We will consider two special families of polynomial perturbations of the linear center. For the resulting perturbed systems, which are generalized Liénard systems, we provide the exact upper bound for the number of limit cycles that bifurcate from the periodic orbits of the linear center.

For a vector eld ξ on R we construct, under certain assumptions on ξ, an ordered model-theoretic structure associated to the ow of ξ. We do this in such a way that the set of all limit cycles of ξ is represented by a de nable set. This allows us to give two restatements of Dulac's Problem for ξ that is, the question whether ξ has nitely many limit cycles in model-theoretic terms, one involving ...

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Al~tract--In this paper the non-linear features of adaptive control algorithms are highlighted. Using a simple "linear" discrete time example, based on classical design, it is demonstrated that in the presence of undermodelling errors, non-linear phenomena in the feedback gain such as limit cycles and even chaos arise. Despite these complicated dynamics, robust stabilization of the plant can st...

We study the effect of additive Brownian noise on an ODE system that has a stable hyperbolic limit cycle, for initial data that are attracted to the limit cycle. The analysis is performed in the limit of small noise – that is, we modulate the noise by a factor ε ց 0 – and on a long time horizon. We prove explicit estimates on the proximity of the noisy trajectory and the limit cycle up to times...

For any given positive integer n we show the existence of a class of discontinuous piecewise linear differential systems with two zones in the plane having exactly n hyperbolic limit cycles. Moreover, all the points on the separation boundary between the two zones are of sewing type, except the origin which is the only equilibrium point.

This paper gives conjecture on the absence of limit cycles of the minimum L2-sensitivity realizations subject to L2scaling constraints for second-order digital filters. We design second-order digital filters with various pole-zero configurations, synthesize the minimum L2-sensitivity realizations subject to L2-scaling constraints, and examine if their coefficient matrices satisfy a sufficient c...

For real numbers a, b > 1, let ab (also written as a b) denote the result of interpreting a in base b instead of base 10. We define “dungeons” (as opposed to “towers”) to be numbers of the form a b c d . . . e, parenthesized either from the bottom upwards (preferred) or from the top downwards. Among other things, we show that the sequences of dungeons with nth terms 10 11 12 . . . (n− 1) n or n...

If Γ1, . . . ,Γn are limit groups and S ⊂ Γ1 × · · · × Γn is of type FPn(Q) then S contains a subgroup of finite index that is itself a direct product of at most n limit groups. This answers a question of Sela.

This AUTO demo shows the use of a set of boundary conditions for the localization and continuation of the transcritical bifurcation of a limit cycle in a four-dimensional Marr-Pirt model.