2017/08/27 by Bernold Fiedler, Fiedler, Bernold, Isabelle Schneider +1
Computer Science · Engineering · Medicine · #34K13 #34K18 #34K20 #34K35 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Dynamics and Pattern Formation #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1708.08101
openalex publication_date 2017/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study scalar delay equations
dotx (t) =
lambda f(x(t-1)) + b-1\n(x(t) + x(t -p/2)) with odd nonlinearity f, real nonzero parameters\n\λ, , b, and two positive time delays 1, p/2. We assume\nsupercritical Hopf~bifurcation from x \≡ 0 in the well-understood\nsingle-delay case b = \∞. Normalizing f' (0)=1, branches of constant\nminimal period pk = 2\π/\ωk are known to bifurcate from eigenvalues\ni\ωk = i(k+ tfrac12)\π at \λk = (-1)k+1\ωk, for any\nnonnegative integer k. The unstable dimension of these rapidly oscillating\nperiodic solutions is k, at the local branch k. We obtain stabilization of\nsuch branches, for arbitrarily large unstable dimension k, and for,\nnecessarily, delicately narrow regions of control amplitudes b < 0.\n For p:= pk the branch k of constant period pk persists as a\nsolution, for any b\≠ 0. Indeed the delayed feedback term controlled by b\nvanishes on branch k: the feedback control is noninvasive there. Following an\nidea of Pyragas (1992), we seek parameter regions \P =\n( underlinebk,\bk) of controls b \≠ 0 such that the branch\nk becomes stable, locally at Hopf~bifurcation. We determine rigorous\nexpansions for \P in the limit of large k. Our analysis is based\non a 2-scale covering lift for the slow and rapid frequencies involved.\n These results complement earlier results by Fiedler and Oliva (2016) which\nrequired control terms b-1 (x(t-
vartheta) + x(t-
vartheta -p/2)) with a\nthird delay \ϑ near 1.\n