On the Solutions of a Class of Discrete PWC Systems Modeled with Caputo-Type Delta Fractional Difference Equations

In this paper, it is shown that a class of discrete Piece Wise Continuous (PWC) systems with Caputo-type delta fractional difference may not have solutions. To overcome this obstacle, the discontinuous problem is restarted as a continuous fractional problem. First, the single-valued PWC problem is transformed into a set-valued one via Filippov’s theory, after which Cellina’s theorem allows the restart of the problem into a single-valued continuous one. A numerical example is proposed and analyzed.

[1]  A. Lopes,et al.  How Many Fractional Derivatives Are There? , 2022, Mathematics.

[2]  J. Jonnalagadda,et al.  An analysis of polynomial sequences and their application to discrete fractional operators , 2021, Journal of Difference Equations and Applications.

[3]  Kamaleldin Abodayeh,et al.  Existence and stability of nonlinear discrete fractional initial value problems with application to vibrating eardrum. , 2021, Mathematical biosciences and engineering : MBE.

[4]  M. Bohner,et al.  Existence and uniqueness of solutions for nonlinear Caputo fractional difference equations , 2020 .

[5]  Guanrong Chen,et al.  Complex dynamics, hidden attractors and continuous approximation of a fractional-order hyperchaotic PWC system , 2018, 1804.10774.

[6]  Marius-F. Danca,et al.  On Numerical Integration of Discontinuous Dynamical Systems , 2017, Int. J. Bifurc. Chaos.

[7]  Viet-Thanh Pham,et al.  The Relationship Between Chaotic Maps and Some Chaotic Systems with Hidden Attractors , 2016, Int. J. Bifurc. Chaos.

[8]  J. Baoguo,et al.  The asymptotic behavior of Caputo delta fractional equations , 2016 .

[9]  Yang Liu,et al.  Path-following analysis of the dynamical response of a piecewise-linear capsule system , 2016, Commun. Nonlinear Sci. Numer. Simul..

[10]  A. Peterson,et al.  Discrete Fractional Calculus , 2016 .

[11]  J. Jonnalagadda.  Analysis of a system of nonlinear fractional nabla difference equations , 2015 .

[12]  Josef Diblík,et al.  Nonexistence of periodic solutions and S-asymptotically periodic solutions in fractional difference equations , 2015, Appl. Math. Comput..

[13]  J. Machado,et al.  A Review of Definitions for Fractional Derivatives and Integral , 2014 .

[14]  Marius-F. Danca,et al.  Continuous Approximations of a Class of Piecewise Continuous Systems , 2014, Int. J. Bifurc. Chaos.

[15]  Joseph Páez Chávez,et al.  Bifurcation analysis of a piecewise-linear impact oscillator with drift , 2014 .

[16]  N. Ford,et al.  A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations , 2013 .

[17]  G. V. S. R. Deekshitulu,et al.  Fractional Order Difference Equations , 2012 .

[18]  Yong Zhou,et al.  Existence Results for Nonlinear Fractional Difference Equation , 2011 .

[19]  M. Yazdani,et al.  On the existence of periodic solutions in time-invariant fractional order systems , 2011, Autom..

[20]  Thabet Abdeljawad,et al.  On Riemann and Caputo fractional differences , 2011, Comput. Math. Appl..

[21]  Marius-F. Danca,et al.  Approach of a Class of Discontinuous Dynamical Systems of fractional Order: existence of solutions , 2011, Int. J. Bifurc. Chaos.

[22]  M. di Bernardo,et al.  Discontinuity-induced bifurcations of piecewise smooth dynamical systems , 2010, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[23]  Johnny Henderson,et al.  Impulsive differential inclusions with fractional order , 2010, Comput. Math. Appl..

[24]  Mohammad Saleh Tavazoei,et al.  A proof for non existence of periodic solutions in time invariant fractional order systems , 2009, Autom..

[25]  P. Eloe,et al.  Initial value problems in discrete fractional calculus , 2008 .

[26]  J. Cortés.  Discontinuous dynamical systems , 2008, IEEE Control Systems.

[27]  Soumitro Banerjee,et al.  Border-Collision bifurcations in One-Dimensional Discontinuous Maps , 2003, Int. J. Bifurc. Chaos.

[28]  Marian Wiercigroch,et al.  Applied nonlinear dynamics and chaos of mechanical systems with discontinuities , 2000 .

[29]  A. G. Ibrahim,et al.  Multivalued fractional differential equations , 1995 .

[30]  Asen L. Dontchev,et al.  Difference Methods for Differential Inclusions: A Survey , 1992, SIAM Rev..

[31]  Chia,et al.  Results for the discontinuous logistic map. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[32]  K. K.,et al.  Stick-slip vibrations and chaos , 1990, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[33]  D. Baleanu,et al.  Positivity analysis for mixed order sequential fractional difference operators , 2022, AIMS Mathematics.

[34]  R. Ferreira.  Discrete Fractional Calculus and Fractional Difference Equations , 2022, SpringerBriefs in Mathematics.

[35]  J. Jonnalagadda.  SOLUTIONS OF FRACTIONAL NABLA DIFFERENCE EQUATIONS - EXISTENCE AND UNIQUENESS , 2016 .

[36]  M. Benchohra,et al.  Existence of solutions to differential inclusions with fractional order and impulses. , 2010 .

[37]  Naima Hamidi,et al.  FRACTIONAL ORDER DIFFERENTIAL INCLUSIONS ON THE HALF-LINE , 2010 .

[38]  Paul W. Eloe,et al.  DISCRETE FRACTIONAL CALCULUS WITH THE NABLA OPERATOR , 2009 .

[39]  Marius-F. Danca,et al.  On a class of discontinuous dynamical systems , 2001 .

[40]  Alois Kastner-Maresch,et al.  Difference methods with selection strategies for differential inclusions , 1993 .

[41]  F. Lempio.  Difference Methods for Differential Inclusions , 1992 .

[42]  K. Deimling.  Nonlinear functional analysis , 1985 .