Quadratic Differential Equations Research Articles

The global behavior of a quadratic difference equation

In this paper we will present the Julia set and the global behavior of a quadratic second order difference equation of type xn+1 = axnxn-1 + ax2n-1 + bxn-1 where a > 0 and 0 ? b < 1 with non-negative initial conditions.

Null-controllability of hypoelliptic quadratic differential equations

We study the null-controllability of parabolic equations associated to a general class of hypoelliptic quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. We consider in this work the class of accretive quadratic operators with zero singular spaces. These possibly degenerate non-selfadjoint differential operators are known to be hypoelliptic and to generate contraction semigroups which are smoothing in specific Gelfand-Shilov spaces for any positive time. Thanks to this regularizing effect, we prove by adapting the Lebeau-Robbiano method that parabolic equations associated to these operators are null-controllable in any positive time from control regions, for which null-controllability is classically known to hold in the case of the heat equation on the whole space. Some applications of this result are then given to the study of parabolic equations associated to hypoelliptic Ornstein-Uhlenbeck operators acting on weighted $L^2$ spaces with respect to invariant measures. By using the same strategy, we also establish the null-controllability in any positive time from the same control regions for parabolic equations associated to any hypoelliptic Ornstein-Uhlenbeck operator acting on the flat $L^2$ space extending in particular the known results for the heat equation or the Kolmogorov equation on the whole space.

Fifth Order Predictor-Corrector Method for Solving Quadratic Riccati Differential Equations

In this paper, fifth order predictor-corrector method is presented for solving quadratic Riccati differential equations. First, the interval is discretized and then the method is formulated by using the Newton’s backward difference interpolation formula. The stability and convergence of the method have been investigated. To validate the applicability of the proposed method, three model examples with exact solutions have been considered and numerically solved by using MATLAB software. The numerical results are presented in tables and figures for different values of mesh size h. Pointwise absolute errors and maximum absolute errors are also estimated. Concisely, the present method gives better result than some existing numerical methods reported in the literature.

Phase portraits of Abel quadratic differential systems of the second kind

ABSTRACTWe provide normal forms and the global phase portraits on the Poincaré disk of some Abel quadratic differential equations of the second kind. Moreover, we also provide the bifurcation diagrams for these global phase portraits.

An analytical approach for variance swaps with an Ornstein-Uhlenbeck process

Pricing variance swaps have become a popular subject recently, and most research of this type come under Heston’s two-factor model. This paper is an extension of some recent research which used the dimension-reduction technique based on the Heston model. A new closed-form pricing formula focusing on a log-return variance swap is presented here, under the assumption that the underlying asset prices can be described by a mean-reverting Gaussian volatility model (Ornstein–Uhlenbeck process). Numerical tests in two respects using the Monte Carlo (MC) simulation are included. Moreover, we discuss a procedure of solving a quadratic differential equation with one variable. Our method can avoid the previously encountered limitations, but requires more time for calculation than other recent analytical discrete models. doi:10.1017/S1446181117000268

Collocated Mixed Discrete Least Squares Meshless (CMDLSM) method for solving quadratic partial differential equations

In this paper, a Collocated Mixed Discrete Least Squares Meshfree (MDLSM) method is proposed and used for efficient solution of engineering problems. Background mesh is not required in the MDLSM method; hence the method is a truly meshfree method. Nodal points are used in the MDLSM methods to construct the shape functions while collocated points are used to form the least squares functional. In the original MDLSM method, the location of the nodal points and collocated points are the same. In the proposed Collocated Mixed Discrete Least Squares Meshfree (CMDLSM) method, a set of additional collocated points is introduced. It is expected that the accuracy of results is improved by using the additional collocated points. It is noted that the size of coefficient matrix is not increased in the proposed CMDLSM method compared with the MDLSM method. The required computational effort for solving the linear algebraic system of equations, therefore, is the same as that of MDLSM method. A set of benchmark numerical examples, cited in the literature, is used to evaluate the performance of proposed method. The results indicate that the accuracy of solutions is improved by using additional collocated points in the proposed CMDLSM method.

Using Constrained Bilinear Quadratic Regulator for the Optimal Semi-Active Control Problem

Semi-active systems provide an attractive solution for the structural vibration problem. A useful approach, aimed to simplify the control design, is to divide the control system into two parts: an actuator and a controller. The actuator generates a force that tracks a command which is generated by the controller. Such approach reduces the complexity of the control law design as it allows for complex properties of the actuator to be considered separately. In this study, the semi-active control design problem is treated in the framework of optimal control theory by using bilinear representation, a quadratic performance index, and a constraint on the sign of the control signal. The optimal control signal is derived in a feedback form by using Krotov's method. To this end, a novel sequence of Krotov functions which suits the multi-input constrained bilinear-quadratic regulator problem is formulated by means of quadratic form and differential Lyapunov equations. An algorithm is proposed for the optimal control computation. A proof outline for the algorithm convergence is provided. The effectiveness of the suggested method is demonstrated by numerical example. The proposed method is recommended for optimal semi-active feedback design of vibrating plants with multiple semi-active actuators.

Quadratic BSDE with $\mathbb{L}^{2}$-terminal data: Krylov’s estimate, Itô–Krylov’s formula and existence results

We establish a Krylov-type estimate and an Ito–Krylov change of variable formula for the solutions of one-dimensional quadratic backward stochastic differential equations (QBSDEs) with a measurable generator and an arbitrary terminal datum. This allows us to prove various existence and uniqueness results for some classes of QBSDEs with a square integrable terminal condition and sometimes a merely measurable generator. It turns out that neither the existence of exponential moments of the terminal datum nor the continuity of the generator are necessary to the existence and/or uniqueness of solutions. We also establish a comparison theorem for solutions of a particular class of QBSDEs with measurable generator. As a byproduct, we obtain the existence of viscosity solutions for a particular class of quadratic partial differential equations (QPDEs) with a square integrable terminal datum.

Gap metric computation for time-varying linear systems on finite horizons

It is shown how to compute the aperture, or gap, between the finite-horizon graphs of linear time-varying dynamical systems, given state-space models of the continuous-time input-output maps. The approach involves four quadratic matrix first-order differential equations, three with the symplectic structure of Riccati equations. All four are subject to single-ended boundary conditions. So standard numerical methods can be used to compute the solutions as required. Three of the differential equations, two Riccati and the other not, need to be solved once to construct normalized graph representations, and to test an invertibility condition, respectively. When this invertibility condition is verified, the directed gaps are equal, and the remaining Riccati differential equation is solved repeatedly in a bisection search to determine one of these. Computation of both directed gaps to find the maximum would, by contrast, involve repeatedly solving two such Riccati differential equations, each in a bisection search.

On the generalized algebraic Riccati equations

Three hundred years have passed since Jacopo Francesco Riccati analyzed a quadratic differential equation that would have been of crucial importance in many fields of engineering and applied mathematics. Indeed, countless variations and generalizations of this equation have been considered as they proved to be the right mathematical tool to address important problems. This paper is focused on a generalized version of the matrix Riccati equation where the matrix that in the classical Riccati equation is inverted can be singular: we analyze the equation obtained by substituting the inverse operator with the Moore-Penrose pseudo-inverse. The equations obtained by this substitution are known as generalized Riccati equations. The relation between these equations — both in continuos-time and in discrete-time — and singular Linear Quadratic (LQ) optimal control problem are examined. A geometric characterization of the set of solutions of the generalized Riccati equation is illustrated. It is shown that in this general setting there are LQ optimal control problems for which the optimal closed-loop system is stable also in cases where the Riccati equation does not possess a stabilizing solution.

AN ANALYTICAL APPROACH FOR VARIANCE SWAPS WITH AN ORNSTEIN–UHLENBECK PROCESS

Pricing variance swaps have become a popular subject recently, and most research of this type come under Heston’s two-factor model. This paper is an extension of some recent research which used the dimension-reduction technique based on the Heston model. A new closed-form pricing formula focusing on a log-return variance swap is presented here, under the assumption that the underlying asset prices can be described by a mean-reverting Gaussian volatility model (Ornstein–Uhlenbeck process). Numerical tests in two respects using the Monte Carlo (MC) simulation are included. Moreover, we discuss a procedure of solving a quadratic differential equation with one variable. Our method can avoid the previously encountered limitations, but requires more time for calculation than other recent analytical discrete models.

On the Geometry of Quadratic Second-Order Abel Ordinary Differential Equations

In this paper, we study the contact geometry of second-order ordinary differential equations that are quadratic in the highest derivative (the so-called quadratic Abel equations). Namely, we realize each quadratic Abel equation as the kernel of some nonlinear differential operator. This operator is defined by a quadratic form on the Cartan distribution in the 1-jet space. This observation makes it possible to establish a one-to-one correspondence between quadratic Abel equations and quadratic forms on Cartan distribution. Using this realization, we construct a contact-invariant {e}-structure associated with a nondegenerate Abel equation (i.e., the basis of vector fields that is invariant under contact transformations). Finally, in terms of this {e}-structure we solve the problem of contact equivalence of nondegenerate Abel equations

Analytical approximate solutions for quadratic Riccati differential equation of fractional order using the Polynomial Least Squares Method

In this paper the Polynomial Least Squares Method is applied in order to compute analytical approximate polynomial solutions for the fractional Riccati type differential equations. The accuracy of the method is tested by means of the comparison with previous results for several applications.

An index-aware parametric model order reduction method for parameterized quadratic differential–algebraic equations

Modeling of sophisticated applications, such as coupled problems arising from nanoelectronics can lead to quadratic differential algebraic equations (DAEs). The quadratic DAEs may also be parameterized, due to variations in material properties, system configurations, etc., and they are usually subject to multi-query tasks, such as optimization, or uncertainty quantification. Model order reduction (MOR), specifically parametric model order reduction (pMOR), is known as a useful tool for accelerating the simulations in a multi-query context. However, pMOR dedicated to this particular structure, has not yet been systematically studied. Directly applying the existing pMOR methods may produce parametric reduced-order models (pROMs) which are less accurate, or may be very difficult to simulate. The same problem was already observed for linear DAEs, and could be eliminated by introducing splitting MOR techniques such as the index-aware MOR (IMOR) methods. We extend the IMOR methods to parameterized quadratic DAEs, thereby producing accurate and easy to simulate index-aware parametric reduced-order models (IpROMs). The proposed approach is so far limited to index-1 one-way coupled problems, but these often appear in computational nanoelectronics. We illustrate the performance of the new approach using industrial models for nanoelectronic structures.

OPTIMAL STOCHASTIC CONTROL PROBLEM UNDER MODEL UNCERTAINTY WITH NONENTROPY PENALTY

In this paper, a stochastic control problem under model uncertainty with general penalty term is studied. Two types of penalties are considered. The first one is of type [Formula: see text]-divergence penalty treated in the general framework of a continuous filtration. The second one called consistent time penalty is studied in the context of a Brownian filtration. In the case of consistent time penalty, we characterize the value process of our stochastic control problem as the unique solution of a class of quadratic backward stochastic differential equation with unbounded terminal condition.

Про математичне моделювання хвороби Паркiнсона методами хаотичної динамiки

An modeling attempt of behavior process of brain electric impulses for some patient by the solutions of 3D system of quadratic differential equations is undertaken. (This system of differential equations was got from a multivariate times series with the help of polynomial averages and least square method.) Transition conditions from a chaotic attractor to a limit cycle (and vice versa) of the system of differential equations are found. Exactly these conditions characterize beginning of process of disease by Parkinson’s illness at the patient.

STOCHASTIC CENTER OF SYSTEMS OF STOCHASTIC DIFFERENTIAL EQUATIONS ON THE PLANE

We study a stochastic analogy of the famous center problem of Dulac for quadratic differential equations in the plane. We introduce the concept of center for systems of stochastic differential equations of Ito's type on the plane, called stochastic center. We derive a criterion for the existence of such a center. We apply it to obtain necessary and sufficient conditions for quadratic stochastic differential equations in dimension 2.

Limit cycles for quadratic and cubic planar differential equations under polynomial perturbations of small degree

In this paper we consider planar systems of differential equations of the form \begin{document}$\left\{ \begin{array}{lcl} \dot x&=&-y+\delta p(x,y)+\varepsilon P_n(x,y), \dot y&=x'>where \begin{document}$δ, \varepsilon$\end{document} are small parameters, $(p, q)$ are quadratic or cubic homogeneous polynomials such that the unperturbed system ($\varepsilon=0$) has an isochronous center at the origin and $P_n, Q_n$ are arbitrary perturbations. Estimates for the maximum number of limit cycles are provided and these estimatives are sharp for $n≤q 6$ (when $p, q$ are quadratic). When $p, q$ are cubic polynomials and $P_n, Q_n$ are linear, the problem is addressed from a numerical viewpoint and we also study the existence of limit cycles.

Application of the quadratic logistic differential equation for the rationalization of methanol electrooxidation dynamics

<p class="PaperAbstract"><span lang="EN-US">The electrooxidation of methanol in both acidic and alkaline media on poly-crystalline platinum under the regime of cyclic voltammetry is analyzed by application of quadratic logistic equation. The current-charge curves in the anodic cycles fit the logistic differential equation reasonably well and are accounted on the basis of the non-linearity of the kinetics and the effect of positive feedback. In the reverse cycle however, no fit is observed, presumably due to the lack of correlation between the net faradaic current and the surface charge of adsorbates.</span></p>

Solving linear and quadratic random matrix differential equations using: A mean square approach. The non-autonomous case

This paper is aimed to extend, the non-autonomous case, the results recently given in the paper Casabán et al. (2016) for solving autonomous linear and quadratic random matrix differential equations. With this goal, important deterministic results like the Abel–Liouville–Jacobi’s formula, are extended to the random scenario using the so-called L p -random matrix calculus. In a first step, random time-dependent matrix linear differential equations are studied and, in a second step, random non-autonomous Riccati matrix differential equations are solved using the hamiltonian approach based on dealing with the extended underlying linear system. Illustrative numerical examples are also included.