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On periodic solutions and global dynamics in a periodic differential delay equation
Several aspects of global dynamics and the existence of periodic solutions are studied for the scalar differential delay equation x′(t)=a(t)f(x([t−K])), where f(x) is a continuous negative feedback function, x⋅f(x)<0x≠0,0≤a(t) is continuous ω-periodic, [⋅] is the integer part function, and the integer K≥0 is the delay. The case of integer period ω allows for a reduction to finite-dimensional difference equations. The dynamics of the latter are studied in terms of corresponding discrete maps, including the partial case of interval maps (K=0).
Observability of systems with delay convoluted observation
This paper analyzes finite dimensional linear time-invariant systems with observation of a delay, where that delay satisfies a particular implicit relation with the state variables, rendering the entire problem nonlinear. The objective is to retrieve the state variables from the measured delay. The first contribution involves the direct inversion of the delay, the second is the design of a finite dimensional state observer, and the third involves the derivation of certain properties of the delay - state relation. Realistic examples treat vehicles with ultrasonic position sensors
Square wave periodic solutions of a differential delay equation
We prove the existence of periodic solutions of the differential delay equation εx˙(t)+x(t)=f(x(t−1)),ε>0 under the assumptions that the continuous nonlinearity f(x) satisfies the negative feedback condition, x⋅f(x)<0,x≠0, has sufficiently large derivative at zero |f′(0)|, and possesses an invariant interval I∋0,f(I)⊆I, as a dimensional map. As ε→0+ we show the convergence of the periodic solutions to a discontinuous square wave function generated by the globally attracting 2-cycle of the map f.