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Sains Malaysiana 55(3)(2026): 517-529
http://doi.org/10.17576/jsm-2026-5503-13 Bifurcation Analysis and Chaos Control in a
Discrete Fractional- Order Model of Immune Response to Abnormal Cells
(Analisis Bifurkasi dan Kawalan Kekacauan dalam Model Tertib Pecahan Diskret bagi Tindak Balas Imun terhadap Sel Abnormal)
ALEH A. AL-SHAMMARI, A.S.
RAMBELY* & N.S. KAMARUDIN
Received: 25 February 2025/Accepted: 19 February
2026
Abstract
This study investigates the complex dynamics and control of a
discrete-time fractional-order model describing the immune response to abnormal
cells. We establish the local asymptotic stability of the equilibrium points
and derive the specific parametric conditions governing the emergence of
Neimark-Sacker and flip bifurcations. Numerical simulations not only confirm
these bifurcations but also show the subsequent onset of chaos. To mitigate
this, we successfully implement and demonstrate the efficacy of three control
strategies state feedback, pole placement, and hybrid control in stabilizing
the chaotic regimes and suppressing undesirable bifurcations. The numerical
results provide strong validation of the theoretical analysis, underscoring the
value of these control techniques for managing pathological dynamics in immune
system models.
Keywords: Discrete immune system model with fractional-order;
flip bifurcation; Hopf bifurcation; N-S bifurcation
Abstrak
Penyelidikan ini mengkaji dinamik kompleks dan kawalan model
tertib pecahan masa diskret yang menggambarkan tindak balas imun terhadap sel
yang tidak normal. Kami mewujudkan kestabilan asimptotik tempatan bagi titik
keseimbangan dan memperoleh keadaan parametrik khusus yang mengawal kemunculan
bifurkasi Neimark-Sacker dan flip. Simulasi berangka bukan sahaja mengesahkan
bifurkasi ini tetapi juga mendedahkan permulaan huru-hara berikutnya. Untuk
mengurangkan ini, kami berjaya melaksana dan menunjukkan keberkesanan tiga strategi
kawalan maklum balas keadaan, penempatan kutub dan kawalan hibrid dalam
menstabilkan rejim huru-hara dan menyekat bifurkasi yang tidak diingini.
Keputusan berangka memberikan pengesahan yang kukuh terhadap analisis teori,
menggariskan nilai teknik kawalan ini untuk mengurus dinamik patologi dalam
model sistem imun.
Kata kunci: Bifurkasi flip; bifurkasi Hopf; bifurkasi N-S; model
sistem imun diskret dengan tertib pecahan
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*Corresponding
author; email: asr@ukm.edu.my
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