Dynamic Cell Mapping Algorithm for Computing Basins of Attraction in Planar Filippov Systems

Erazo, Christian; Homer, Martin; Piiroinen, Petri T.; Bernardo, Mario di

doi:10.1142/s0218127417300415

Cited by 10 publications

(4 citation statements)

References 42 publications

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“…However, for the stochastic non-smooth impact systems under harmonic excitation, the FPK equations are non-homogeneous, and it is difficult to solve the stationary FPK response of the systems [32]. Specially, as an efficient numerical method, the generalized cell mapping (GCM) method has been applied in stochastic smooth systems [33][34][35]. And in the Ref [36], we proposed an improved GCM method to calculate the stochastic response of non-smooth systems.…”

Section: The Non-smooth R-d Oscillator and The Response Calculation M...mentioning

confidence: 99%

P-bifurcation phenomena of the non-smooth modified rayleigh-duffing oscillator under the combined action of harmonic excitation and noise perturbation

Ma¹,

Wang

Zhang³

et al. 2023

Phys. Scr.

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In this paper, the stochastic dynamics of a modified Rayleigh-Duffing oscillator with Coulomb frictional damping and elastic impact is investigated under combined harmonic and noise excitations. On the premise of retaining the non-smooth properties, a non-smooth steady-state probability density response numerical calculation method is introduced by taking advantage of Markov process. Utilizing this method, the stochastic P-bifurcation phenomena of oscillators without and with externally excitation are discussed in detail by inscribing changes in the topology of the steady-state probability density function. It is displayed that certain nonlinear damping coefficient and external excitation amplitude change the structure of the response, and that both the friction coefficient and the elastic coefficient of the contact surface induce stochastic P-bifurcation phenomena in systems without and with harmonic excitation, respectively. This study reveals the effect of non-smooth factors on the stability of the Rayleigh-Duffing oscillator.

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Section: The Non-smooth R-d Oscillator and The Response Calculation M...mentioning

confidence: 99%

P-bifurcation phenomena of the non-smooth modified rayleigh-duffing oscillator under the combined action of harmonic excitation and noise perturbation

Ma¹,

Wang

Zhang³

et al. 2023

Phys. Scr.

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“…Deriving a mathematical expression of this surface is useful to determine the relative position of points with respect to the separatrices and to establish the behaviour (reaching the crossing or sliding region) of trajectories starting from a selected point. Similar techniques have been used in for approximating the basins of attraction for equilibrium points of dynamical systems (see for example [13,26]).…”

Section: 3mentioning

confidence: 99%

Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems

Colombo¹,

Buono²,

Lopez³

et al. 2018

Discrete &Amp; Continuous Dynamical Systems - B

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Several models in the applied sciences are characterized by instantaneous changes in the solutions or discontinuities in the vector field. Knowledge of the geometry of interaction of the flow with the discontinuities can give new insights on the behaviour of such systems. Here, we focus on the class of the piecewise smooth systems of Filippov type. We describe some numerical techniques to locate crossing and sliding regions on the discontinuity surface, to compute the sets of attraction of these regions together with the mathematical form of the separatrices of such sets. Some numerical tests will illustrate our approach.

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“…On the applications of Filippov systems, the works have been mainly oriented to friction oscillators [31,[37][38][39][40][41], neural networks activated by discontinuous functions [42][43][44][45][46], memristor-based neural networks [47][48][49][50][51][52][53], neural networks with switching control using the Filippov system with delay [54][55][56][57], and electronic converters [58]. On issues related to Sustainable Development, the number of papers is much more limited, with approaches from the analysis of communities [59], from the analysis of companies [60] and others that touch on close issues such as energy systems [61][62][63][64], pest or disease control [65][66][67], HIV behavior [68,69], behavior longterm communities [70], or communications security [71].…”

Section: Introductionmentioning

confidence: 99%

Cooperation‐Based Modeling of Sustainable Development: An Approach from Filippov’s Systems

et al. 2021

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The concept of Sustainable Development has given rise to multiple interpretations. In this article, it is proposed that Sustainable Development should be interpreted as the capacity of territory, community, or landscape to conserve the notion of well-being that its population has agreed upon. To see the implications of this interpretation, a Brander and Taylor model, to evaluate the implications that extractivist policies have over an isolated community and cooperating communities, is proposed. For an isolated community and through a bifurcation analysis in which the Hopf bifurcation and the heteroclinic cycle bifurcation are detected, 4 prospective scenarios are found, but only one is sustainable under different extraction policies. In the case of cooperation, the exchange between communities is considered by coupling two models such as the one defined for the isolated community, with the condition that their transfers of renewable resources involve conservation policies. Since human decisions do not occur in a continuum, but rather through jumps, the mathematical model of cooperation used is a Filippov System, in which the dynamics could involve two switching manifolds of codimension one and one switching manifold of codimension two. The exchange in the cooperation model, for specific parameter arrangements, exhibits n -periodic orbits and chaos. It is notable that, in the cases in which the system shows sliding, it could be interpreted as a recovery delay related to the time needed by the deficit community to recover, until its dependence on the other community stops. It is concluded (1) that a sustainability analysis depends on the way well-being is defined because every definition of well-being is not necessarily sustainable, (2) that sustainability can be visualized as invariant sets in the nonzero region of the space of states (equilibrium points, n -periodic orbits, and strange attractors), and (3) that exchange is key to the prevalence of the human being in time. The results question us on whether Sustainable Development is only to keep us alive or if it also implies doing it with dignity.

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Dynamic Cell Mapping Algorithm for Computing Basins of Attraction in Planar Filippov Systems

Cited by 10 publications

References 42 publications

P-bifurcation phenomena of the non-smooth modified rayleigh-duffing oscillator under the combined action of harmonic excitation and noise perturbation

P-bifurcation phenomena of the non-smooth modified rayleigh-duffing oscillator under the combined action of harmonic excitation and noise perturbation

Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems

Cooperation‐Based Modeling of Sustainable Development: An Approach from Filippov’s Systems

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