Dynamics and Bifurcations

Convergence under dynamical thresholds with delays

“…Thus we are motivated to look for conditions ensuring the nonexistence of periodic solutions of (28). We By the well-known Poincare-Bendixson negative criterion (see [12]), it follows that (28) has no nontrivial periodic solutions; now, since solutions of (28) remain bounded for , by the discussion above, the conclusion (35) follows and this completes the proof.…”

“…We consider the system (1) with first and subsequently discuss the behavior of the system when ; when , we have from (1) (9) We note that the delay differential equation (9) is supplemented with an initial condition of the form where is assumed to be a continuous real valued function on . We let (10) and obtain from (9) that (11) If denotes an equilibrium of (11), then satisfies (12) We assume that are such that (13) It is now elementary to note that (12) has a unique solution under (13) denoted by . Thus (11) has the trivial resting state on as its only equilibrium.…”

Homoclinic orbits of non-autonomous maps and their approximation

“…by assumption A7, the variational equation (5) possesses an exponential dichotomy on − and on + with projectors of equal rank fulfilling (7). This is a consequence of the Roughness-Theorem 16, cf.…”

Quasiperiodic motions in dynamical systems: Review of a renormalization group approach

“…Moreover γ(v, t) > 0 for ε small enough (recall that γ = 1/ε), in a neighbourhood U of the origin. Hence we can apply Barbashin-Krasokvsky's theorem [62] (or Lasalle's invariance principle [55]) to conclude that the origin is asymptotically stable and U is contained in its basin of attraction. In some cases, for instance if g(x) = x 2p+1 , p ∈ N, and f 0 = 0 (so that ∂ x g(c 0 ) > 0), the response solution turns out to be a global attractor [4], but of course in general it is only locally attracting.…”