Oscillatory modes in a nonlinear second-order differential equation with delay

Abstract

Solution properties of the nonlinear second-order delay-differential equation 2(0 = -ax(t) +f [x(t -3)] are studied where f is a piecewise constant function which mimics negative feedback. We show that the solutions can be obtained by a simple geometrical construction which, in principle, can be implemented using a ruler and a compass. Analytical results guarantee the existence and stability properties of limit cycle solutions. Computer-aided constructions reveal a remarkable richness of different types of dynamical behaviors including a variety of unconventional bifurcation schemes. KEY WORDS:Nonlinear differential equations of second order with deviating argument; oscillations; periodic solutions.

Modelling human balance using switched systems with linear feedback control

“…This scenario can explain noise-induced switchings between two coexisting attractors reported in Eurich & Milton [20]. Finally, we should note that multi-stability as well as the presence of periodic oscillations have already been reported in the literature on switched delay differential equations, see [28][29][30][31]. However, the delay differential equations studied in these works contain discrete time delay in the position feedback only.…”

Quantization improves stabilization of dynamical systems with delayed feedback

“…7,11 Historically, the effect of piecewise constant, time-delayed feedback was extensively studied in early investigations into the dynamics of time-delayed feedback control. [24][25][26][27][28][29][30] Since experimental paradigms could be readily developed, it was possible to directly compare prediction with observation. [30][31][32][33] It is important to note that in these models the feedback switching times could be precisely computed analytically and thus the solutions were obtained by piecing together exponential segments or spiral arcs.…”

“…where h is the quantization step. When jxj ) h, then Ch Intð xðtÀsÞ h Þ % Cxðt À sÞ and hence the global dynamics are still governed by (26). Alternatively, one can write (28) Figure 4 summarizes the behavior of (28) as a function of C and s when h ¼ 1.…”

On Super-high Frequencies in Discontinuous 1st-Order Delay–Differential Equations

“…Similarly X decreases from h to the first time h+t where X(h+t)=0, so t=F & (X(h)). This proves that the length of the time intervals on which X is positive is equal to T + , defined by (10). Similarly the length of the time intervals on which X is negative is equal to T & , defined by (11).…”

“…Finally, some partial results for second order systems have been obtained by An der Heiden and Reichard [10] and Fridman et al [6].…”