Unpredictable tunneling in a retarded bistable potential.

We have studied the rich dynamics of a damped particle inside an external double-well potential under the influence of state-dependent time-delayed feedback. In certain regions of the parameter space, we observe multistability with the existence of two different attractors (limit cycle or strange attractor) with well separated mean Lyapunov energies forming a two-level system. Bifurcation analysis reveals that, as the effects of the time-delay feedback are enhanced, chaotic transitions emerge between the two wells of the double-well potential for the attractor corresponding to the fundamental energy level.

Average conservative chaos in quantum dusty plasmas.

We consider a hydrodynamic model of a quantum dusty plasma. We prove mathematically that the resulting dust ion-acoustic plasma waves present the property of being conservative on average. Furthermore, we test this property numerically, confirming its validity.

The dose-dense principle in chemotherapy.

Chemotherapy is a cancer treatment modality that uses drugs to kill tumor cells. A typical chemotherapeutic protocol consists of several drugs delivered in cycles of three weeks. We present mathematical analyses demonstrating the existence of a maximum time between cycles of chemotherapy for a protocol to be effective.

Dynamics of the cell-mediated immune response to tumour growth.

Using a hybrid cellular automaton, we investigate the transient and asymptotic dynamics of the cell-mediated immune response to tumour growth. We analyse the correspondence between this dynamics and the three phases of the theory of immunoedition: elimination, equilibrium and escape. Our results demonstrate that the immune system can keep a tumour dormant for long periods of time, but that this dormancy is based on a frail equilibrium between the mechanisms that spur the immune response and the growth of the tumour.

Decay Dynamics of Tumors.

The fractional cell kill is a mathematical expression describing the rate at which a certain population of cells is reduced to a fraction of itself. We investigate the mathematical function that governs the rate at which a solid tumor is lysed by a cell population of cytotoxic lymphocytes. We do it in the context of enzyme kinetics, using geometrical and analytical arguments.

A validated mathematical model of tumor growth including tumor-host interaction, cell-mediated immune response and chemotherapy.

We consider a dynamical model of cancer growth including three interacting cell populations of tumor cells, healthy host cells and immune effector cells. The tumor-immune and the tumor-host interactions are characterized to reproduce experimental results. A thorough dynamical analysis of the model is carried out, showing its capability to explain theoretical and empirical knowledge about tumor development.

Avoiding healthy cells extinction in a cancer model.

We consider a dynamical model of cancer growth including three interacting cell populations of tumor cells, healthy host cells and immune effector cells. For certain parameter choice, the dynamical system displays chaotic motion and by decreasing the response of the immune system to the tumor cells, a boundary crisis leading to transient chaotic dynamics is observed. This means that the system behaves chaotically for a finite amount of time until the unavoidable extinction of the healthy and immune cell populations occurs.

Effective suppressibility of chaos.

Suppression of chaos is a relevant phenomenon that can take place in nonlinear dynamical systems when a parameter is varied. Here, we investigate the possibilities of effectively suppressing the chaotic motion of a dynamical system by a specific time independent variation of a parameter of our system. In realistic situations, we need to be very careful with the experimental conditions and the accuracy of the parameter measurements.