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Albert C J Luo - One of the best experts on this subject based on the ideXlab platform.
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a periodically forced piecewise linear system part ii the fragmentation mechanism of strange attractors and grazing
Communications in Nonlinear Science and Numerical Simulation, 2007Co-Authors: Albert C J LuoAbstract:Abstract In the first part of this work, the local singularity of non-Smooth Dynamical Systems was discussed and the criteria for the grazing bifurcation were presented mathematically. In this part, the fragmentation mechanism of strange attractors in non-Smooth Dynamical Systems is investigated. The periodic motion transition is completed through grazing. The concepts for the initial and final grazing, switching manifolds are introduced for six basic mappings. The fragmentation of strange attractors in non-Smooth Dynamical Systems is described mathematically. The fragmentation mechanism of the strange attractor for such a non-Smooth Dynamical system is qualitatively discussed. Such a fragmentation of the strange attractor is illustrated numerically. The criteria and topological structures for the fragmentation of the strange attractor need to be further developed as in hyperbolic strange attractors. The fragmentation of the strange attractors extensively exists in non-Smooth Dynamical Systems, which will help us better understand chaotic motions in non-Smooth Dynamical Systems.
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an analytical prediction of sliding motions along discontinuous boundary in non Smooth Dynamical Systems
Nonlinear Dynamics, 2007Co-Authors: Albert C J Luo, Brandon C GeggAbstract:This paper presents a method for the analytical prediction of sliding motions along discontinuous boundaries in non-Smooth Dynamical Systems. The methodology is demonstrated through investigation of a periodically forced linear oscillator with dry friction. The switching conditions for sliding motions in non-Smooth Dynamical Systems are given. The generic mappings for the friction-induced oscillator are introduced. From the generic mappings, the corresponding criteria for the sliding motions are presented through the force product conditions. The analytical prediction of the onset and vanishing of the sliding motions is illustrated. Finally, numerical simulations of sliding motions are carried out to verify the analytical prediction. This analytical prediction provides an accurate prediction of sliding motions in non-Smooth Dynamical Systems. The switching conditions developed in this paper are expressed by the total force of the oscillator, and the nonlinearity and linearity of the spring and viscous damping forces in the oscillator cannot change such switching conditions. Therefore, the achieved force criteria can be applied to the other Dynamical Systems with nonlinear friction forces processing a C0-discontinuity.
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On grazing and strange attractors fragmentation in non-Smooth Dynamical Systems
Communications in Nonlinear Science and Numerical Simulation, 2006Co-Authors: Albert C J LuoAbstract:Abstract This paper presents some new ideas to understand the strange attractor fragmentation caused by grazing in non-Smooth dynamic Systems. The sufficient and necessary conditions for grazing bifurcations in non-Smooth dynamic Systems are presented. The initial sets of grazing mapping are introduced and the corresponding initial grazing manifolds are discussed. The grazing-induced fragmentation of strange attractors of chaotic motions in non-Smooth Dynamical Systems is presented. The mathematical theory for such a fragmentation of strange attractors should be further developed.
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The symmetry of steady-state solutions of non-Smooth Dynamical Systems with two constraints
Proceedings of the Institution of Mechanical Engineers Part K: Journal of Multi-body Dynamics, 2005Co-Authors: Albert C J LuoAbstract:In this paper, the symmetry of steady-state solutions in non-Smooth Dynamical Systems with two symmetrical constraints are investigated to obtain all possible stable and unstable motions. An invariant transformation exists in regular and chaotic motions relative to skew-symmetrical mapping pairs in symmetrical Systems with harmonic excitations. This investigation provides a mathematical foundation for the symmetry of solutions in this class of non-Smooth Dynamical Systems. The group structure of mapping combinations should be further investigated.
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Imaginary, sink and source flows in the vicinity of the separatrix of non-Smooth dynamic Systems
Journal of Sound and Vibration, 2004Co-Authors: Albert C J LuoAbstract:In this letter, the real and imaginary flows for non-Smooth Dynamical Systems are described, and the δ-layer of the separation boundary is introduced as well. The onset, existence and vanishing of the sink and source flows in the δ-layer are presented. The switching between the two semi-passable flows and the switching between the sink and source flows are investigated through the singular gluing points. Finally, the necessary and sufficient conditions for the onset, vanishing and switching are presented. These conditions provide the criteria to determine the sliding motion on the separatrix, which can be very easily applied to non-Smooth Dynamical Systems.
Marco Antonio Teixeira - One of the best experts on this subject based on the ideXlab platform.
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generic singularities of 3d piecewise Smooth Dynamical Systems
arXiv: Dynamical Systems, 2018Co-Authors: Marco Antonio Teixeira, Otavio M L GomideAbstract:The aim of this paper is to provide a discussion on the current directions of research involving typical singularities of 3D nonSmooth vector fields. A brief survey of known results is also presented.
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on the birth of limit cycles for non Smooth Dynamical Systems
Bulletin Des Sciences Mathematiques, 2015Co-Authors: Jaume Llibre, Douglas D Novaes, Marco Antonio TeixeiraAbstract:Abstract The main objective of this work is to develop, via Brower degree theory and regularization theory, a variation of the classical averaging method for detecting limit cycles of certain piecewise continuous Dynamical Systems. In fact, overall results are presented to ensure the existence of limit cycles of such Systems. These results may represent new insights in averaging, in particular its relation with non-Smooth Dynamical Systems theory. An application is presented in careful detail.
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Sliding vector fields for non-Smooth Dynamical Systems having intersecting switching manifolds
Nonlinearity, 2015Co-Authors: Jaume Llibre, Paulo R. Silva, Marco Antonio TeixeiraAbstract:We consider a differential equation , with discontinuous right-hand side and discontinuities occurring on a set Σ. We discuss the dynamics of the sliding mode which occurs when, for any initial condition near p ∈ Σ, the corresponding solution trajectories are attracted to Σ. Firstly we suppose that Σ = H−1(0), where H is a Smooth function and is a regular value. In this case Σ is locally diffeomorphic to the set . Secondly we suppose that Σ is the inverse image of a non-regular value. We focus our attention to the equations defined around singularities as described in Gutierrez and Sotomayor (1982 Proc. Lond. Math. Soc 45 97–112). More precisely, we restrict the degeneracy of the singularity so as to admit only those which appear when the regularity conditions in the definition of Smooth surfaces of in terms of implicit functions and immersions are broken in a stable manner. In this case Σ is locally diffeomorphic to one of the following algebraic varieties: (double crossing); (triple crossing); (cone) or (Whitney's umbrella).
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On the birth of limit cycles for non-Smooth Dynamical Systems
Bulletin des Sciences Mathématiques, 2015Co-Authors: Jaume Llibre, Douglas D Novaes, Marco Antonio TeixeiraAbstract:Agraïments: The first author is partially supported by a FEDER-UNAB10-4E-378. The second author is partially supported by a FAPESP-BRAZIL grant 2013/16492-0. The second and third authors are partially supported by a FAPESP-BRAZIL grant 2012/18780-0. The three authors are also supported by a CAPES CSF-PVE grant 88881.030454/2013-01.The main objective of this work is to develop, via Brower degree theory and regularization theory, a variation of the classical averaging method for detecting limit cycles of certain piecewise continuous Dynamical Systems. In fact, overall result are presented to ensure the existence of limit cycles of such Systems. These result may represent new insights in averaging, in particular its relation with non Smooth Dynamical Systems theory. An application is presented in careful detail
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Stability conditions in piecewise Smooth Dynamical Systems at a two-fold singularity
Journal of Dynamical and Control Systems, 2013Co-Authors: Alain Jacquemard, Marco Antonio Teixeira, Durval José TononAbstract:Some qualitative and geometric aspects of 3-dimensional non-Smooth vector fields theory are discussed. Our main aim is to study the dynamics near typical singularities of piecewise Smooth Dynamical Systems, the so-called two-fold singularities. More specifically, we are interested in discussing stability problems of such Systems around these singularities.
K. W. Wang - One of the best experts on this subject based on the ideXlab platform.
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Predicting non-stationary and stochastic activation of saddle-node bifurcation in non-Smooth Dynamical Systems
Nonlinear Dynamics, 2018Co-Authors: Jinki Kim, K. W. WangAbstract:Saddle-node bifurcation can cause Dynamical Systems undergo large and sudden transitions in their response, which is very sensitive to stochastic and non-stationary influences that are unavoidable in practical applications. Therefore, it is essential to simultaneously consider these two factors for estimating critical system parameters that may trigger the sudden transition. Although many Systems exhibit non-Smooth Dynamical behavior, estimating the onset of saddle-node bifurcation in them under the dual influence remains a challenge. In this work, a new theoretical framework is developed to provide an effective means for accurately predicting the probable time at which a non-Smooth system undergoes saddle-node bifurcation while the governing parameters are swept in the presence of noise. The stochastic normal form of non-Smooth saddle-node bifurcation is scaled to assess the influence of noise and non-stationary factors by employing a single parameter. The Fokker–Planck equation associated with the scaled normal form is then utilized to predict the distribution of the onset of bifurcations. Experimental efforts conducted using a double-well Duffing analog circuit successfully demonstrate that the theoretical framework developed in this study provides accurate prediction of the critical parameters that induce non-stationary and stochastic activation of saddle-node bifurcation in non-Smooth Dynamical Systems.
Jinki Kim - One of the best experts on this subject based on the ideXlab platform.
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Predicting non-stationary and stochastic activation of saddle-node bifurcation in non-Smooth Dynamical Systems
Nonlinear Dynamics, 2018Co-Authors: Jinki Kim, K. W. WangAbstract:Saddle-node bifurcation can cause Dynamical Systems undergo large and sudden transitions in their response, which is very sensitive to stochastic and non-stationary influences that are unavoidable in practical applications. Therefore, it is essential to simultaneously consider these two factors for estimating critical system parameters that may trigger the sudden transition. Although many Systems exhibit non-Smooth Dynamical behavior, estimating the onset of saddle-node bifurcation in them under the dual influence remains a challenge. In this work, a new theoretical framework is developed to provide an effective means for accurately predicting the probable time at which a non-Smooth system undergoes saddle-node bifurcation while the governing parameters are swept in the presence of noise. The stochastic normal form of non-Smooth saddle-node bifurcation is scaled to assess the influence of noise and non-stationary factors by employing a single parameter. The Fokker–Planck equation associated with the scaled normal form is then utilized to predict the distribution of the onset of bifurcations. Experimental efforts conducted using a double-well Duffing analog circuit successfully demonstrate that the theoretical framework developed in this study provides accurate prediction of the critical parameters that induce non-stationary and stochastic activation of saddle-node bifurcation in non-Smooth Dynamical Systems.
Claudio A. Buzzi - One of the best experts on this subject based on the ideXlab platform.
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Generic bifurcation of refracted Systems
Advances in Mathematics, 2013Co-Authors: Claudio A. Buzzi, Joao C. Medrado, Marco Antonio TeixeiraAbstract:Abstract In this article we discuss some qualitative and geometric aspects of non-Smooth Dynamical Systems theory. Our goal is to study the diagram bifurcation of typical singularities that occur generically in one parameter families of certain piecewise Smooth vector fields named Refracted Systems. Such Systems has a codimension-one submanifold as its discontinuity set.
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slow fast Systems on algebraic varieties bordering piecewise Smooth Dynamical Systems
Bulletin Des Sciences Mathematiques, 2012Co-Authors: Claudio A. Buzzi, Paulo R. Silva, Marco Antonio TeixeiraAbstract:Abstract This article extends results contained in Buzzi et al. (2006) [4] , Llibre et al. (2007, 2008) [12] , [13] concerning the dynamics of non-Smooth Systems. In those papers a piecewise C k discontinuous vector field Z on R n is considered when the discontinuities are concentrated on a codimension one submanifold. In this paper our aim is to study the dynamics of a discontinuous system when its discontinuity set belongs to a general class of algebraic sets. In order to do this we first consider F : U → R a polynomial function defined on the open subset U ⊂ R n . The set F − 1 ( 0 ) divides U into subdomains U 1 , U 2 , … , U k , with border F − 1 ( 0 ) . These subdomains provide a Whitney stratification on U . We consider Z i : U i → R n Smooth vector fields and we get Z = ( Z 1 , … , Z k ) a discontinuous vector field with discontinuities in F − 1 ( 0 ) . Our approach combines several techniques such as e-regularization process, blowing-up method and singular perturbation theory. Recall that an approximation of a discontinuous vector field Z by a one parameter family of continuous vector fields is called an e-regularization of Z (see Sotomayor and Teixeira, 1996 [18] ; Llibre and Teixeira, 1997 [15] ). Systems as discussed in this paper turn out to be relevant for problems in control theory (Minorsky, 1969 [16] ), in Systems with hysteresis (Seidman, 2006 [17] ) and in mechanical Systems with impacts (di Bernardo et al., 2008 [5] ).
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Slow–fast Systems on algebraic varieties bordering piecewise-Smooth Dynamical Systems
Bulletin des Sciences Mathématiques, 2012Co-Authors: Claudio A. Buzzi, Paulo R. Silva, Marco Antonio TeixeiraAbstract:Abstract This article extends results contained in Buzzi et al. (2006) [4] , Llibre et al. (2007, 2008) [12] , [13] concerning the dynamics of non-Smooth Systems. In those papers a piecewise C k discontinuous vector field Z on R n is considered when the discontinuities are concentrated on a codimension one submanifold. In this paper our aim is to study the dynamics of a discontinuous system when its discontinuity set belongs to a general class of algebraic sets. In order to do this we first consider F : U → R a polynomial function defined on the open subset U ⊂ R n . The set F − 1 ( 0 ) divides U into subdomains U 1 , U 2 , … , U k , with border F − 1 ( 0 ) . These subdomains provide a Whitney stratification on U . We consider Z i : U i → R n Smooth vector fields and we get Z = ( Z 1 , … , Z k ) a discontinuous vector field with discontinuities in F − 1 ( 0 ) . Our approach combines several techniques such as e-regularization process, blowing-up method and singular perturbation theory. Recall that an approximation of a discontinuous vector field Z by a one parameter family of continuous vector fields is called an e-regularization of Z (see Sotomayor and Teixeira, 1996 [18] ; Llibre and Teixeira, 1997 [15] ). Systems as discussed in this paper turn out to be relevant for problems in control theory (Minorsky, 1969 [16] ), in Systems with hysteresis (Seidman, 2006 [17] ) and in mechanical Systems with impacts (di Bernardo et al., 2008 [5] ).
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Fold-Saddle Bifurcation in Non-Smooth Vector Fields on the Plane
arXiv: Dynamical Systems, 2010Co-Authors: Claudio A. Buzzi, Tiago Carvalho, Marco Antonio TeixeiraAbstract:This paper presents results concerning bifurcations of 2D piecewise-Smooth Dynamical Systems governed by vector fields. Generic three parameter families of a class of Non-Smooth Vector Fields are studied and its bifurcation diagrams are exhibited. Our main result describes the unfolding of the so called Fold-Saddle singularity.
