On the dynamics of a class of difference equations with continuous arguments and its singular perturbation
The dynamical properties of a class of difference equations with continuous arguments of the form x(t)=g(x(t-r1),x(t-r2)) and its singularly perturbed counterpart ∊dxdt=-x(t)+g(x(t-r1),x(t-r2)) are investigated here. We discuss the effect of the time delays r1 and r2 on the qualitative behavior of t...
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| Format: | Article |
| Language: | English |
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Elsevier
2023-03-01
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| Series: | Alexandria Engineering Journal |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S1110016822007025 |
| _version_ | 1811162718832427008 |
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| author | A.M.A. EL-Sayed S.M. Salman A.M.A. Abo-Bakr |
| author_facet | A.M.A. EL-Sayed S.M. Salman A.M.A. Abo-Bakr |
| author_sort | A.M.A. EL-Sayed |
| collection | DOAJ |
| description | The dynamical properties of a class of difference equations with continuous arguments of the form x(t)=g(x(t-r1),x(t-r2)) and its singularly perturbed counterpart ∊dxdt=-x(t)+g(x(t-r1),x(t-r2)) are investigated here. We discuss the effect of the time delays r1 and r2 on the qualitative behavior of the considered dynamical systems. The local stability of the fixed points is studied. It is proved that the systems exhibit Hopf bifurcation which means that periodic orbits can be created from a fixed point by varying the delays. We compare the results of the singularly perturbed equation with those of the associated difference equation with continuous arguments when the perturbation parameter ∊⟶0 and with those of the corresponding delay differential equation when ∊⟶1. By letting the perturbation parameter ∊⟶0, we find that the singularly perturbed equation exhibits the same qualitative behavior as its corresponding difference equation. Furthermore, the singularly perturbed equation behaves qualitatively the same as its corresponding delay differential equation when ∊⟶1. Finally, we discuss that how this work can be generalized in the fractional differential calculus sense. |
| first_indexed | 2024-04-10T06:34:49Z |
| format | Article |
| id | doaj.art-20a093d58b0546e49a1a1558ff507e74 |
| institution | Directory Open Access Journal |
| issn | 1110-0168 |
| language | English |
| last_indexed | 2024-04-10T06:34:49Z |
| publishDate | 2023-03-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Alexandria Engineering Journal |
| spelling | doaj.art-20a093d58b0546e49a1a1558ff507e742023-03-01T04:30:50ZengElsevierAlexandria Engineering Journal1110-01682023-03-0166739749On the dynamics of a class of difference equations with continuous arguments and its singular perturbationA.M.A. EL-Sayed0S.M. Salman1A.M.A. Abo-Bakr2Faculty of Science, Alexandria University, Alexandria, EgyptFaculty of Education, Alexandria University, Alexandria, EgyptFaculty of Science, Alexandria University, Alexandria, EgyptThe dynamical properties of a class of difference equations with continuous arguments of the form x(t)=g(x(t-r1),x(t-r2)) and its singularly perturbed counterpart ∊dxdt=-x(t)+g(x(t-r1),x(t-r2)) are investigated here. We discuss the effect of the time delays r1 and r2 on the qualitative behavior of the considered dynamical systems. The local stability of the fixed points is studied. It is proved that the systems exhibit Hopf bifurcation which means that periodic orbits can be created from a fixed point by varying the delays. We compare the results of the singularly perturbed equation with those of the associated difference equation with continuous arguments when the perturbation parameter ∊⟶0 and with those of the corresponding delay differential equation when ∊⟶1. By letting the perturbation parameter ∊⟶0, we find that the singularly perturbed equation exhibits the same qualitative behavior as its corresponding difference equation. Furthermore, the singularly perturbed equation behaves qualitatively the same as its corresponding delay differential equation when ∊⟶1. Finally, we discuss that how this work can be generalized in the fractional differential calculus sense.http://www.sciencedirect.com/science/article/pii/S1110016822007025Difference equationsSingular perturbationDelay differential equationsHopf bifurcationChaos |
| spellingShingle | A.M.A. EL-Sayed S.M. Salman A.M.A. Abo-Bakr On the dynamics of a class of difference equations with continuous arguments and its singular perturbation Alexandria Engineering Journal Difference equations Singular perturbation Delay differential equations Hopf bifurcation Chaos |
| title | On the dynamics of a class of difference equations with continuous arguments and its singular perturbation |
| title_full | On the dynamics of a class of difference equations with continuous arguments and its singular perturbation |
| title_fullStr | On the dynamics of a class of difference equations with continuous arguments and its singular perturbation |
| title_full_unstemmed | On the dynamics of a class of difference equations with continuous arguments and its singular perturbation |
| title_short | On the dynamics of a class of difference equations with continuous arguments and its singular perturbation |
| title_sort | on the dynamics of a class of difference equations with continuous arguments and its singular perturbation |
| topic | Difference equations Singular perturbation Delay differential equations Hopf bifurcation Chaos |
| url | http://www.sciencedirect.com/science/article/pii/S1110016822007025 |
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