1,940 research outputs found
Reduction of dimension for nonlinear dynamical systems
We consider reduction of dimension for nonlinear dynamical systems. We
demonstrate that in some cases, one can reduce a nonlinear system of equations
into a single equation for one of the state variables, and this can be useful
for computing the solution when using a variety of analytical approaches. In
the case where this reduction is possible, we employ differential elimination
to obtain the reduced system. While analytical, the approach is algorithmic,
and is implemented in symbolic software such as {\sc MAPLE} or {\sc SageMath}.
In other cases, the reduction cannot be performed strictly in terms of
differential operators, and one obtains integro-differential operators, which
may still be useful. In either case, one can use the reduced equation to both
approximate solutions for the state variables and perform chaos diagnostics
more efficiently than could be done for the original higher-dimensional system,
as well as to construct Lyapunov functions which help in the large-time study
of the state variables. A number of chaotic and hyperchaotic dynamical systems
are used as examples in order to motivate the approach.Comment: 16 pages, no figure
Activation of effector immune cells promotes tumor stochastic extinction: A homotopy analysis approach
In this article we provide homotopy solutions of a cancer nonlinear model
describing the dynamics of tumor cells in interaction with healthy and effector
immune cells. We apply a semi-analytic technique for solving strongly nonlinear
systems - the Step Homotopy Analysis Method (SHAM). This algorithm, based on a
modification of the standard homotopy analysis method (HAM), allows to obtain a
one-parameter family of explicit series solutions. By using the homotopy
solutions, we first investigate the dynamical effect of the activation of the
effector immune cells in the deterministic dynamics, showing that an increased
activation makes the system to enter into chaotic dynamics via a
period-doubling bifurcation scenario. Then, by adding demographic stochasticity
into the homotopy solutions, we show, as a difference from the deterministic
dynamics, that an increased activation of the immune cells facilitates cancer
clearance involving tumor cells extinction and healthy cells persistence. Our
results highlight the importance of therapies activating the effector immune
cells at early stages of cancer progression
Generalized models as a universal approach to the analysis of nonlinear dynamical systems
We present a universal approach to the investigation of the dynamics in
generalized models. In these models the processes that are taken into account
are not restricted to specific functional forms. Therefore a single generalized
models can describe a class of systems which share a similar structure. Despite
this generality, the proposed approach allows us to study the dynamical
properties of generalized models efficiently in the framework of local
bifurcation theory. The approach is based on a normalization procedure that is
used to identify natural parameters of the system. The Jacobian in a steady
state is then derived as a function of these parameters. The analytical
computation of local bifurcations using computer algebra reveals conditions for
the local asymptotic stability of steady states and provides certain insights
on the global dynamics of the system. The proposed approach yields a close
connection between modelling and nonlinear dynamics. We illustrate the
investigation of generalized models by considering examples from three
different disciplines of science: a socio-economic model of dynastic cycles in
china, a model for a coupled laser system and a general ecological food web.Comment: 15 pages, 2 figures, (Fig. 2 in color
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