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In this paper, a singularly perturbed linear time-dependent controlled system with a point-wise delay in the input (the control variable) is considered. For this system, the problem of minimum energy control is formulated. An approximate solution to this problem is derived. An illustrative example is presented. [ABSTRACT FROM AUTHOR]

The integrodifferential equations of mathematical physics are objects of research, and construction of interpolating polynomials to obtain approximate solutions of such equations is the subject of investigation. This paper lays out a technique for constructing approximate expressions for functionals on solutions of integrodifferential equations which are an analog of the Hermite polynomial used to interpolate functions. In the example of the diffusion equation, it is shown that the use of such basis solutions allows a substantial increase in the accuracy of the approximate representation of the functionals in comparison to the first approximation of perturbation theory with practically the same computational costs. [ABSTRACT FROM AUTHOR]

In this paper, we continue the study of singularly perturbed differential-difference equations with small shifts, which is motivated by the problem of determination of the expected time for generation of action potentials in nerve cells by random synaptic inputs in the dendrites [1]. We consider a more general boundary-value problem which contains both convection and reaction terms with both type of shifts (negative as well as positive) than the problem discussed in paper [2,7]. We consider the case when the solution of such type of boundary-value problem exhibits boundary layer behavior. An ℇ-uniform convergent scheme based on fitting operator is derived for boundary value problems for singularly perturbed differential-difference equations with small shifts. We introduce an exponential fitting parameter to the standard finite difference scheme which reflects the singularly perturbed nature of differential operator. The method is analyzed for convergence. Several numerical experiments are carried in support of theoretical results and to show the effect of small shifts on the boundary layer solution. [ABSTRACT FROM AUTHOR]

The Hamiltonian and the equations of motion for various modes involving spontaneous and stimulated Raman processes arc derived. The coupled differential equations involving these modes are solved analytically by using an intuitive approach (IA). The solution of the present paper docs not require any short time approximation. As a matter of fact, the present solution for field operators and hence the squeezing for various modes are valid for all interaction time t. In this way, the present paper is more general compared to the earlier investigations where short-time approximations were found a must. By exploiting the solutions for the field operators, we obtain the squeezing effects of input coherent light for pure and for mixed field modes. The IA employs the perturbation theory and hence we came across the secular terms in the solution. Of course, for small coupling constants these secular terms could be summed for all orders. To establish our claim, we remove the secular terms at least for one occasion. [ABSTRACT FROM AUTHOR]

In this paper we present a new method to solve differential inclusions in Hilbert spaces. This method is a Galerkin-like method where we approach the original problem by project- ing the state into a n-dimensional Hilbert space but not the velocity. We prove that the approached problem always has a solution and that, under some compactness conditions, the approached prob- lems have a subsequence which converges strongly pointwisely to a solution of the original differential inclusion. We apply this method to the generalized perturbed sweeping process governed by nonregu- lar sets (equi-uniformly subsmooth or positively α-far). This differential inclusion includes Moreau's sweeping process, the state-dependent sweeping process, and second-order sweeping process for which we give very general existence results. [ABSTRACT FROM AUTHOR]

This paper considers a perturbed Markov-modulated risk model with two-sided jumps, where both the upward and downward jumps follow arbitrary distribution. We first derive a system of differential equations for the Gerber-Shiu function. Furthermore, a numerical result is given based on Chebyshev polynomial approximation. Finally, an example is provided to illustrate themethod. [ABSTRACT FROM AUTHOR]

This paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of ideas and methods of singular perturbation theory. The work done in this area during the periods 1984-2000 and 2000-2005 has already been surveyed in 2002 and 2007 but our main objective is to produce a collection of important research articles of physical significance. In this paper, the crux of research articles published by numerous researchers during 2006-2010 in referred journals has been presented, and this leads to conclusions and recommendations about what methods to use on singular perturbation problems. [ABSTRACT FROM AUTHOR]

We study in this paper an M/ M/1 queue whose server rate depends upon the state of an independent Ornstein–Uhlenbeck diffusion process ( X( t)) so that its value at time t is μ φ( X( t)), where φ( x) is some bounded function and μ>0. We first establish the differential system for the conditional probability density functions of the couple ( L( t), X( t)) in the stationary regime, where L( t) is the number of customers in the system at time t. By assuming that φ( x) is defined by φ( x)=1− ε(( x ∧ a/ ε) ∨(− b/ ε)) for some positive real numbers a, b and ε, we show that the above differential system has a unique solution under some condition on a and b. We then show that this solution is close, in some appropriate sense, to the solution to the differential system obtained when φ is replaced with Φ( x)=1− ε x for sufficiently small ε. We finally perform a perturbation analysis of this latter solution for small ε. This allows us to check at the first order the validity of the so-called reduced service rate approximation, stating that everything happens as if the server rate were constant and equal to $\mu(1-\varepsilon {\mathbb{E}}(X(t)))$ . [ABSTRACT FROM AUTHOR]

This survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of ideas and methods of singular perturbation theory. In continuation of a survey performed earlier, this paper limits its coverage to some standard numerical methods developed by numerous researchers between 2000 and 2005. A summary of the results of some recent methods is presented and this leads to conclusions and recommendations regarding methods to use on singular perturbation problems. Because of space constraints, we considered one-dimensional singularly perturbed boundary value problems only. [ABSTRACT FROM AUTHOR]

In this paper we study the free boundary problem arising as a limit as ɛ → 0 of the singular perturbation problem $${\textrm{div}(A(x)\nabla u) = \Gamma(x) \beta_\varepsilon(u)}$$ , where A = A( x) is Holder continuous, β ɛ converges to the Dirac delta δ0. By studying some suitable level sets of u ɛ, uniform geometric properties are obtained and show to hold for the free boundary of the limit function. A detailed analysis of the free boundary condition is also done. At last, using very recent results of Salsa and Ferrari, we prove that if A and Γ are Lipschitz continuous, the free boundary is a C 1,γ surface around $${\mathcal{H}^{N-1}}$$ a.e. point on the free boundary. [ABSTRACT FROM AUTHOR]

The computational singular perturbation (CSP) method of Lam and Goussis is an iterative method to reduce the dimensionality of systems of ordinary differential equations with multiple time scales. In [J. Nonlinear Sci., 14 (2004), pp. 59-91], the authors of this paper showed that each iteration of the CSP algorithm improves the approximation of the slow manifold by one order. In this paper, it is shown that the CSP method simultaneously approximates the tangent spaces to the fast fibers along which solutions relax to the slow manifold. Again, each iteration adds one order of accuracy. In some studies, the output of the CSP algorithm is postprocessed by linearly projecting initial data onto the slow manifold along these approximate tangent spaces. These projections, in turn, also become successively more accurate. [ABSTRACT FROM AUTHOR]

A class of singularly perturbed two point boundary value problems (BVPs) for third order ordinary differential equations is considered. The BVP is reduced to a weakly coupled system of one first order Ordinary Differential Equation (ODE) with a suitable initial condition and one second order singularly perturbed ODE subject to boundary conditions. In order to solve this system, a computational method is suggested in this paper. This method combines an exponentially fitted finite difference scheme and a classical finite difference scheme. The proposed method is distinguished by the fact that, first we divide the domain of definition of the differential equation into three subintervals called inner and outer regions. Then we solve the boundary value problem over these regions as two point boundary value problems. The terminal boundary conditions of the inner regions are obtained using zero order asymptotic expansion approximation of the solution of the problem. The present method can be extended to system of two equations, of which, one is a first order ODE and the other is a singularly perturbed second order ODE. Examples are presented to illustrate the method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a Van der Pol–Duffing oscillator is studied. Governing equations are solved by using a newly proposed method, namely; the “Homotopy Perturbation Transform Method” (HPTM). The beauty of the paper is error analysis between the exact solutions, approximate solutions and numerical solutions, which shows that our approximate solutions converge very rapidly to the exact solutions. HPTM is not limited to the small parameter, such as in the classical perturbation method. The method gives an analytical solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. Results indicate that this technique is very effective and simple for solving nonlinear oscillatory systems. The solution procedure confirms that this method can be easily extended to other kinds of nonlinear oscillator equations. [Copyright &y& Elsevier]

The radial differential equations associated with separable perturbations of Kerr-Newman black holes are known to admit Liouvillian (closed-form) solutions for constrained frequencies and black hole parameters. In this paper, we show that the parameter constraints are satisfied exactly in the case of no rotation and thereby obtain a countable infinity of exact purely damped quasi-normal modes of fields on a Reissner-Nordstrom background at special values of the black hole charge-mass ratio. We show that with rotation the parameter constraints for Liouvillian quasi-normal modes are satisfied approximately in two distinct physical scenarios, where analytical approximations for angular eigenvalues are known. We arrive at functional expressions for quasi-normal frequencies and wave-functions in the case of near-extremal slow rotation and in a particular case of highly damped scalar modes of Kerr and Kerr-Newman. In the near-extremal case, our formulas extend a recent result of Hod to electromagnetic and gravitational perturbations. [ABSTRACT FROM AUTHOR]

In this paper, in order to solve a class of nonlinear bilevel programming problems, we equivalently transform the nonlinear bilevel programming problems into corresponding single level nonlinear programming problems by using the Karush-Kuhn-Tucker optimality condition. Then, based on penalty function theory, we construct a smooth approximation method for obtaining optimal solutions of classic l1-exact penalty function optimality problems, which is equivalent to the single level nonlinear programming problems. Furthermore, using ∈-approximate optimal solution theory, we prove convergence of a simple ∈-approximate optimal algorithm. Finally, through adding parameters in the constraint set of objective function, we prove some perturbation convergence results for solving the nonlinear bilevel programming problems. [ABSTRACT FROM AUTHOR]

In this paper, the problem of an exponential stability for time-delay systems with interval time-varying delays and nonlinear perturbations is investigated. Based on the Lyapunov method, a new delay-dependent criterion for exponential stability is established in terms of LMI (linear matrix inequalities). Numerical examples are carried out to support the effectiveness of our results. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the age-structured SEIR epidemic model which is described by partial differential equations. Using theory in function analysis and small perturbation, we obtain the existence and stability of steady states under some conditions. [ABSTRACT FROM AUTHOR]

The paper is a survey of the development of the direct Lyapunov method based on matrix-valued functions for systems with structural perturbations. Sufficient conditions for stability in the large, uniform asymptotic stability in the large, exponential stability in the large, and instability of nonlinear continuous systems under structural perturbations are presented. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Camassa–Holm equation, which is a model for shallow water waves. We first establish the existence of solitary wave solutions for the equation without delay. And then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the method of dynamical system, especially the geometric singular perturbation theory and invariant manifold theory. According to the relationship between solitary wave and homoclinic orbit, the Camassa–Holm equation is transformed into the ordinary differential equations with fast variables by using the variable substitution. It is proved that the equation with disturbance also possesses homoclinic orbit, and there exists solitary wave solution of the delayed Camassa–Holm equation. [ABSTRACT FROM AUTHOR]

We study singular perturbation of impulsive system with a proportional-integral-derivative controller (PID controller) and solve an optimal control problem. The perturbation system comprises two important variables, a fast variable and a slow variable. Because of the complexity of the system, it is difficult to find its exact solution. This paper presents an approximation method for solving it. The aim of the approximation method is to reduce the complexity of the system by eliminating the fast variable. The solution of the method is expressed in an integral form, and it is called an approximated mild solution of the perturbed system. An example is provided to illustrate our result. [ABSTRACT FROM AUTHOR]

This paper presents a modified Laplace transform homotopy perturbation method with finite boundary conditions (MLT-HPM) designed to improve the accuracy of the approximate solutions obtained by LT-HPM and other methods. To this purpose, a suitable initial approximation will be introduced, in addition, the residual error in several points of the interest interval (RECP) will be canceled. In order to prove the efficiency of the proposed method a couple of nonlinear ordinary differential equations with mixed boundary conditions, indeed, difficult to approximate, are proposed. The square residual error (S.R.E) of the proposed solutions will result to be of hundredths and tenths, requiring only a first order approximation of MLT-HPM, unlike LT-HPM, which will require more iterations for the same cases study. [ABSTRACT FROM AUTHOR]

The transverse vibrations of axially moving strings with time dependent velocity and its instability are examined in this paper. The velocity function is assumed to be a harmonically varying function about a constant low mean speed. Approximate solutions are sought using the two timescales perturbation method in conjunction with the method of characteristic coordinates. Moreover, it will turn out that infinitely many resonances in the system can arise, when the velocity fluctuation frequency is equal (or close) to an odd multiple of the lowest natural frequency of the constant velocity system. The dynamic displacement-response and the energy of the belt system obtained using the method of characteristic coordinates are compared numerically. It is shown that the energy of the belt system can become large in a resonance case. In addition, the amplitude of vibration is also found to be growing on long timescales, that is, on a timescale of order ε −1 . [ABSTRACT FROM AUTHOR]

This article deals with the asymptotic behavior of nonoscillatory solutions of fourth order linear differential equation where the coefficients are perturbations of constants. We define a change of variable and deduce that the new variable satisfies a third order nonlinear differential equation. We assume three hypotheses. The first hypothesis is related to the constant coefficients and set up that the characteristic polynomial associated with the fourth order linear equation has simple and real roots. The other two hypotheses are related to the behavior of the perturbation functions and establish asymptotic integral smallness conditions of the perturbations. Under these general hypotheses, we obtain four main results. The first two results are related to the application of a fixed point argument to prove that the nonlinear third order equation has a unique solution. The next result concerns with the asymptotic behavior of the solutions of the nonlinear third order equation. The fourth main theorem is introduced to establish the existence of a fundamental system of solutions and to precise the formulas for the asymptotic behavior of the linear fourth order differential equation. In addition, we present an example to show that the results introduced in this paper can be applied in situations where the assumptions of some classical theorems are not satisfied. [ABSTRACT FROM AUTHOR]

In the present paper, an averaging perturbation technique leads to the determination of a time-explicit analytic approximate solution for the motion of a low-Earth-orbiting satellite . The two dominant perturbations are taken into account: the Earth oblateness and the atmospheric drag. The proposed orbit propagation algorithm comprises the Brouwer-Lyddane transformation (direct and inverse), coupled with the analytic solution of the averaged equations of motion. This solution, based on equinoctial elements, is singularity-free, and therefore it stands for low inclinations and small eccentricities as well. The simplifying assumption of a constant atmospheric density is made, which is reasonable for near-circular orbits and short-time orbit propagation. Two sets of time-explicit equations are provided, for moderate and small eccentricities ( $$\mathcal {O} ( e^{4}) =0$$ and $$\mathcal {O}( e^{2}) =0,$$ respectively), and they are obtained by performing (1) a regularization of the original averaged differential equations of motion for the vectorial orbital elements, and (2) Taylor series expansions of the aforementioned equations with respect to the eccentricity. The numerical simulations show that the errors due to the use of the proposed analytic model in the presence of drag are almost the same as the errors of the Brouwer first-order approximation in the absence of drag. [ABSTRACT FROM AUTHOR]

The delay-range-dependent stochastic stability for uncertain neutral Markovian jump systems with interval time-varying delays and nonlinear perturbations is investigated. The perturbations under consideration are time-varying and norm-bounded. By delay interval dividing, a novel augmented Lyapunov functional which contains triple-integral terms to reduce the conservativeness is introduced. Based on the Lyapunov functional approach and the nature of convex combination, some improved delay-range-dependent stochastic stability criteria are obtained in terms of linear matrix inequalities without introducing any free-weighting matrices. Finally, numerical examples are given to illustrate the effectiveness of the developed techniques. [ABSTRACT FROM AUTHOR]

This paper applies a new modification of the Homotopy Perturbation Method that is called Rational Homotopy Perturbation Method (RHPM) to obtain an analytic approximation of stiff systems of ordinary differential equations. The procedure of the method will be explained briefly and some examples are presented to illustrate the method. The results of RHPM, are compared with those of traditional Homotopy Perturbation Method. The results show the accuracy and fast convergence of the proposed method. [ABSTRACT FROM AUTHOR]

This paper deals with the singularly perturbed boundary value problem for a linear second order differential-difference equation of convection-diffusion type. In the numerical treatment of such type of problems, first we use Taylor’s approximation to tackle the term containing the small shift. A fitting parameter has been introduced in a tridiagonal finite difference method and is obtained from the theory of singular perturbations. Thomas algorithm is used to solve the tridiagonal system. The method is analysed for convergence. Several numerical examples are solved to demonstrate the applicability of the method. Graphs are plotted for the solutions of these problems to illustrate the effect of small shift on the boundary layer solution. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the combined Laplace transform and new homotopy perturbation methods (LTNHPM) is employed to obtain the closed form solutions of the coupled Burgers equation. The results obtained are in good agreement with the exact solution. These results show that the technique introduced here is accurate and easy to apply. [Copyright &y& Elsevier]

In this paper, the Homotopy Perturbation Method (HPM) has been implemented for solving the extended Fisher-Kolomogrov-Petrovskii-Piskunov (FKPP) equation. The HPM is an approximate method which yields an analytical solution in the form of a convergent power series with components, which can be easily computed. We show that the HPM can generate highly accurate solutions for the FKPP equation. [ABSTRACT FROM AUTHOR]

Regularization is used in order to obtain a reasonable estimate of the solution to an ill-posed inverse problem. One common form of regularization is to use a filter to reduce the influence of components corresponding to small singular values, perhaps using a Tikhonov least squares formulation. In this work, we break the problem into subproblems with narrower bands of singular values using spectrally defined windows, and we regularize each subproblem individually. We show how to use standard parameter-choice methods, such as the discrepancy principle and generalized cross-validation, in a windowed regularization framework. A perturbation analysis gives sensitivity estimates. We demonstrate the effectiveness of our algorithms on deblurring images and on the backward heat equation. [ABSTRACT FROM AUTHOR]

We apply the well-known Kovacic algorithm to find closed form, i.e., Liouvillian solutions, to the differential equations governing perturbations of black holes. Our analysis includes the full gravitational perturbations of Schwarzschild and Kerr, the full gravitational and electromagnetic perturbations of Reissner-Nordstrom, and specialized perturbations of the Kerr-Newman geometry. We also include the extreme geometries. We find all frequencies ω, in terms of black hole parameters and an integer n, which allow Liouvillian perturbations. We display many classes of black hole parameter values and their corresponding Liouvillian perturbations, including new closed-form perturbations of Kerr and Reissner-Nordstrom. We also prove that the only type 1 Liouvillian perturbations of Schwarzschild are the known algebraically special ones and that type 2 Liouvillian solutions do not exist for extreme geometries. In cases where we do not prove the existence or nonexistence of Liouvillian perturbations we obtain sequences of Diophantine equations on which decidability rests. [ABSTRACT FROM AUTHOR]

Consideration was given to stability of the solution of the system of differential equations to the function of a locally bounded variation added to the right-hand side of the derivative (understood in the general sense). Simple attributes of stability and asymptotic stability were established for different classes of these equations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a unifying approach to the problems of computing of stability radii of positive linear systems. First, we study stability radii of linear time-invariant parameter-varying differential systems. A formula for the complex stability radius under multi perturbations is given. Then, under hypotheses of positivity of the system matrices, we prove that the complex, real and positive stability radii of the system under multi perturbations (or affine perturbations) coincide and they are computed via simple formulae. As applications, we consider problems of computing of (strong) stability radii of linear time-invariant time-delay differential systems and computing of stability radii of positive linear functional differential equations under multi perturbations and affine perturbations. We show that for a class of positive linear time-delay differential systems, the stability radii of the system under multi perturbations (or affine perturbations) are equal to the strong stability radii. Next, we prove that the stability radii of a positive linear functional differential equation under multi perturbations (or affine perturbations) are equal to those of the associated linear time-invariant parameter-varying differential system. In particular, we get back some explicit formulas for these stability radii which are given recently in [P.H.A. Ngoc, Strong stability radii of positive linear time-delay systems, Internat. J. Robust Nonlinear Control 15 (2005) 459–472; P.H.A. Ngoc, N.K. Son, Stability radii of positive linear functional differential equations under multi perturbations, SIAM J. Control Optim. 43 (2005) 2278–2295]. Finally, we give two examples to illustrate the obtained results. [Copyright &y& Elsevier]

Representing surface tension by nonlinear relationship on temperature, the boundary value problem of linear stability differential equation on small perturbation is derived. Under the condition of the isothermal wall the effects of nonlinear surface tension on stability of heat transfer in saturated liquid film of different liquid low boiling point gases are investigated as wall temperature is varied. [ABSTRACT FROM AUTHOR]

Two complex matrix pairs (A, B) and (A', B') are contragrediently equivalent if there are nonsingular S and R such that (A', B') = (S-1AR, R-1BS). M.I. García-Planas and V.V. Sergeichuk (1999) constructed a miniversal deformation of a canonical pair (A, B) for contragredient equivalence; that is, a simple normal form to which all matrix pairs (A + Ã, B + ...) close to (A, B) can be reduced by contragredient equivalence transformations that smoothly depend on the entries of à and .... Each perturbation (Ã, ...) of (A, B) defines the first order induced perturbation A... + Ã... of the matrix AB, which is the first order summand in the product (A + Ã)(B + ...) = AB + A... + ÃB + Ã.... We find all canonical matrix pairs (A, B), for which the first order induced perturbations Ae B+ e AB are nonzero for all nonzero perturbations in the normal form of García-Planas and Sergeichuk. This problem arises in the theory of matrix differential equations ẋ = Cx, whose product of two matrices: C = AB; using the substitution x = Sy, one can reduce C by similarity transformations S-1CS and (A, B) by contragredient equivalence transformations (S-1AR, R-1BS). [ABSTRACT FROM AUTHOR]

By the variational methods, the existence criteria of infinitely many nontrivial solutions for fractional differential equations with impulses and perturbation are established. An example is given to illustrate main results. Recent results in the literature are generalized and improved. [ABSTRACT FROM AUTHOR]

We use the reduction method, which allows one to reduce the study of perturbations of multiple eigenvalues to perturbations of simple eigenvalues, to analyze the general perturbation problem for Fredholm points of the discrete spectrum of linear operator functions analytically depending on the spectral parameter. The same method is used to study a perturbation of multiple Fredholm points of the discrete Schmidt spectrum ( s-numbers) of a linear operator. We present an example of a problem on a perturbation of the domain of the Sturm-Liouville problem for a second-order differential operator. [ABSTRACT FROM AUTHOR]

In this communication, we describe the Homotopy Perturbation Method with Laplace Transform (LT-HPM), which is used to solve the Lane-Emden type differential equations. It's very difficult to solve numerically the Lane-Emden types of the differential equation. Here we implemented this method for two linear homogeneous, two linear nonhomogeneous, and four nonlinear homogeneous Lane-Emden type differential equations and use their appropriate comparisons with exact solutions. In the current study, some examples are better than other existing methods with their nearer results in the form of power series. The Laplace transform used to accelerate the convergence of power series and the results are shown in the tables and graphs which have good agreement with the other existing method in the literature. The results show that LT-HPM is very effective and easy to implement. [ABSTRACT FROM AUTHOR]

We focus on a subdiffusion–reaction system in which substances are separated at the initial moment. This system is described by nonlinear differential subdiffusion–reaction equations with a fractional time derivative. These equations are very difficult to solve but there exist methods which allow us to solve them approximately. We discuss how useful such methods are, in particular, the quasistatic approximation method and the perturbation method in analytical solving subdiffusion–reaction equations. [ABSTRACT FROM AUTHOR]

Homotopy perturbation method is used to extend the approximate analytical solutions of non-linear reaction equations describing enzyme kinetics for combinations of parameters for which solutions obtained in previous works are not valid. Also, by constructing a new homotopy, alternative approximate analytical expressions for substrate, substrate-enzyme complex and product concentrations are found. These first-order approximate solutions give more accurate results than the second-order approximations derived in previous works. [ABSTRACT FROM AUTHOR]

We consider the (repulsive) Euler–Poisson system for the electrons in two dimensions and prove that small smooth perturbations of a constant background exist for all time and remain smooth (never develop shocks). This extends to 2D the work of Guo 6. [ABSTRACT FROM AUTHOR]

With the help of second-order nonlinear interactions and hence Hamiltonian, we construct the equations of motion corresponding to pump and signal (second harmonic) fields in a second harmonic generation. The corresponding coupled differential equations involving the non-commuting field operators are not solvable in closed analytical forms. With the help of a perturbation method, we obtain analytical solutions of these field operators up to cubic orders in the interaction constant. In an appropriate limit (truncating the solution up to the second order in the interaction constant), these solutions lead to the existing solutions under the short-time approximation. The present analytical solutions are exploited to investigate the squeezing and the antibunching of photons of the input coherent light coupled to the said second-order nonlinear medium. We report the squeezing and antibunching effects of the pump field even for solutions up to the second order and hence the present results are consistent with earlier results. However, for the second harmonic mode, we report the squeezing involving the leading cubic term in the interaction constant. [ABSTRACT FROM PUBLISHER]

We consider a system of differential equations that consists of two parts, a regularly perturbed and a singularly perturbed one. We assume that the matrix of the linear part of the regularly perturbed system has pure imaginary eigenvalues, while the matrix of the singularly perturbed part is hyperbolic; i.e., all of its eigenvalues have nonzero real parts. We derive the so-called determining equation, to each of whose positive solutions there corresponds an invariant torus. We show that, in general position, there is an m-dimensional invariant torus bifurcating from the equilibrium as the small parameter passes through the critical zero point; here m is the number of pure imaginary eigenvalues. In addition, in the degenerate case, we find conditions for the coexistence of two- and three-dimensional invariant tori. [ABSTRACT FROM AUTHOR]