The mathematical nature of the jump phenomenon associated with the damped, harmonically forced Duffing equation is investigated, as regards the amplitude of the harmonic response of the system. The occurring discontinuities are treated as a bifurcation induced phenomenon, where the critical values of the respective control parameter are determined in relation to the solution of a cubic equation. Moreover, graphs illustrating the phenomenon, as well as bifurcation diagrams in the parameter planes of the system are presented. Finally, application of the presented analysis to phenomena of this kind as regards the behavior of living organisms is also discussed.
In the present paper some aspects of the global dynamical behavior of a new SEIRS epidemic model with nonlinear incidence rate, non-permanent immunity and an additional epidemic-induced death rate are presented. More specifically, a point-to-point homoclinic connecting orbit to an endemic saddle equilibrium is numerically located, as a bifurcation with respect to the active parameter. Moreover, we compute members of the families of different types of heteroclinic connections, be them point-to-cycle, as well as, point-to-point connecting orbits. The physical meaning of these orbits in relation to the physical system is also discussed.
A new SEIRS epidemic model with nonlinear incidence rate and nonpermanent immunity is presented in the present paper. The fact that the incidence rate per infective individual is given by a nonlinear function and product of rational powers of two state variables, as well as the introduction of an epidemic-induced death rate, leads to a more realistic modeling of the physical problem itself. A stability analysis is performed and the features of Hopf bifurcation are investigated. Both the corresponding critical regions in the parameter space and their stability characteristics are presented. Furthermore, by using algorithms based on a new symbolic form as regards the restriction of an n-dimensional nonlinear parametric system to the center manifold and the normal forms of the corresponding Hopf bifurcation, as well, the associated bifurcation diagram is derived, and finally various emerging limit cycles are numerically obtained by appropriate implemented methods.
In the present paper, a custom algorithm based on the method of orthogonal collocation on finite elements is presented and used for the location of global homoclinic point-to-point asymptotic connecting orbits. This kind of global bifurcation occurs in a large variety of problems in Applied Sciences, being associated to specific, significant physical aspects of the problem under consideration. In order to confront the difficulties faced when the location of such orbits is attempted, high order boundary conditions are constructed through scale order approximations, and used instead of the more common first order ones. The effectiveness of the implemented algorithm is justified by means of the specific applications and the figures presented.
In this paper, a methodology for the numerical location of a global point-to-point (P2P for short) homoclinic asymptotically connecting orbit is applied to a modified version of Shimizu-Morioka system, which models a semiconductor laser. This type of global bifurcation can be considered as a stylized mathematical description of self-pulsation in this laser type, associ-ated with saturation. The location is achieved by use of a custom algorithm based on the method of orthogonal collocation on finite elements with fourth order boundary conditions, constructed through scale order approximations. The effectiveness of the algorithm and the superiority of high-order boundary conditions over the widely used first order ones are justified throughout the obtained graphical results.
The center manifold theory with respect to the simple Hopf bifurcation of a n-dimensional nonlinear multi-parametric system is treated via a proper symbolic form. Analytical expressions of the involved quantities are obtained as functions of the parameters of the system via effective algorithms based on the followed procedure and carried out using a symbolic computation software. Moreover the normal form of a codimension 1 Hopf bifurcation, as well as the corresponding Lyapunov coefficient and bifurcation portrait, can be computed for any system under consideration. Here the computational procedure is applied to two nonlinear three-dimensional, three-parametric systems and graphical results are obtained as concerns the stability regions, the bifurcation portraits, as well as emerged limit cycles with respect to both the supercritical and the subcritical case of bifurcation. (C) 2017 Elsevier Inc. All rights reserved.
The restriction of an n-dimensional nonlinear parametric system on the center manifold is treated via a new proper symbolic form and analytical expressions of the involved quantities are obtained as functions of the parameters by lengthy algebraic manipulations combined with computer assisted calculations. Normal forms regarding degenerate Hopf bifurcations up to codimension 3, as well as the corresponding Lyapunov coefficients and bifurcation portraits, can be easily computed for any system under consideration.
Paper constructs approximate analytical expressions of periodic disequilibrium fluctuations business cycles occurring in connection with Hopf bifurcations in nonlinear problems of economic interactions described by four-dimensional continuous time dynamical systems. Two nonlinear macrodynamic models are employed as tests models. The region of equilibrium stability in parameter space is obtained in each case and a Hopf bifurcation curve is identified as a boundary of the region. Validity of the analytical approximations obtained for the cycles generated by the loss of equilibrium stability on this curve is confirmed by comparison to numerically determined cycles. Explicit analytical description of such limit cycles is of particular interest in the case of subcritical bifurcation, due to the difficulty of the numerical determination of the generated unstable cycles. Mathematics Subject Classification: 34A34, 34C07, 34C23, 37G10
Paper constructs approximate analytical expressions of periodic disequilibrium fluctuations – business cycles – occurring in connection with Hopf bifurcations in economic dynamics. The Kaldorian open economy macrodynamic model with money proposed earlier by the first author is employed as the test model. Its region of equilibrium stability in parameter space is obtained and a Hopf bifurcation curve is identified as a boundary of the region. Validity of the analytical approximations obtained for the cycles generated by the loss of equilibrium stability on this curve is confirmed by comparison to numerically determined cycles. Explicit analytical description of such disequilibrium fluctuations is of particular interest in the case of subcritical bifurcations when the cycles are unstable and their numerical determination is difficult, as in the case of the test model, and may conceivably be useful in testing for subsequent bifurcation of the cycles.
Certain nonlinear autonomous ordinary differential equations of the second order are reduced to Abel equations of the first kind ((Ab-1) equations). Based on the results of a previous work, concerning a closed-form solution of a general (Ab-1) equation, and introducing an arbitrary function, exact one-parameter families of solutions are derived for the original autonomous equations, for the most of which only first integrals (in closed or parametric form) have been obtained so far. Two-dimensional autonomous systems of differential equations of the first order, equivalent to the considered herein autonomous forms, are constructed and solved by means of the developed analysis.
Through a suitable ad hoc assumption, a nonlinear PDE governing a three-dimensional weak, irrotational, steady vector field is reduced to a system of two nonlinear ODEs: the first of which corresponds to the two-dimensional case, while the second involves also the third field component. By using several analytical tools as well as linear approximations based on the weakness of the field, the first equation is transformed to an Abel differential equation which is solved parametrically. Thus, we obtain the two components of the field as explicit functions of a parameter. The derived solution is applied to the two-dimensional small perturbation frictionless flow past solid surfaces with either sinusoidal or parabolic geometry, where the plane velocities are evaluated over the body's surface in the case of a subsonic flow.
By using appropriate transformations in combination with specific Abel equations solvable in closed form containing arbitrary functions, an implicit solution as well as the associated sufficient condition are derived for certain differential equations of the Abel class of the first kind.
We introduce a new version of Hill's problem that incorporates the effects of radiation of the primary and oblateness of the secondary and study the basic dynamical features of this new model-problem. This formulation is more appropriate for some astronomical applications as an approximation to the corresponding restricted three-body problem. We use iterative methods for deriving approximate expressions of the equilibrium point locations and study their stability properties by using a linear stability analysis. All equilibrium points are unstable. We also employ singular perturbations methods for obtaining approximate expressions of the Lyapunov families emanating from equilibrium points, in both coplanar and spatial case, and numerical techniques for their continuation.
We establish an analytical method leading to a more general form of the exact solution of a nonlinear ODE of the second order due to Gambier. The treatment is based on the introduction and determination of a new function, by means of which the solution of the original equation is expressed. This treatment is applied to another nonlinear equation, subjected to the same general class as that of Gambier, by constructing step by step an appropriate analytical technique. The developed procedure yields a general exact closed form solution of this equation, valid for specific values of the parameters involved and containing two arbitrary (free) parameters evaluated by the relevant initial conditions. We finally verify this technique by applying it to two specific sets of parameter values of the equation under consideration.
The particular ordinary differential equations (ODEs) governing some specific non-linear oscillators are reduced to equivalent Abel equations by means of a series of functional transformations. These equivalent equations do not accept analytical solutions in terms of tabulated functions. So, the results of this work combined with the non-existence of exact solutions in the literature lead to the speculation that we need to construct new analytical functions in order to confront particular non-linearities in mechanical oscillations and physical problems generally.
In Ref. [6] the authors constructed analytical solutions including one arbitrary function for the problem of nonlinear, unsteady, supersonic flow analysis concerning slender bodies of revolution due to small amplitude oscillations. An application describing a flow past a right circular cone was presented and the constructed solutions were given in the form of infinite series through a set of convenient boundary and initial conditions in accordance with the physical problem. In the present paper we develop an appropriate convergence analysis concerning the before mentioned series solutions for the specific geometry of a rigid right circular cone. We succeed in estimating the limiting values of the series producing velocity and acceleration resultants of the problem under consideration. Several graphics for the velocity and acceleration flow fields are presented. We must underline here that the proposed convergence technique is unique and can be applied to any other geometry of the considered body of revolution.
We construct analytical solutions for the problem of nonlinear supersonic flow past slender bodies of revolution due to small amplitude oscillations. The method employed is based on the splitting of the time dependent small perturbation equation to a nonlinear time independent partial differential equation (P.D.E.) concerning the steady flow, and a linear time dependent one, concerning the unsteady flow. Solutions in the form of three parameters family of surfaces for the first equation are constructed, while solutions including one arbitrary function for the second equation are extracted. As an application the evaluation of the small perturbation velocity resultants for a flow past a right circular cone is obtained making use of convenient boundary and initial conditions in accordance with the physical problem.
Making use of convenient ad hoc assumptions we construct closed-form solutions of the non-linear two-dimensional irrotational steady small perturbation equation appearing in fluid mechanics. The methodologies developed succeed in giving the above solutions expressed in the form of fewer arbitrary functions than needed for general solutions. As an application we specify the above mentioned solutions in the case of the simplified non-linear transonic equation governing the boundary value problem of a two-dimensional flow past a wave shaped wall.
Making use of convenient ad hoc assumptions we construct closed-form solutions of the non-linear two-dimensional irrotational steady small perturbation equation appearing in fluid mechanics. The methodologies developed succeed in giving the above solutions expressed in the form of fewer arbitrary functions than needed for general solutions. As an application we specify the above mentioned solutions in the case of the simplified non-linear transonic equation governing the boundary value problem of a two-dimensional flow past a wave shaped wall.
In the present paper analytical solutions concerning the stress state at the tip of a crack in an elastic-perfectly plastic body, subjected to mixed mode loadings under plane strain conditions, are presented. Analytical solutions of the nonlinear ordinary differential equations are obtained and the dominant singularity is completely determined with the aid of suitable boundary conditions. The obtained results are in perfect agreement with those given by other investigators, both analytical and numerical. The novel aspect here is the methodology used for the solution, as well as the direct determination of the plastic zones. As a consequence, the resulting analytical solutions cover many more problems in the mathematical theory of plasticity compared to similar existing methods and they may be proved of importance in various applications.
