Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories

Dynamical equations are formulated and a numerical study is provided for self-oscillatory model systems based on the triple linkage hinge mechanism of Thurston -- Weeks -- Hunt -- MacKay. We consider systems with holonomic mechanical constraint of three rotators as well as systems, where three rotators interact by potential forces. We present and discuss some quantitative characteristics of the chaotic regimes (Lyapunov exponents, power spectrum). Chaotic dynamics of the models we consider are associated with hyperbolic attractors, at least, at relatively small supercriticality of the self-oscillating modes; that follows from numerical analysis of the distribution for angles of intersection of stable and unstable manifolds of phase trajectories on the attractors. In systems based on rotators with interacting potential the hyperbolicity is violated starting from a certain level of excitation.Comment: 30 pages, 18 figure

Distributional properties of exponential functionals of Levy processes

We study the distribution of the exponential functional I(\xi,\eta)=\int_0^{\infty} \exp(\xi_{t-}) \d \eta_t, where $\xi$ and $\eta$ are independent L\'evy processes. In the general setting using the theories of Markov processes and Schwartz distributions we prove that the law of this exponential functional satisfies an integral equation, which generalizes Proposition 2.1 in Carmona et al "On the distribution and asymptotic results for exponential functionals of Levy processes". In the special case when $\eta$ is a Brownian motion with drift we show that this integral equation leads to an important functional equation for the Mellin transform of $I(\xi,\eta)$, which proves to be a very useful tool for studying the distributional properties of this random variable. For general L\'evy process $\xi$ ($\eta$ being Brownian motion with drift) we prove that the exponential functional has a smooth density on $\r \setminus \{0\}$, but surprisingly the second derivative at zero may fail to exist. Under the additional assumption that $\xi$ has some positive exponential moments we establish an asymptotic behaviour of \p(I(\xi,\eta)>x) as $x\to +\infty$, and under similar assumptions on the negative exponential moments of $\xi$ we obtain a precise asympotic expansion of the density of $I(\xi,\eta)$ as $x\to 0$. Under further assumptions on the L\'evy process $\xi$ one is able to prove much stronger results about the density of the exponential functional and we illustrate some of the ideas and techniques for the case when $\xi$ has hyper-exponential jumps.Comment: In this version we added a remark after Theorem 1 about extra conditions required for validity of equation (2.3

Behavior of tumors under nonstationary theraphy

We present a model for the interaction dynamics of lymphocytes-tumor cells population. This model reproduces all known states for the tumor. Futherly,we develop it taking into account periodical immunotheraphy treatment with cytokines alone. A detailed analysis for the evolution of tumor cells as a function of frecuency and theraphy burden applied for the periodical treatment is carried out. Certain threshold values for the frecuency and applied doses are derived from this analysis. So it seems possible to control and reduce the growth of the tumor. Also, constant values for cytokines doses seems to be a succesful treatment.Comment: 6 pages, 7 figure