s identity for these quaternions and real representation of these quaternions are given.

In this paper the chaotic behavior of second-order, discontinuous systems with a pseudo-equilibrium point on a discontinuity surface is analyzed. The discontinuous system is piecewise linear and approximated to a non-smooth continuous system. The discontinuous term is represented by a sign function that is replaced by a saturation function with high slope. Some of the conditions that determine the chaotic behavior of the approximate system are formally established. Besides, the convergence of its chaotic solutions to those of the discontinuous system is shown. Several bifurcation diagrams of both systems show the similarity of their dynamical behavior in a wide parameter range, and particularly for a parameter region determined from the application of the Melnikov technique to non-smooth systems, where a chaotic behavior can be displayed.

The vaccination, latent and relapse period are three important factors affecting the whole disease development. In this paper, we propose an SVEIR epidemic model with continuous age-dependent vaccination, latency and relapse, at the same time, the nonlinear incidence rate is also considered. Uniform persistence of the model is proved by reformulating it as the so called Volterra integral equations. The basic reproduction number R 0 , which completely determines the global dynamics of the model, is derived. By using Lyapunov functionals, the global stability of the equilibria is obtained. Namely, the disease-free equilibrium is globally asymptotically stable if R 0 1 , while if R 0 > 1 the endemic equilibrium is globally asymptotically stable. Finally, two numerical examples support our main analytical results.

In this paper, an age structured SIRS epidemic model with age of recovery is studied which allows the removed individuals to become susceptible again when they lose the protection property as the time goes. The age structured model is reformulated as an abstract non-densely defined Cauchy problem and the expression of the basic reproduction number, R 0 , is obtained. If R 0 1 , the model only has the disease-free steady state and it is global stability. If R 0 > 1 , besides the disease-free steady state the model also has an endemic steady state. By analyzing the associated characteristic equation, the existence of a local Hopf bifurcation is proved under certain conditions. Also, we considered the global continuation of the local Hopf bifurcation. Finally, some numerical simulations are carried out to illustrate the theoretical results.

In this paper, the invariance properties of time fractional coupled Drinfeld–Sokolov–Satsuma–Hirota (DSSH) equations have been investigated using the Lie group analysis method. In this regard a systematic research to derive Lie point symmetries of time fractional coupled DSSH equations is performed. Using the Lie group analysis method, the vector fields and the symmetry reduction of the time fractional coupled DSSH equations are obtained. It is shown that, the time fractional coupled DSSH equations can be reduced to the fractional coupled ordinary differential equations by using fractional Erdelyi–Kober differential operator with Riemann–Liouville derivative. Finally using new conservation theorem with formal Lagrangian, the new conserved vectors are well constructed with a detailed derivation, which present the conservation analysis for time fractional coupled Drinfeld–Sokolov–Satsuma–Hirota (DSSH) equations.

In this paper, based on quasi steady-state assumption, a phenomenological model of viral infections with intracellular delay is proposed to explore viral infection process. The existence as well as stability of equilibria and bifurcation are studied. The results indicate that the basic reproduction number is a sharp threshold parameter to determine the outcomes of infection when the intracellular delay is omitted. Otherwise, Hopf bifurcation may occur under some conditions. Combining with numerical simulations, we observe that the explicit dynamics of free virus can affect the dynamics behavior, the fluctuation scenario may appears in the case of full logistic proliferation term even if there is no intracellular delay. In addition, there are different patterns of bifurcation under different drug effectiveness and patient’s age interacting with intracellular delay, which indicates that intracellular delay may induce complicated kinetics of virus infection.

Let M be a closed connected smooth Riemannian manifold with dimM ≥ 2 and let f: M → M be a diffeomorphism. In the paper, we show that C1 generically, if a diffeomorphism f does not present a homoclinic tangency then it is weak Lebesgue measure expansive and, as an example, we find a partially hyperbolic diffeomorphism which is not weak measure expansive. Moreover, for a surface, if a diffeomorphism f has a homoclinic tangency then there is a diffeomorphism g C1 close to f such that g is not weak measure expansive.

This paper is concerned with existence, uniqueness and asymptotic behavior of traveling wave fronts for a generalized Fisher equation with nonlocal delay. The existence of traveling wave fronts is established by linear chain trick and geometric singular perturbation theory. The strategy is to reformulate the problem as the existence of a heteroclinic connection in R 4 . The problem is then tackled by using Fenichel’s invariant manifold theory. The asymptotic behavior and uniqueness of traveling wave fronts are also obtained by using standard asymptotic theory and sliding method.

In this paper, we establish sufficient conditions for the existence and uniqueness of the solution in functional weighted Sobolev space for a class of initial-boundary value problems with integral condition for a class of nonlinear partial fractional reaction-diffusion (RD) equations. The results are established by using a priori estimate in Bouziani fractional spaces and applying an iterative process based on results obtained for the linear problem, we prove the existence, uniqueness of the weak generalized solution of the nonlinear problem.

In this paper, we consider the existence of periodic solutions for discrete time periodic time-varying coupled systems on networks (DPTCSN). Some novel sufficient conditions are obtained to guarantee the existence of periodic solutions for DPTCSN, which have a close relation to the topology property of the corresponding network. Our approach is based on the continuation theorem of coincidence degree theory, generalized Kirchhoff’s matrix tree theorem in graph theory, Lyapunov method and some new analysis techniques. The approach is applied to the existence of periodic solutions for discrete time Cohen–Grossberg Neural Networks (CGNN). Finally, an example and numerical simulations are provided to illustrate the effectiveness of our theoretical results.

In this paper, we study the Chebyshev’s property of the three-dimensional vector space E = J 0 , J 1 , J 2 > , where J i ( h ) = ∫ H = h x i d x y and H ( x , y ) = 1 2 y 2 + V ( x ) is a hyperelliptic Hamiltonian of degree 7. Our main result asserts that in two specific cases, namely (a) V ′ ( x ) = x 3 ( x − 4 7 ) ( x − 1 ) 2 and (b) V ′ ( x ) = x ( x − 2 7 ) ( x − 1 ) 4 , E is an extended complete Chebyshev system. To this end, we use the criterion and the tools developed by Grau–Manosas–Villadelprat in Trans. Amer. Math. Soc. in 2011.

Consider the following stochastic partial differential equation, \begin{equation*} \partial_t u_t(x)= \mathcal{L}u_t(x)+ \xi\sigma (u_t(x)) \dot F(t,x), \end{equation*} where $\xi$ is a positive parameter and $\sigma$ is a globally Lipschitz continuous function. The stochastic forcing term $\dot F(t,x)$ is white in time but possibly colored in space. The operator $\mathcal{L}$ is a non-local operator. We study the behaviour of the solution with respect to the parameter $\xi$, extending the results in \cite{FoonNual} and \cite{Bin}, Comment: 14 pages; updated versionof the original paper titled "Large time behaviour for the non-linear fractional stochastic heat equation"

In this paper, we present the finite difference method for Caputo-type parabolic equation with fractional Laplacian, where the time derivative is in the sense of Caputo with order in (0, 1) and the spatial derivative is the fractional Laplacian. The Caputo derivative is evaluated by the L 1 approximation, and the fractional Laplacian with respect to the space variable is approximated by the Caffarelli–Silvestre extension. The difference schemes are provided together with the convergence and error estimates. Finally, numerical experiments are displayed to verify the theoretical results.

In this paper, we study the existence of infinitely many solutions to a class of boundary value problems for impulsive fractional Hamiltonian systems. The main tool is the use of variant Fountain theorems, which allow to give some sufficient conditions to guarantee that the impulsive problems object of our study have infinitely many solutions.

In this paper we make use of energy methods to study the Navier–Stokes equations with time-fractional derivative. Such equations can be used to simulate anomalous diffusion in fractal media. In the first step, we construct a regularized equation by using a smoothing process to transform unbounded differential operators into bounded operators and then obtain the approximate solutions. The second part describes a procedure to take a limit in the approximation program to present a global solution to the objective equation.

This paper introduces a new concept called impulsive control inclusion condition, i.e., the impulsive condition is presented, in the first time, as inclusion related to multivalued maps and controls. The notion of approximate controllability of a class of semilinear Hilfer fractional differential control inclusions in Banach spaces is established. For the main results, we use fractional calculus, fixed point technique, semigroup theory and multivalued analysis. An appropriate set of sufficient conditions for the considered system to be approximately controllable is studied. Finally, we give an illustrated example to provide the obtained theory.

This paper derives the stochastic solution of a Cauchy problem for the distribution of a fractional diffusion process. The governing equation involves the Bessel-Riesz derivative (in space) to model heavy tails of the distribution, and the Caputo-Djrbashian derivative (in time) to depicts the memory of the diffusion process. The solution is obtained as Brownian motion with time change in terms of the Bessel-Riesz subordinator on the inverse stable subordinator. This stochastic solution, named fractional Bessel-Riesz motion, provides a method to simulate a large class of stochastic motions with memory and heavy tails.

In the present survey paper, we prove some results concerning the existence of solutions and weak solutions for some classes of Hadamard and Hilfer fractional differential equations. Some attractivity and Ulam stability results are presented. Our results are provided by applying the fixed point theory, and the technique of strong and weak measures of noncompactness.

In this paper we use the tools from harmonic analysis to study the Cauchy problem for Navier–Stokes equations with time-fractional derivative of order α ∈ (0, 1), which can be used to simulate the anomalous diffusion in fractal media. Two main results concerning the local existence of solutions for the given problem in Sobolev space are addressed.

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.

This paper is purported to investigate a cross-diffusion system arising in a predator-prey population model including Holling type-II functional response in a bounded domain with Dirichlet boundary condition. The asymptotical stabilities are investigated to this system by using the method of eigenvalue. Moreover, the existence of positive steady states are considered by using fixed points index theory, bifurcation theory, energy estimates and the differential method of implicit function.

Let X be a compact metric space and f: X → X be a continuous map. In [14], it was shown that if a dynamical system (X, f) has strictly coupled-expanding property, then the Hyperspace dynamical system ( K ( X ) , f ¯ ) , induced by (X, f), has a subsystem which is topologically semi-conjugated to a full shift (Σk, σ). In this paper, we show that under some conditions more weaker than those of [14] , ( K ( X ) , f ¯ ) has a subsystem which not only is topologically semi-conjugated to a subshift of finite type (ΣA, σA), but also is bigger than the subsystem builded in [14] . Furthermore, we expand above results to the fuzzy dynamical system, extended by (X, f).

In this paper, we give the multifractal analysis of the weighted local entropies for arbitrary invariant measures. Our result is applied to self-affine systems.

A nonlocal delayed reaction-diffusive two-species model with Dirichlet boundary condition and general functional response is investigated in this paper. Based on the Lyapunov–Schmidt reduction, the existence, bifurcation direction and stability of Hopf bifurcating periodic orbits near the positive spatially nonhomogeneous steady-state solution are obtained, where the time delay is taken as the bifurcation parameter. Moreover, the general results are applied to a diffusive Lotka–Volterra type food-limited population model with nonlocal delay effect, and it is found that diffusion and nonlocal delay can also affect the other dynamic behavior of the system by numerical experiments.

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 this paper, the behavior of rank one strange attractors in a predator - prey system with discrete and distributed delay is studied. This investigation is conducted from the perspective of a recent chaos theory of rank one maps developed by us for delayed systems. We specifically choose the well - known predator - prey system with discrete and distributed delay to demonstrate the applicability of the theory to the case of ecological systems, and prove the appearance of chaotic behavior under reasonable conditions. The results of the numerical simulations presented are in close agreement with the expectations of the theory.