Search

A new 3D chaotic system with only one nonlinear term is constructed in this paper, which has conservative and dissipative features respectively by choosing different parameters. The complex dynamic behaviors of the system are analyzed theoretically and numerically by using nonlinear analytical tools, such as Lyapunov exponents, phase portraits, and bifurcation diagrams. In a certain range of parameters, the new system is a non-Hamiltonian conservative chaotic system, which can produce infinitely many coexisting conservative flows. With the variation of parameters, the divergence of the system is less than zero, and the system can also generate hidden and self-excited attractors with multi-scroll. To verify the feasibility of the new system, it has been implemented by field-programmable gate array; phase diagrams obtained in the experiment are consistent with the simulation of the data.

In this paper we gave a general formula for particular solution of the non-homogenous linear ordinary differential equation in a method of variation of parameters of kth order. We solved the linear...

This study proposed an efficient numerical technique for coupled differential equations (CDEs) using the clique polynomials of the Complete graph. Recently, Graph theory has dragged the attention of many mathematicians due to its wide applications. Here, some problems have been considered to examine the proposed scheme proficiency. Some theorems on convergence are discussed. Here, the CDEs are rehabilitated into an algebraic equation system using the clique polynomials and collocation technique. The proposed scheme results are compared with the undetermined coefficients (UDCM) and variation of parameters method (VPM) solutions through tables and graphs. Obtained results reveal that the current approach is more accurate.

In this study, we propose a numerical scheme for stochastic oscillators with additive noise obtained by the method of variation of constants formula using generalized numerical integrators. For both of the displacement and the velocity components, we show that the scheme has an order of 3/2 in one step convergence and a first order in overall convergence. Theoretical statements are supported by numerical experiments.

The article is devoted to the problem of obtaining computationally efficient mathematical models of dynamic objects with viscoelastic properties on the basis of equivalent transformations of primary mathematical descriptions in the form of integro-differential equations to Volterra integral equations. This takes into account the significantly growing in recent years, the possibilities of computer engineering and software engineering, as well as the positive properties of integral models in their numerical implementation: computational stability, good convergence, resistance to high-frequency spectra of noise, a computational efficiency. Methods of transformation of dynamic models are given, which allow to reduce integro-differential equations to integral equations with Volterra operator, which simplifies the problem of creating modelling algorithms. In particular, the application of integral models in computer modelling problems, which are obtained from primary integro-differential models using the following methods of equivalent transformations: the method of variation of constants, the method of senior derivative, the modified method of substituting variables, which in numerical implementation using quadrature formulas allows to obtain recurrent formulas. An example of application of this method is given. The possibility of solving the direct and inverse problem of determining the impact on a multimass system using the integral form of the Cauchy problem is demonstrated. It is shown that the problem of developing more complete and thorough mathematical models of dynamical systems, namely in the form of integro-differential equations, is important and relevant, and their numerical implementation is necessary for a wide range of applied modelling problems.

Made available in DSpace on 2020-12-12T01:23:01Z (GMT). No. of bitstreams: 0 Previous issue date: 2020-01-01 In this paper, an abstract degenerate hyperbolic equation is considered that includes the semilinear Blackstock–Crighton–Westervelt equation. By proposing a new approach based on strongly continuous semigroups and resolvent families of operators, we prove an explicit representation of the strong and mild solutions for the linearized model by means of a kind of variation of parameters formula. Moreover, we show that under nonlocal initial conditions, the existence of a mild solution of the semilinear equation can be established. Institute of Biosciences Humanities and Exact Sciences (Ibilce) São Paulo State University (Unesp), Campus São José do Rio Preto Departamento de Matemática y Ciencia de la Computación Universidad de Santiago de Chile, las Sophoras 173, Estación central Institute of Biosciences Humanities and Exact Sciences (Ibilce) São Paulo State University (Unesp), Campus São José do Rio Preto

A simple, yet accurate modified multi-scale method (MMSM) for an approximately analytical solution in nonlinear oscillators with two time scales under forced harmonic excitation is proposed. This method depends on the classical multi-scale method (MSM) and the method of variation of parameters. Assuming that the forced excitation is a constant, one could easily obtain the approximate analytical solution of the simplified system based on the traditional MSM. Then, this solution for the oscillator under forced harmonic excitation could be established after replacing the harmonic excitation by the constant excitation. To certify the correctness and precision of the proposed analytical method, the van der Pol system with two scales subject to slowly periodic excitation is investigated; this system presents rich dynamical phenomena such as spiking (SP), spiking-quiescence (SP-QS), and quiescence (QS) responses. The approximate analytical expressions of the three types of responses are given by the MMSM, and it can be found that the precision of the new analytical method is higher than that of the classical MSM and better than that of the harmonic balance method (HBM). The results obtained by the present method are considerably better than those obtained by traditional methods, quantitatively and qualitatively, particularly when the excitation frequency is far less than the natural frequency of the system.

B-method is a novel method developed by Beck et al. (SIAM J. Sci. Comput. 37(5), A2998–A3029, 2015), and has been shown theoretically to be very advantageous in time discretization of the second-order parabolic equations with blow-up solutions. In this paper, we extend the B-method to approximate the blow-up solution of a class of fourth-order parabolic equations, which plays very important role in many engineering applications. First, by following the systematic means of constructing numerical schemes based on the technique of variation of constants proposed by Beck et al., we give some B-method schemes for the fourth-order semilinear parabolic equations. Second, we perform a truncation error analysis to show when and why the B-method scheme is advantageous over its classical counterpart. Third, we take one of the constructed numerical schemes as an example to show the well-posedness using the technique of upper and lower solutions. Last, we carry out numerical experiments to approximate the blow-up solutions and illustrate the efficiency of our numerical schemes.

In order to study the dissemination mechanism of knowledge, a SIRI dynamics model with the learning rate, the forgetting rate and the recalling rate is constructed in this paper. Stability of equilibria and global dynamics of the SIRI model are analyzed. Two thresholds that determine whether knowledge is disseminated are given. We describe the stability of the equilibria for the SIRI model in which there are an equilibrium and a line of equilibria. In particular, we find the dividing curve function which is used to partition invariant set in order to discuss the local stability, and obtain the equation of the wave peak value or wave trough value in the process of knowledge dissemination. Numerical simulations are provided to support the theoretical results. The complicated dynamics properties exhibit that the model is very sensitive to variation of parameters, which play an important role on controlling and administering the knowledge dissemination.

The Aim of the paper is to develop an algorithm of prompt detection of the moment of dependability characteristics variation in a system that consists of a set of homogeneous elements, assuming that failures of such elements occur at random moments in time, are a Poisson flow of events and, consequently, the time intervals between them are an exponential probability distribution. In order to solve the problem, it is suggested using one of the classical algorithms of detection of “imbalance” of a discrete random process, i.e. spontaneous change of one of its probabilistic characteristics. As such a characteristic, the exponential distribution parameter θ was chosen, that is uniquely associated with the mean time between failures Тmn: θ = 1/Тmn. It is believed that the imbalance consists in the discontinuous variation of parameter θ from the initial steady state θ = θ0 to the level of minimal (expected, maximum allowable, critical) imbalance, when θ = θ1 > θ0. In this paper, the imbalance is detected using the cumulative sum algorithm (CUSUM) as it has certain optimal properties and is widely used in practice. For this algorithm, the required design ratios, descriptions of its properties and features are provided. The paper proposes a procedure for synthesizing the control algorithm with desired properties, in the course of which, based on the user-selected values of desired mean time between false alarms , initial basic level θ0 and nominal imbalance θ1 > θ0, the value of decision boundary Н is identified, the speed of algorithm action is estimated trough the calculation of the average lag in the detection of nominal imbalance , along with its efficiency for various values of d, that quantitatively characterize the value of imbalance: d=θ1/ θ0. For the purpose of practical implementation of the synthesis procedure, the paper cites reference data, that was obtained by means of simulation and that ensures the development of the control algorithm with required characteristics. It is noted that the presented synthesis procedure can, in principle, also be used for cases of gradual (continuous) change of parameter θ. However, the statistical properties of the control procedure will remain unclear as they require sufficiently intense additional research.

The use of partition function zeros in the study of phase transition is growing in the last decade mainly due to improved numerical methods as well as novel formulations and analysis. In this paper the impact of different parameters choice for the energy probability distribution (EPD) zeros recently introduced by Costa et al is explored in search for optimal values. Our results indicate that the EPD method is very robust against parameter variations and only small deviations on estimated critical temperatures are found even for large variation of parameters, allowing to obtain accurate results with low computational cost. A proposal to circumvent potential convergence issues of the original algorithm is presented and validated for the case where it occurs.

Instead of a single model for one discipline, thermoelastic analysis involves at least a thermal model and a structural model. In many cases, these two analysis steps are followed by a RF or optical simulation to determine the impact on the performance of the instrument of the thermoelastic responses. Each analysis step is accompanied with uncertainties. Besides the thermal and structural analyses, the transfer of temperature data introduces additional uncertainties. A method based on factors of safety, as is used in structural analysis to cover uncertainties, is discussed. Although the method can be pragmatic, this approach is often lacking a physical basis, may not be compatible with the requirements for specific cases, and the values used for these factors are to a large extent ambiguous. Stochastic approaches are known to be powerful for estimating ranges in the responses due to variation of parameters. Although the Monte Carlo method is well known and robust, it is computationally expensive. Promising alternative methods to quantify uncertainties of the responses, among many others, are the Latin hypercube sampling (LHS) method and the Rosenblueth \(2k+1\) Point Estimates Moments (PEM) method. The significance of the random design variables on the random output is detected through a sensitivity analysis, i.e. by a regression analysis method or as an intermediate result of the Rosenblueth PEM method. The section concludes with several worked out examples.

Supercapacitor (SC) is one of the most trending energy storage solutions. The SCs equivalent circuit models have been extensively applied to energy management because of their accuracy and simplicity. However, there is a difficulty in solving the conventional differential equation-based model used to characterize the electrical behavior of SCs operating in constant power applications. Hence, numerical or metaheuristic techniques have been used in the literature to derive SC's internal voltage at any time. In this work, a thorough mathematical analysis that enables a precise calculation of the electrical variables implied in the charge/discharge processes of SCs operated at constant power as a function of time is presented. First, the transcendental discharge voltage expression of SCs operating at constant power is formulated, and then it was solved using the Special Trans Function theory (STFT). The precision of calculation of the method used for solving the transcendental expression is presented and discussed. Second, the transcendental voltage of charging expression of SCs operating at constant power as a function of time is also formulated, and then it was solved using Lambert W equation. Third, a comparison of different methods for solving mentioned equations is presented. Fourth, the electrical variables involved in the charge/discharge processes of SCs – voltage, current, power, energy, state of charge, and power loss are investigated. Furthermore, the results obtained for the variation of parameters and their operating conditions demonstrate the proposed equations’ applicability and accuracy. Finally, the results obtained validate that the closed-form expressions suggested in this paper are accurate and straightforward, which can contribute to proper modeling, investigation, sizing, regulation, and control of constant power SCs in modern energy systems.

This paper deals with the complete controllability for piecewise linear time-varying impulsive systems with multiple time delays in input. The problem is addressed based on algebraic and geometric methods. First, an explicit solution for such systems is established via variation of parameters and mathematical induction. Then, based on the derived solution, complete controllability is investigated and several necessary and sufficient controllability criteria are established analytically for the underlying system. Finally, the obtained controllability conditions are verified through one numerical example. Compared with previous studies, the considered model is more general and the results obtained in this paper are less conservative.

Solutions to nonlinear boundary value problems modelling physical phenomena in engineering applications have traditionally been approximated using numerical methods. More recently, several semi-analytical methods have been developed and used extensively in diverse engineering applications. This work compares the method of variation of parameters to semi-analytical methods for solving nonlinear boundary value problems arising in engineering. The accuracy and efficiency of the method of variation of parameters are compared to those of two widely used semi-analytical methods, the Adomian decomposition method and the differential transformation method, for three practical engineering boundary value problems: (1) the deflection of a cantilevered beam with a concentrated load, (2) an adiabatic tubular chemical reactor which processes an irreversible exothermal reaction, and (3) the electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylinder conduit. The accuracy and convergence of each method is investigated using the error remainder function. The method of variation of parameters is significantly more efficient than both the semi-analytical methods and traditional numerical methods while maintaining comparable accuracy. Unlike the semi-analytical methods, the efficiency of variation of parameters is independent of the nonlinearity. Variation of parameters is shown to be an attractive alternative to semi-analytical methods and traditional numerical methods for solving boundary value problems encountered in engineering applications in which solution efficiency is important.

We introduce the fractional delayed matrix cosine and sine functions and use the variation of constants method to obtain an explicit formula for a solution to the initial value problem for a Riemann–Liouville sequential fractional linear time-delay system of order 1 2 α ≤ 2 . Our results extend those for fractional linear time-delay systems and novel for the classical case α = 1 .

This paper discusses Green’s matrix of nonlinear boundary value problem of first-order differential system with rectangular coeffisients, especially about its properties. In this case, the differential equation of the form with boundary conditions of the form and which is a real matrix with whose entries are continuous on and . , are nonsingular matrices such that and are constant vectors. To get the Green’s matrix and the assosiated generalized Green’s matrix, we change the boundary condition problem into an equivalent differential equation by using the properties of the Moore-Penrose generalized inverse, then its solution is found by using method of variation of parameters. The last we prove that the defined matrices satisfy the properties of green’s function. The result is the corresponding the Green’s matrix and the assosiated generalized Green’s matrix have the property of Green’s functions with the jump-discontinuity.

In this paper, we give rank conditions for controllability and observability of a discrete fractional time-invariant linear system of order ν (0

Lane-Emden type equation is a nonlinear differential equation appears in many fields such as stellar structure, radioactive cooling and modeling of clusters of galaxies. In this work, this equation is investigated using a semi-analytical method called the Variation of parameters method with an auxiliary parameter. In the applied technique, an unknown auxiliary parameter is inserted in Variation of Parameters Method to solve some special cases of these equations. The used algorithm is easy to implement and very effective. The obtained solutions are also fairly accurate.

Analytic solutions in implicit-form are derived for a linear stochastic partial differential equation (SPDE) with fractional derivative terms, which can model the dynamics of a stochastically excited Euler–Bernoulli beam resting on a viscoelastic foundation. Specifically, the original initial–boundary value problem of the SPDE is reduced to an initial value problem of a second-order stochastic differential equation in an appropriate Hilbert space. Next, addressing the abstract Cauchy problem, employing cosine and sine families of operators, and representing the fractional derivative term in a suitable form, a variation of parameters treatment yields the solution in implicit-form. The limiting purely viscous and purely elastic modeling cases are also studied within the same framework. The herein proposed technique and derived implicit-form solutions can be construed as an extension of available results in the literature to account for fractional derivative terms. This generalization is of significant importance given the vast utilization of fractional calculus modeling in engineering mechanics, and in viscoelastic material behavior in particular. In this regard, the herein proposed analytical treatment also supplements existing more numerically oriented solution schemes available in the engineering mechanics literature.

This paper derives a representation of a solution to the initial value problem for a linear fractional delay differential equation with Riemann–Liouville derivative. We apply the method of variation of constants to obtain the representation of a solution via a delayed Mittag-Leffler type matrix function.

This paper is focused on the numerical solution of stochastic harmonic damped oscillators, characterized by a nonlinear high-oscillating time-varying forcing term, coupled with a random force driven by an additive Wiener noise. The scheme is based on asymptotic quadratures together with the application of the variation of constants formula. Theoretical issues are provided and the numerical evidence on selected popular related physical models is also given.

In this paper, we study the problem of exponential stability for the Hopfield neural network with time-varying delays. Different from the existing results, we establish new stability criteria by employing the method of variation of constants and Gronwall’s integral inequality. Finally, we give several examples to show the effectiveness and applicability of the obtained criterion.

In this paper, we provide solution of first order linear intuitionistic fuzzy differential equations by using variation of constant formula and we present the general form of their solutions. Finally, we present some examples to illustrate this work.

In this paper, our purpose is to study the approximate controllability of abstract nonlocal neutral integro-differential equations with finite delay in a Hilbert space using the resolvent operator theory. We derive a variation of parameters formula for representing a solution of the given neutral integro-differential system in the form of resolvent operators and then define a mild solution of the system. We also study the existence of a mild solution of the system with the help of resolvent operator theory. The fractional power theory, $$\alpha $$ -norm, resolvent operator theory, semigroup theory and Krasnoselskii’s fixed point theorem are used to prove the approximate controllability of the system. Finally, we illustrate our results with the help of an example.

The aim of this paper is to develop a center manifold theory for a class of stochastic partial differential equations with a non-dense domain through the Lyapunov-Perron method. We construct a novel variation of constants formula by the resolvent operator to formulate the integrated solutions. Moreover, we impose an additional condition involving a non-decreasing map to deduce the required estimate since Young's convolution inequality is not applicable. As an application, we present a stochastic parabolic equation to illustrate the obtained results.

This work presents new approximate analytical solutions for the Riccati equation (RE) resulting from the application of the method of variation of parameters. The original equation is solved using another RE explicitly dependent on the independent variable. The solutions obtained are easy to implement and highly applicable, which is confirmed by solving several examples corresponding to REs whose solution is known, as well as optimizing the method to determine the density of the members that make up a population. In this way, new perspectives are open in the study of the phenomenon of pattern formation.

We present a new method for computing orbits in the perturbed two-body problem: the position and velocity vectors of the propagated object in Cartesian coordinates are replaced by eight orbital elements, i.e., constants of the unperturbed motion. The proposed elements are uniformly valid for any value of the total energy. Their definition stems from the idea of applying Sundman's time transformation in the framework of the projective decomposition of motion, which is the starting point of the Burdet-Ferr\'andiz linearisation, combined with the Stumpff functions. In analogy with Deprit's ideal elements, the formulation relies on a special reference frame that evolves slowly under the action of external perturbations. We call it the intermediate frame, hence the name of the elements. Two of them are related to the radial motion, the next four, given by Euler parameters, fix the orientation of the intermediate frame. The total energy and a time element complete the state vector. All the necessary formulae for extending the method to orbit determination and uncertainty propagation are provided. For example, the partial derivatives of the position and velocity with respect to the intermediate elements are obtained explicitly together with the inverse partial derivatives. Numerical tests are included to assess the performance of the proposed special perturbation method when propagating the orbit of comets C/2003~T4 (LINEAR) and C/1985~K1 (Machholz).

In this chapter, we apply the concept of resolvent that we developed in Sect. 1.4.1 for vector Volterra integral dynamic equations and show the boundedness of solutions. The resolvent is an abstract term which makes it difficult, if not impossible, to make efficient use of it. However, by the help of Lyapunov functionals and variation of parameters, we will be able to verify all the conditions that are related to the resolvent. In Sect. 6.1.1 we make use of the resolvent along with fixed point theory and show the existence and uniqueness of solutions of nonlinear Volterra dynamic equations. Later in the chapter, we consider nonlinear Volterra integral dynamic equations and construct suitable Lyapunov functional to obtain results regarding boundedness and stability of the zero solution. Contents of this chapter are totally new and not published anywhere else except those of Sect. 6.3 that can be found in Adivar and Raffoul (Appl Math Comput 273:258–266, 2016).

Analytic solutions in implicit form are derived for a nonlinear partial differential equation (PDE) with fractional derivative elements, which can model the dynamics of a deterministically excited Euler-Bernoulli beam resting on a viscoelastic foundation. Specifically, the initial-boundary value problem for the corresponding PDE is reduced to an initial value problem for a nonlinear ordinary differential equation in a Hilbert space. Next, by employing the cosine and sine families of operators, a variation of parameters representation of the solution map is introduced. Due to the presence of a nonlinear term, a local fixed point theorem is employed to prove the local existence and uniqueness of the solution. Relying on the regularity properties of cosine and sine families, taking into account the form of the nonlinear term, and considering the properties of the fractional derivative, the solution map of the abstract problem is cast into a derivative-free analytic solution in implicit-form for the initial boundary value problem. Results corresponding to the limiting purely elastic and purely viscous cases are also provided. The herein developed technique and derived implicit form solutions can be construed as generalizations of available results in the literature to account for fractional derivative elements. This is of significant importance given the vast utilization of fractional calculus modeling in modern engineering mechanics, and in viscoelastic material behavior in particular.

In the case of nonlinear problems, whether in differential or difference equations, it is difficult and in some cases impossible to invert the problem and obtain a suitable mapping that can be effectively used in fixed point theory to qualitatively analyze its solutions. In this paper we consider the existence of a positive sequence and utilize it in the capacity of integrating factor to obtain a new variation of parameters formula. Then, we will use the obtained new variation of parameters formula and revert to the contraction principle to arrive at results concerning, boundedness, periodicity and stability. The author is working on parallel results for the continuous case. Resumen En el caso de problemas no-lineales, ya sea en ecuaciones diferenciales o en diferencias, es difícil y en algunos casos imposible invertir el problema y obtener una aplicación apropiada que pueda ser efectivamente usada en teoría de punto fijo para analizar quantitativamente sus soluciones. En este paper consideramos la existencia de una sucesión positiva y la usamos en la capacidad del factor de integración para obtener una nueva fórmula de variación de parámetros. Luego, usaremos esta nueva fórmula de variación de parámetros y volver al principio de contracción para obtener resultados que involucran acotamiento, periodicidad y estabilidad. El autor se encuentra trabajando en resultados paralelos para el caso continuo

In this paper, we derive the solutions of homogeneous and non-homogeneous nth-order linear general quantum difference equations based on the general quantum difference operator $D_{\beta }$ which is defined by $D_{\beta }{f(t)}= (f(\beta (t))-f(t) )/ (\beta (t)-t )$ , $\beta (t)\neq t$ , where β is a strictly increasing continuous function defined on an interval $I\subseteq \mathbb{R}$ that has only one fixed point $s_{0}\in {I}$ . We also give the sufficient conditions for the existence and uniqueness of solutions of the β-Cauchy problem of these equations. Furthermore, we present the fundamental set of solutions when the coefficients are constants, the β-Wronskian associated with $D_{\beta }$ , and Liouville’s formula for the β-difference equations. Finally, we introduce the undetermined coefficients, the variation of parameters, and the annihilator methods for the non-homogeneous β-difference equations.

The existence and smoothness of centre manifolds and a reduction principle are proven for impulsive delay differential equations. Several intermediate results of theoretical interest are developed, including a variation of constants formula for linear equations in the phase space of right-continuous regulated functions, linear variational equation and smoothness of the nonautonomous process, and a Floquet theorem for periodic systems. Three examples are provided to illustrate the results.

This paper is concerned with the stability issues of stochastic functional differential equations with impulsive effects. Several unified Lyapunov-type stability criteria are established based on the formula of the variation of parameters and the comparison principle of the impulsive systems. The sufficient conditions proposed in this paper depend mainly on the integral average value of the time-varying coefficients and the average impulsive interval, which are less conservative than previous literature. Two examples are provided to demonstrate the effectiveness of the theoretical results.

In this paper, we proposed a new analytical hybrid methods for the solution of conformable fractional differential equations (CFDE), which are based on the recently proposed conformable fractional derivative (CFD) in R. Khalil, M. Al Horani, A. Yusuf and M. Sababhed, A New definition of fractional derivative, J. Comput. Appl. 264 (2014). Moreover, we use the method of variation of parameters and reduction of order based on CFD, for the CFDE. Furthermore, to show the efficiency of the proposed analytical hybrid method, some examples are also presented.

This paper deals with the Ulam stability of linear differential equations by using the method of variation of parameters, which provides a unified method to study the Ulam stability problem of linear differential equations of n-th order with constant and nonconstant coefficients. As an application of the main results, we also obtain the Hyers–Ulam stability of the Cauchy–Euler differential equations of second order, third order and n-th order. Our results make up to some deficiencies in the relevant literature.

In this paper, we introduce a concept of delayed two parameters Mittag-Leffler type matrix function, which is an extension of the classical Mittag-Leffler matrix function. With the help of the delayed two parameters Mittag-Leffler type matrix function, we give an explicit formula of solutions to linear nonhomogeneous fractional delay differential equations via the variation of constants method. In addition, we prove the existence and uniqueness of solutions to nonlinear fractional delay differential equations. Thereafter, we present finite time stability results of nonlinear fractional delay differential equations under mild conditions on nonlinear term. Finally, an example is presented to illustrate the validity of the main theorems.

This paper attempts to construct a new 3-D chaotic system which is easily hardware realisable and fulfil the requirement of a real-life application. The proposed system is relatively more chaotic (based on the first Lyapunov exponent) and has larger bandwidth than 50 available chaotic systems. Lyapunov spectrum and bifurcation diagram of the system reveal that it has chaotic behaviour for a wider range of its parameters. Such characteristic is helpful for an easy hardware realisation of the system. It is to be noted that the reported systems with hidden attractors are not considered here for the comparison. The proposed system has more complexity and disorder due to several unique properties like asymmetry to principle coordinates, dissimilar and asymmetrical equilibria, and non-uniform contraction and expansion of volume in phase space. The proposed system also exhibits asymmetric pairs of coexisting attractors during its operation in two modes. The new system has different routes to chaos including crisis, an inverse crisis, period-doubling and reverse period-doubling routes to chaos with the variation of parameters. MATLAB simulation results confirm the claims, and the results of hardware circuit realisation validate the simulation results. An application of the new system is shown by masking and retrieving an information signal. It is also shown that the proposed system is better than a well-known Lorenz chaotic system for this application. A system with the above unique properties is rare in the literature.

In this paper, we use the modified variation of parameters method (MVPM), an elegant coupling of variation of parameters method (VPM) and Adomian's decomposition method (ADM), for finding the analytical solution of system of nonlinear fractional boundary value problems associated with obstacle. Caputo sense of fractional derivative is used to coup up with fractional term. The results are calculated in terms of series with easily computable components. The used technique is quite easy and convenient for such type problems because it has been previously applied over several nonlinear obstacle systems.

This paper gives a representation of a solution to the Cauchy problem for a fractional linear system with pure delay. We introduce the fractional delayed matrices cosine and sine of a polynomial of degree and establish some properties. Then, we use the variation of constants method to obtain the solution and our results extend those for second order linear system with pure delay. As an application, the representation of a solution is used to obtain a finite time stability result.

We obtain in this article a solution of sequential differential equation involving the Hadamard fractional derivative and focusing the orders in the intervals (1,2) and (2,3). Firstly, we obtain the solution of the linear equations using variation of parameter technique, and next we investigate the existence theorems of the corresponding nonlinear types using some fixed-point theorems. Finally, some examples are given to explain the theorems.

Objectives of the paper are (1) to design two new real and complex no equilibrium point hyperchaotic systems, (2) to design synchronisation technique for the new systems using the contraction theory and (3) to validate the results by using circuit realisation. First a new no equilibrium point hyperchaotic system is developed using a 3-D generalised Lorenz system; then using the new system a new complex no equilibrium point hyperchaotic system is reported. Both the new systems have hidden chaotic attractors. Various dynamical behaviours are observed in the new systems like chaotic, periodic, quasi-periodic and hyperchaotic. Both the systems have inverse crisis route to chaos with the variation of parameter a and crisis route to chaos with the variation of parameters $$b,\ c$$ and d. These phenomena along with hidden attractors in a complex hyperchaotic system are not seen in the literature. Synchronisation between the identical new hyperchaotic systems is achieved using the contraction theory. Further the synchronisation between the identical new complex hyperchaotic systems is achieved using adaptive contraction theory. The proposed synchronisation strategies are validated using the MATLAB simulation and circuit implementation results. Further, an application of the proposed system is shown by transmitting and receiving an audio signal.

In this paper, we consider the asymptotic stability of solutions to impulsive multi-delayed differential equations with linear parts defined by pairwise permutable matrices. First, we introduce the concept for an impulsive multi-delayed Cauchy matrix and then use it to obtain the representation of solutions to linear impulsive Cauchy problems via the variation of constants principle. Next, we give a norm estimate of the impulsive multi-delayed Cauchy matrix and establish sufficient conditions to guarantee that the trivial solutions are asymptotically stable when the nonlinear terms satisfy appropriate conditions. Finally, two numerical examples are given to illustrate the effectiveness of the results.

SummaryIn the method of variation of parameters, a sufficient condition is often imposed in order to obtain a particular solution for a higher order linear nonhomogeneous differential equation. We ...

In this paper, the dynamics of cholera disease, using a mathematical model, have been investigated. The basic reproduction number, using the next generation matrix method, has been computed which informs us about disease spread or control among people. There are two types of equilibria: disease absent and disease present. Stabilities of both equilibria have been explored. The key factors which significantly involve in spreading the disease have been pointed out by calculating the sensitivity indices of the threshold parameter and examining the graphical results of basic reproduction number for different parameters. Intensity of parameters has also been observed through the variation of parameters. A control problem is designed to overcome the scattering of ailment. To get over the disease, the schemes are devised by considering the outcomes of sensitivity analysis. The problem was revisited to see the comparison of analytical and numerical results. It was discovered that numerical results support the analytical results.

In this paper, we have inquired pine wilt disease (PWD) governed by a mathematical model to access its dynamics. By using next generation matrix method, we worked out for basic reproduction number R ∘ , which apprises us about the disease dissemination or control in the community. Two kinds of equilibria, disease absent and disease present, have been established. Stability of both equilibria has been discussed. Lypunove functional theory and graph theoretic approach are used for disease free and endemic equilibrium, respectively. The parameters, expressed in the model, captured the growth in case onsets and the estimated results are almost compatible with the studied actual reported cases. By using the estimated parameters, we found the sensitivity indices of the basic reproduction number through the calculation of ratio of relative change in the parameter to the relative change in R ∘ . Influence of the parameters on the number of infectious pines and bark beetles has also been observed by the variation of parameters. Keeping in mind those vital factors, calculated through sensitivity analysis, that can help to overcome the disease, an effective control strategy has been designed and optimal control problem has been formulated. To get in sight the comparison of analytical results with numerical ones, the problem has been reconsidered, and it has been seen that numerical results, obtained by using estimated parameters, express effectiveness of the applied controls to reduce pine wilt infection.

We aim to find exact solutions to some generalized KP- and BKP-type equations by the Wronskian technique and the linear superposition principle. By applying Wronskian identities of the bilinear KP hierarchy, three Wronskian formulations are first furnished, with all generating functions for matrix entries satisfying a system of combined linear partial differential equations. Second, we apply the linear superposition principle to exponential traveling wave solutions of the introduced generalized KP and BKP equations. Each exponential wave in the Wronskian and N-wave solutions satisfies the corresponding nonlinear dispersion relation.

In this paper, the synchronization problem is studied for a class of stochastic discrete-time complex networks with partial mixed impulsive effects. The involving impulsive effects, called partial mixed impulses, can be regarded as local and time-varying impulses, which means that impulses are not only injected into a fraction of nodes in networks but also contain synchronizing and desynchronizing impulses at the same time. In order to handle this case, several mathematical techniques are proposed to tackle mixed impulsive effects in discrete-time dynamical systems. Based on the variation of parameters formula, several sufficient criteria are derived to ensure that synchronization of the addressed networks can be achieved in mean square. The obtained criteria not only rely on the strengths of mixed impulses and the impulsive intervals, but also can reduce conservativeness. Finally, a numerical example is presented to show the effectiveness of our results for neural networks.

With the aid of multivariate transformation technique, we obtain periodic and rational solutions for variable-coefficient modified Korteweg–de Vries (vc-mKdV) equation. The rational solutions can be obtained as limiting cases of periodic solutions in analogy with the rational solutions of nonlinear Schrodinger equation (NLSE). Further, the effect of variation of parameters on solutions is studied for physically relevant periodic form. Unlike the effect of variation of parameters on solutions of NLSE, where both amplitude and width can be controlled, only path or trajectory of solutions of vc-mKdV equation can be modulated.

The aim of this study is to develop a fractional version of Duhamel’s principle for a class of fractional partial differential equations. Also we establish the existence of unique solution.