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On Dissipative Quadratic Stochastic Operators  [PDF]
Farruh Shahidi
Mathematics , 2007,
Abstract: In present paper we introduce the notion of dissipative quadratic stochastic operator and cubic stochastic operator. We prove necessary conditions for dissipativity of quadratic stochastic operators. Besides, it is studied certain limit behavior of such operators. Finally we prove ergodic theorem for dissipative operators.
One Dimensional Relativistic Free Particle in a Quadratic Dissipative Medium  [PDF]
G. V. López, G. C. Montes, J. G. T. Zanudo
Journal of Modern Physics (JMP) , 2015, DOI: 10.4236/jmp.2015.62016
Abstract: The deduction of a constant of motion, a Lagrangian, and a Hamiltonian for relativistic particle moving in a dissipative medium characterized by a force which depends on the square of the velocity of the particle is done. It is shown that while the trajectories in the space (x,v), defined by the constant of motion, look as one might expected, the trajectories in the space (x,p), defined by the Hamiltonian, have an odd behavior.<
One dimensional relativistic free particle in a quadratic dissipative medium  [PDF]
G. V. Lopez,G. C. Montes,J. G. T. Zenudo
Physics , 2009,
Abstract: The deduction of a constant of motion, a Lagrangian, and a Hamiltonian for relativistic particle moving in a dissipative medium characterized by a force which depends on the square of the velocity of the particle is done. It is shown that meanwhile the trajectories in the space (x,v), defined by the constant of motion, look as one might expected, the trajectories in the space (x,p), defined by the Hamiltonian, have a odd behavior.
WKB quantization for completely bound quadratic dissipative systems
González, G;Iván Piza, H;
Revista mexicana de física , 2011,
Abstract: we study the energy quantization for completely bound dissipative systems over a full cycle of motion. we approach the problem by means of an effective phenomenological hamiltonian and the wkb quantization rule to obtain the energy levels in the system. an example of this approach is given for the quantum bouncer with quadratic dissipation.
The Limit Behavior Of The Trajectories of Dissipative Quadratic Stochastic Operators on Finite Dimensional Simplex  [PDF]
F. A. Shahidi,M. T. Abu Osman
Mathematics , 2011, DOI: 10.1142/S201019451200534X
Abstract: The limit behavior of trajectories of dissipative quadratic stochastic operators on a finite-dimensional simplex is fully studied. It is shown that any dissipative quadratic stochastic operator has either unique or infinitely many fixed points. If dissipative quadratic stochastic operator has a unique point, it is proven that the operator is regular at this fixed point. If it has infinitely many fixed points, then it is shown that $\omega-$ limit set of the trajectory is contained in the set of fixed points.
Dissipative quadratic solitons supported by a localized gain  [PDF]
Valery E. Lobanov,Olga V. Borovkova,Boris A. Malomed
Physics , 2014, DOI: 10.1103/PhysRevA.90.053820
Abstract: We propose two models for the creation of stable dissipative solitons in optical media with the $\chi^{(2)}$ (quadratic) nonlinearity. To compensate spatially uniform loss in both the fundamental-frequency (FF) and second-harmonic (SH) components of the system, a strongly localized "hot spot", carrying the linear gain, is added, acting either on the FF component, or on the SH one. In both systems, we use numerical methods to find families of dissipative $\chi^{(2)}$ solitons pinned to the "hot spot". The shape of the existence and stability domains may be rather complex. An existence boundary for the solitons, which corresponds to the guided mode in the linearized version of the systems, is obtained in an analytical form. The solitons demonstrate noteworthy features, such as spontaneous symmetry breaking (of spatially symmetric solitons) and bistability.
On point-dissipative systems of differential equations with quadratic nonlinearity  [cached]
Anile K. Bose,Alan S. Cover,James A. Reneke
International Journal of Mathematics and Mathematical Sciences , 1991, DOI: 10.1155/s0161171291000108
Abstract: The system x ¢ € 2=Ax+f(x) of nonlinear vector differential equations, where the nonlinear term f(x) is quadratic with orthogonality property xTf(x)=0 for all x, is point-dissipative if uTAu<0 for all nontrivial zeros u of f(x).
On point dissipative N-dimensional systems of differential equations with quadratic nonlinearity
Anil K. Bose,Alan S. Cover,James A. Reneke
International Journal of Mathematics and Mathematical Sciences , 1993, DOI: 10.1155/s016117129300016x
Abstract: Known sufficient conditions for quadratic dynamical system x ¢ € 2=Ax+f(x) to be point dissipative given in terms of A and f for dimensions 2 and 3 are extended to allow for more general forms for the nonlinear term f(x). Furthermore, the conditions extend to n dimensions when f is quadratic with zero set an (n ¢ ’1)-dimensional hyperplane.
The action principle for dissipative systems  [PDF]
Matheus J. Lazo,Cesar E. Krumreich
Physics , 2014, DOI: 10.1063/1.4903991
Abstract: In the present work we redefine and generalize the action principle for dissipative systems proposed by Riewe by fixing the mathematical inconsistencies present in the original approach. In order to formulate a quadratic Lagrangian for non-conservative systems, the Lagrangian functions proposed depend on mixed integer order and fractional order derivatives. As examples, we formulate a quadratic Lagrangian for a particle under a frictional force proportional to the velocity, and to the classical problem of an accelerated point charge.
ON THE STABILITY OF DISSIPATIVE MECHANICAL SYSTEMS WITH CONSTRAINT DAMPING
有约束阻尼的非保守力学系统的稳定性

Zhaolin Wang,
李俊峰
,王照林

力学学报 , 1997,
Abstract: The paper investigates the stability of a nonlinear dissipative mechanical system subjected to potential force, gyroscopic force, Rayleigh dampingand constraint damping, using direct Liapunov method. Assuming that the gyroscopic force depends on a parameter, two asymptotically stability theorems are obtained, and the lower bounds of the parameter are evaluated.
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