The dynamic behaviors of a nonautonomous system for migratory birds with Hassell-Varley type functional response and the saturation incidence rate are studied. Under quite weak assumptions, some sufficient conditions are obtained for the permanence and extinction of the disease. Moreover, the global attractivity of the model is discussed by constructing a Lyapunov function. Numerical simulations which support our theoretical analysis are also given.

We propose a new fractional-order chaotic complex system and study its dynamical properties including symmetry, equilibria and their stability, and chaotic attractors. Chaotic behavior is verified with phase portraits, bifurcation diagrams, the histories, and the largest Lyapunov exponents. And we find that chaos exists in this system with orders less than 5 by numerical simulation. Additionally, antisynchronization of different fractional-order chaotic complex systems is considered based on the stability theory of fractional-order systems. This new system and the fractional-order complex Lorenz system can achieve antisynchronization. Corresponding numerical simulations show the effectiveness and feasibility of the scheme.

The purpose of this paper is to investigate the fixed points of solutions $f(z)$ of some $q$-difference equations and obtain some results about the exponents of convergence of fixed points of $f(z)$ and $f(q^{j}z)(j\in {\mathbb{N}}_{+})$, $q$-differences ${\mathrm{\Delta }}_{q}f(z)=f(qz)-f(z)$, and $q$-divided differences ${\mathrm{\Delta }}_{q}f(z)/f(z)$.

By using the variant fountain theorem, we study the existence of multiple solutions for a class of superquadratic fourth-order elliptic problem with Navier boundary value condition.

The classical Wazewski theorem established that nonpositivity of all nondiagonal elements $p_{ij}$ $(i\ne j,\mathrm{ }\mathrm{ }i,j=1,\dots ,n)$ is necessary and sufficient for nonnegativity of the fundamental (Cauchy) matrix and consequently for applicability of the Chaplygin approach of approximate integration for system of linear ordinary differential equations $x_i^{\mathrm{\prime }}\left(t\right)+{\sum }_{j=1}^n\mathrm{‍}{p}_{ij}\left(t\right)x_j\left(t\right)=f_i\left(t\right),\mathrm{ }\mathrm{}i=1,\dots ,n.$ Results on nonnegativity of the Cauchy matrix for system of delay differential equations $x_i^{\mathrm{\prime }}\left(t\right)+{\sum }_{j=1}^n\mathrm{‍}{p}_{ij}\left(t\right)x_j\left(h_{ij}\left(t\right)\right)=f_i\left(t\right),\mathrm{ }\mathrm{}i=1,\dots ,n,$ which were based on nonpositivity of all diagonal elements, were presented in the previous works. Then examples, which demonstrated that nonpositivity of nondiagonal coefficients $p_{ij}$ is not necessary for systems of delay equations, were found. In this paper first sufficient results about nonnegativity of the Cauchy matrix of the delay system without this assumption are proven. A necessary condition of nonnegativity of the Cauchy matrix is proposed. On the basis of these results on nonnegativity of the Cauchy matrix, necessary and sufficient conditions of the exponential stability of the delay system are obtained.

This paper investigates relatively integral stability in terms of two measures for two differential systems with “maxima” by employing Lyapunov functions, Razumikhin method, and comparison principle. An example is given to illustrate our result.

We review the results concerning types of solutions of boundary value problems for the second order nonlinear equation $(l_{2}x)(t)=f(t,x,x^{\mathrm{\prime }}),$ where $(l_{2}x)(t)$ is the second order linear differential form. The existence results and the multiplicity results are stated in terms of types of solutions.

We establish some oscillation criteria for the following certain even order neutral delay differential equations with mixed nonlinearities: ${\left(r\left(t\right){\left|z^{\left(n-1\right)}\left(t\right)\right|}^{\alpha -1}{z}^{\left(n-1\right)}\left(t\right)\right)}^{\text{'}}+q_0(t){\left|(x\left({\tau }_0\left(t\right)\right)\right|}^{\alpha -1}x({\tau }_0(t))+q_1\left(t\right){\left|(x({\tau }_1(t))\right|}^{\beta -1}x({\tau }_1(t))+q_2\left(t\right){\left|(x({\tau }_2(t))\right|}^{\gamma -1}x({\tau }_2(t))=0,\mathrm{ }t\ge t_0,$ where $z(t)=x(t)+p(t)x(\sigma (t)),$ $n$ is even integer, and $\gamma >\alpha >\beta >0.$ Our results generalize and improve some known results for oscillation of certain even order neutral delay differential equations with mixed nonlinearities.

We establish the oscillation criteria of Philos type for second-order half-linear neutral delay dynamic equations with damping on time scales by the generalized Riccati transformation and inequality technique. Our results are new even in the continuous and the discrete cases.

A new method, homoclinic (heteroclinic) breather limit method (HBLM), for seeking rogue wave solution to nonlinear evolution equation (NEE) is proposed. (3+1)-dimensional Yu-Toda-Sasa-Fukuyama (YTSF) equation is used as an example to illustrate the effectiveness of the suggested method. A new family of two-wave solution, rational breather wave solution, is obtained by extended homoclinic test method, and it is just a rogue wave solution. This result shows rogue wave can come from extreme behavior of breather solitary wave for (3+1)-dimensional nonlinear wave fields.

We study two discrete predator-prey models in patchy environment, one without dispersal corridors and one with dispersal corridors. Dispersal corridors are passes that allow the migration of species from one patch to another and their existence may influence the permanence of the model. We will offer sufficient conditions to guarantee the permanence of the two predator-prey models. By comparing the two permanence criteria, we discuss the effects of dispersal corridors on the permanence of the predator-prey model. It is found that the dispersion of the prey from one patch to another is helpful to the permanence of the prey if the population growth of the prey is density dependent; however, this dispersion of the prey could be disadvantageous or advantageous to the permanence of the predator. Five numerical examples are presented to confirm the theoretical results obtained and to illustrate the effects of dispersal corridors on the permanence of the predator-prey model.

Two ellipsoidal ultimate boundary regions of a special three-dimensional (3D) chaotic system are proposed. To this chaotic system, the linear coefficient of the $i$th state variable in the $i$th state equation has the same sign; it also has two one-order terms and one quadratic cross-product term in each equation. A numerical solution and an analytical expression of the ultimate bounds are received. To get the analytical expression of the ultimate boundary region, a new result of one maximum optimization question is proved. The corresponding ultimate boundary regions are demonstrated through numerical simulations. Utilizing the bounds obtained, a linear controller is proposed to achieve the complete chaos synchronization. Numerical simulation exhibits the feasibility of the designed scheme.

This paper deals with the oscillatory behavior of forced second-order integrodynamic equations on time scales. The results are new for the continuous and discrete cases and can be applied to Volterra integral equation on time scale. We also provide a numerical example in the continuous case to illustrate the results.

We use the method of coincidence degree and construct suitable Lyapunov functional to investigate the existence and global exponential stability of antiperiodic solutions of impulsive Cohen-Grossberg neural networks with delays on time scales. Our results are new even if the time scale $\mathbb{T}=\mathbb{R}$ or $\mathbb{Z}$. An example is given to illustrate our feasible results.

We prove that the existence of the mean values of coefficients is sufficient for second-order half-linear Euler-type differential equations to be conditionally oscillatory. We explicitly find an oscillation constant even for the considered equations whose coefficients can change sign. Our results cover known results concerning periodic and almost periodic positive coefficients and extend them to larger classes of equations. We give examples and corollaries which illustrate cases that our results solve. We also mention an application of the presented results in the theory of partial differential equations.

By using of the Gronwall inequality, we prove the Hyers-Ulam stability of differential equations of second order with initial conditions.

We prove the existence and uniqueness of solution for a system of fractional differential equations. Our results are based on the nonlinear alternative of Leray-Schauder type and Banach’s fixed-point theorem.

To solve the charge planning problem involving charges and the orders in each charge, a traveling salesman problem based charge planning model and the improved cross entropy algorithm are proposed. Firstly, the charge planning problem with unknown charge number is modeled as a traveling salesman problem. The objective of the model is to minimize the dissimilarity costs between each order and its charge center order, the open order costs, and the unselected order costs. Secondly, the improved cross entropy algorithm is proposed with the improved initial state transition probability matrix which is constructed according to the differences of steel grades and order widths between orders. Finally, an actual numerical example shows the effectiveness of the model and the algorithm.

Circulant matrices may play a crucial role in solving various differential equations. In this paper, the techniques used herein are based on the inverse factorization of polynomial. We give the explicit determinants of the RFP$r$L$r$R circulant matrices and RLP$r$F$r$L circulant matrices involving Fibonacci, Lucas, Pell, and Pell-Lucas number, respectively.

Circulant type matrices have become an important tool in solving differential equations. In this paper, we consider circulant type matrices, including the circulant and left circulant and $g$-circulant matrices with the sum and product of Fibonacci and Lucas numbers. Firstly, we discuss the invertibility of the circulant matrix and present the determinant and the inverse matrix by constructing the transformation matrices. Furthermore, the invertibility of the left circulant and $g$-circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the left circulant and $g$-circulant matrices by utilizing the relation between left circulant, and $g$-circulant matrices and circulant matrix, respectively.

Block circulant and circulant matrices have already become an ideal research area for solving various differential equations. In this paper, we give the definition and the basic properties of FLS $R$-factor block circulant (retrocirculant) matrix over field $\mathbb{F}$. Fast algorithms for solving systems of linear equations involving these matrices are presented by the fast algorithm for computing matrix polynomials. The unique solution is obtained when such matrix over a field $\mathbb{F}$ is nonsingular. Fast algorithms for solving the unique solution of the inverse problem of $\mathcal{A}X=b$ in the class of the level-2 FLS $(R,r)$-circulant(retrocirculant) matrix of type $(m,n)$ over field $\mathbb{F}$ are given by the right largest common factor of the matrix polynomial. Numerical examples show the effectiveness of the algorithms.

We establish four new oscillation criteria of Grace-type for the second-order nonlinear dynamic equations with damping. These criteria extend known criteria for corresponding dynamic equations. Our results are new even in the continuous and the discrete cases.

Circulant matrices have important applications in solving various differential equations. The level-k scaled factor circulant matrix over any field is introduced. Algorithms for finding the minimal polynomial of this kind of matrices over any field are presented by means of the algorithm for the Gröbner basis of the ideal in the polynomial ring. And two algorithms for finding the inverses of such matrices are also presented. Finally, an algorithm for computing the inverse of partitioned matrix with level-k scaled factor circulant matrix blocks over any field is given by using the Schur complement, which can be realized by CoCoA 4.0, an algebraic system, over the field of rational numbers or the field of residue classes of modulo prime number.

The energy management strategy (EMS) for a plugin hybrid electric vehicle (PHEV) is proposed based on the driving cycle model and dynamic programming (DP) algorithm. A driving cycle model is constructed by collecting and processing the driving data of a certain school bus. The state of charge (SOC) profile can be obtained by the DP algorithm for the whole driving cycle. In order to optimize the energy management strategy in the hybrid power system, the optimal motor torque control sequence can be calculated using the DP algorithm for the segments between the traffic intersections. Compared with the traditional charge depleting-charge sustaining (CDCS) strategy, the test results on the ADVISOR platform show a significant improvement in fuel consumption using the EMS proposed in this paper.

Circulant matrix family is used for modeling many problems arising in solving various differential equations. The RSFPLR circulant matrices and RSLPFL circulant matrices are two special circulant matrices. The techniques used herein are based on the inverse factorization of polynomial. The exact determinants of these matrices involving Perrin, Padovan, Tribonacci, and the generalized Lucas number are given, respectively.

The aim of this work is to discuss asymptotic properties of a class of third-order nonlinear neutral functional differential equations. The results obtained extend and improve some related known results. Two examples are given to illustrate the main results.

We investigate the periodic nature of the positive solutions of the fuzzy max-difference equation $x_{n+1}=\text{max}\left\{A_n/x_{n-m},x_{n-k}\right\},n=\mathrm{0,1},\dots$, where $k,m\in \{\mathrm{1,2},\dots \}$, $A_n$ is a periodic sequence of fuzzy numbers, and $x_{-d},x_{-d+1},\dots ,x_0$ are positive fuzzy numbers with $d=\left\{m,k\right\}$. We show that every positive solution of this equation is eventually periodic with period $k+1$.

Circulant matrices play an important role in solving ordinary and partial differential equations. In this paper, by using the inverse factorization of polynomial of degree n, the explicit determinants of circulant and left circulant matrix involving Tribonacci numbers or generalized Lucas numbers are expressed in terms of Tribonacci numbers and generalized Lucas numbers only. Furthermore, four kinds of norms and bounds for the spread of these matrices are given, respectively.

The skew-circulant matrix has been used in solving ordinary differential equations. We prove that the set of skew-circulants with complex entries has an idempotent basis. On that basis, a skew-cyclic group of automorphisms and functional equations on the skew-circulant algebra is introduced. And different operators on linear vector space that are isomorphic to the algebra of $n\times n$ complex skew-circulant matrices are displayed in this paper.

We introduce a type of fully nonlinear path-dependent (parabolic) partial differential equation (PDE) in which the path ${\omega }_t$ on an interval [0, $t$] becomes the basic variable in the place of classical variables $\left(t,x\right)\in [0,T]\times {ℝ}^d$. Then we study the comparison theorem of fully nonlinear PPDE and give some of its applications.

We present new oscillation criteria for the second order nonlinear dynamic equation ${[r(t){\varphi }_{\gamma }(x^{\mathrm{\Delta }}(t))]}^{\mathrm{\Delta }}+q_{\mathrm{0}}(t){\varphi }_{\gamma }(x(g_{\mathrm{0}}(t)))+{\int }_a^b\mathrm{‍}q(t,s){\varphi }_{\alpha (s)}(x(g(t,s)))\mathrm{\Delta }\zeta (s)=\mathrm{0}$ under mild assumptions. Our results generalize and improve some known results for oscillation of second order nonlinear dynamic equations. Several examples are worked out to illustrate the main results.

This paper is concerned with second-order nonlinear damped dynamic equations on time scales of the following more general form ${(p(t)k_1(x(t),x^{\Delta }(t)))}^{\Delta }+r(t)k_2(x(t),x^{\Delta }(t))x^{\Delta }(t)+f(t,x(\sigma (t)))=0$. New oscillation results are given to handle some cases not covered by known criteria. An illustrative example is also presented.

We set up a generalized Solow-Swan model to study the exogenous impact of population, saving rate, technological change, and labor participation rate on economic growth. By introducing generalized exogenous variables into the classical Solow-Swan model, we obtain a nonautomatic differential equation. It is proved that the solution of the differential equation is asymptotically stable if the generalized exogenous variables converge and does not converge when one of the generalized exogenous variables persistently oscillates.

We consider a class of two-dimensional nonlinear dynamic system with a forced term on a time scale $𝕋$ and obtain sufficient conditions for all solutions of the system to be oscillatory. Our results not only unify the oscillation of two-dimensional differential systems and difference systems but also improve the oscillation results that have been established by Saker, 2005, since our results are not restricted to the case where $b(t)\ne 0$ for all $t\in 𝕋$ and $g(u)=u$. Some examples are given to illustrate the results.

This paper discusses oscillatory and asymptotic behavior of solutions of a class of third-order nonlinear functional differential equations. By using the generalized Riccati transformation and the integral averaging technique, three new sufficient conditions which insure that the solution is oscillatory or converges to zero are established. The results obtained essentially generalize and improve the earlier ones.

A class of third-order nonlinear delay dynamic equations on time scales is studied. By using the generalized Riccati transformation and the inequality technique, four new sufficient conditions which ensure that every solution is oscillatory or converges to zero are established. The results obtained essentially improve earlier ones. Some examples are considered to illustrate the main results.

This paper investigates the stability of switched nonlinear (SN) systems in two cases: (1) all subsystems are globally asymptotically stable (GAS), and (2) both GAS subsystems and unstable subsystems coexist, and it proposes a number of new results on the stability analysis. Firstly, an improved average dwell time (ADT) method is presented for the stability of such switched system by extending our previous dwell time method. In particular, an improved mode-dependent average dwell time (MDADT) method for the switched systems whose subsystems are quadratically stable (QS) is also obtained. Secondly, based on the improved ADT and MDADT methods, several new results to the stability analysis are obtained. It should be pointed out that the obtained results have two advantages over the existing ones; one is that the improved ADT method simplifies the conditions of the existing ADT method, the other is that the obtained lower bound of ADT (${\tau }_a^{*}$) is also smaller than that obtained by other methods. Finally, illustrative examples are given to show the correctness and the effectiveness of the proposed methods.

We consider the following difference equation $x_{n+1}=x_{n-1}g(x_n)$, $n=\mathrm{0,1},\dots ,$ where initial values $x_{-1},x_0\in [0,+\infty )$ and $g:[0,+\infty )\to (\mathrm{0,1}]$ is a strictly decreasing continuous surjective function. We show the following. (1) Every positive solution of this equation converges to $a,0,a,0,\dots ,$ or $0,a,0,a,\dots$ for some $a\in [0,+\infty )$. (2) Assume $a\in (0,+\infty )$. Then the set of initial conditions $(x_{-1},x_0)\in (0,+\infty )\times (0,+\infty )$ such that the positive solutions of this equation converge to $a,0,a,0,\dots ,$ or $0,a,0,a,\dots$ is a unique strictly increasing continuous function or an empty set.