On a possible approximation of discontinuous dynamical systems

Danca, Marius‐F.; Codreanu, S.

doi:10.1016/s0960-0779(01)00002-9

Cited by 28 publications

(18 citation statements)

References 8 publications

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“…A possible approximation (selection) of F 1 in the neighborhood of (0, x 2 ), for x 2 ∈ R, can be for example a smooth cubic polynomial function h (x 1 , x 2 ) = ax 3 1 + bx 2 1 + cx 1 + 9x 2 [Danca & Codreanu, 2002] where a, b, c and d have to be determined from the continuity and differentiability conditions in x 1 = ±ε ( Fig. 1(b).…”

Section: Existence Of the Solutions For Fractional Equationsmentioning

confidence: 99%

Approach of a Class of Discontinuous Dynamical Systems of Fractional Order: Existence of Solutions

Danca

2011

Int. J. Bifurcation Chaos

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In this letter we are concerned with the possibility to approach the existence of solutions to a class of discontinuous dynamical systems of fractional order. In this purpose, the underlying initial value problem is transformed into a fractional set-valued problem. Next, the Cellina's Theorem is applied leading to a single-valued continuous initial value problem of fractional order. The existence of solutions is assured by a Péano like theorem for ordinary differential equations of fractional order.

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Section: Existence Of the Solutions For Fractional Equationsmentioning

confidence: 99%

Approach of a Class of Discontinuous Dynamical Systems of Fractional Order: Existence of Solutions

Danca

2011

Int. J. Bifurcation Chaos

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“…The utilized step size is h = 0.005, the number of time steps N =10,000 and the initial data were the same for each case. To approximate the first component of the right-hand side of (8), denoted by f 1 (x 1 , x 2 ), a cubic surface was utilized following the algorithm proposed in [21]. The graph of the set-valued function F 1 , before and after his approximation, is presented in Fig.…”

Section: Remarkmentioning

confidence: 99%

Numerical approximation of a class of discontinuous systems of fractional order

Danca

2010

Nonlinear Dyn

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In this paper we investigate the possibility to formulate an implicit multistep numerical method for fractional differential equations, as a discrete dynamical system to model a class of discontinuous dynamical systems of fractional order. For this purpose, the problem is continuously transformed into a setvalued problem, to which the approximate selection theorem for a class of differential inclusions applies. Next, following the way presented in the book of Stewart and Humphries (Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996) for the case of continuous differential equations, we prove that a variant of AdamsBashforth-Moulton method for fractional differential equations can be considered as defining a discrete dynamical system, approximating the underlying discontinuous fractional system. For this purpose, the existence and uniqueness of solutions are investigated. One example is presented.

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“…Several properties of switch dynamical systems were analyzed by the author as continuous approximation [12], synchronization [13] and anticontrol of chaos [14].…”

Section: Switch Dynamical Systemsmentioning

confidence: 99%

“…1(a)) (see also [12] for some mathematical characteristics of the system). For for d ¼ 0:01 and k ¼ À0:002 two controlled trajectories reach two stable fixed points ( Fig.…”

Section: Chua Circuitmentioning

confidence: 99%