Fast cycles detecting in non-linear discrete systems

In the paper below we consider a problem of stabilization of a priori unknown unstable periodic orbits in non-linear autonomous discrete dynamical systems. We suggest a generalization of a non-linear DFC scheme to improve the rate of detecting T-cycles. Some numerical simulations are presented

On the stability of cycles by delayed feedback control

We present a delayed feedback control (DFC) mechanism for stabilizing cycles of one dimensional discrete time systems. In particular, we consider a delayed feedback control for stabilizing $T$-cycles of a differentiable function $f: \mathbb{R}\rightarrow\mathbb{R}$ of the form $$x(k+1) = f(x(k)) + u(k)$$ where $$u(k) = (a_1 - 1)f(x(k)) + a_2 f(x(k-T)) + ... + a_N f(x(k-(N-1)T))\;,$$ with $a_1 + ... + a_N = 1$. Following an approach of Morg\"ul, we construct a map $F: \mathbb{R}^{T+1} \rightarrow \mathbb{R}^{T+1}$ whose fixed points correspond to $T$-cycles of $f$. We then analyze the local stability of the above DFC mechanism by evaluating the stability of the corresponding equilibrum points of $F$. We associate to each periodic orbit of $f$ an explicit polynomial whose Schur stability corresponds to the stability of the DFC on that orbit. An example indicating the efficacy of this method is provided

Fejér Polynomials and Chaos

We show that given any μ \u3e 1, an equilibrium x of a dynamic system xn+1=f(xn) (1) can be robustly stabilized by a nonlinear control u=−∑j=1N−1εj(f(xn−j+1)−f(xn−j)), |εj| \u3c 1, j=1,…,N−1, (2) for f ′ (x) ∈ (−μ, 1). The magnitude of the minimal value N is of order √μ. The optimal explicit strength coefficients are found using extremal nonnegative Fejér polynomials. The case of a cycle as well as numeric examples and applications to mathematical biology are considered

Vorapaxar in the secondary prevention of atherothrombotic events

Item does not contain fulltextBACKGROUND: Thrombin potently activates platelets through the protease-activated receptor PAR-1. Vorapaxar is a novel antiplatelet agent that selectively inhibits the cellular actions of thrombin through antagonism of PAR-1. METHODS: We randomly assigned 26,449 patients who had a history of myocardial infarction, ischemic stroke, or peripheral arterial disease to receive vorapaxar (2.5 mg daily) or matching placebo and followed them for a median of 30 months. The primary efficacy end point was the composite of death from cardiovascular causes, myocardial infarction, or stroke. After 2 years, the data and safety monitoring board recommended discontinuation of the study treatment in patients with a history of stroke owing to the risk of intracranial hemorrhage. RESULTS: At 3 years, the primary end point had occurred in 1028 patients (9.3%) in the vorapaxar group and in 1176 patients (10.5%) in the placebo group (hazard ratio for the vorapaxar group, 0.87; 95% confidence interval [CI], 0.80 to 0.94; P<0.001). Cardiovascular death, myocardial infarction, stroke, or recurrent ischemia leading to revascularization occurred in 1259 patients (11.2%) in the vorapaxar group and 1417 patients (12.4%) in the placebo group (hazard ratio, 0.88; 95% CI, 0.82 to 0.95; P=0.001). Moderate or severe bleeding occurred in 4.2% of patients who received vorapaxar and 2.5% of those who received placebo (hazard ratio, 1.66; 95% CI, 1.43 to 1.93; P<0.001). There was an increase in the rate of intracranial hemorrhage in the vorapaxar group (1.0%, vs. 0.5% in the placebo group; P<0.001). CONCLUSIONS: Inhibition of PAR-1 with vorapaxar reduced the risk of cardiovascular death or ischemic events in patients with stable atherosclerosis who were receiving standard therapy. However, it increased the risk of moderate or severe bleeding, including intracranial hemorrhage. (Funded by Merck; TRA 2P-TIMI 50 ClinicalTrials.gov number, NCT00526474.)