We search for limit cycles in the dynamical model of two-species chemical reactions that contain seven reaction rate coefficients as parameters and at least one third-order reaction step, that is, the induced kinetic differential equation of the reaction is a planar cubic differential system. Symbolic calculations were carried out using the Mathematica computer algebra system, and it was also used for the numerical verifications to show the following facts: the kinetic differential equations of these reactions each have two limit cycles surrounding the stationary point of focus type in the positive quadrant. In the case of Model 1, the outer limit cycle is stable and the inner one is unstable, which appears in a supercritical Hopf bifurcation. Moreover, the oscillations in a neighborhood of the outer limit cycle are slow-fast oscillations. In the case of Model 2, the outer limit cycle is unstable and the inner one is stable. With another set of parameters, the outer limit cycle can be made stable and the inner one unstable
COBISS.SI-ID: 32061699
Novel mathematical models of three different repressilator topologies are introduced. As designable transcription factors have been shown to bind to DNA non-cooperatively, we have chosen models containing non-cooperative elements. The extended topologies involve three additional transcription regulatory elements—which can be easily implemented by synthetic biology—forming positive feedback loops. This increases the number of variables to six, and extends the complexity of the equations in the model. To perform our analysis we had to use combinations of modern symbolic algorithms of computer algebra systems Mathematica and Singular. The study shows that all the three models have simple dynamics that can also be called regular behaviour: they have a single asymptotically stable steady state with small amplitude damping oscillations in the 3D case and no oscillation in one of the 6D cases and damping oscillation in the second 6D case. Using the program QeHopf we were able to exclude the presence of Hopf bifurcation in the 3D system.
COBISS.SI-ID: 6549786
For a given family of real planar polynomial systems of ordinary differential equations depending on parameters, we consider the problem of how to find the systems in the family which become time-reversible after some affine transformation. We first propose a general computational approach to solve this problem, and then demonstrate its usage for the case of the family of quadratic systems.
COBISS.SI-ID: 21654550
We propose an approach to study small limit cycle bifurcations on a center manifold in analytic or smooth systems depending on parameters. We then apply it to the investigation of limit cycle bifurcations in a model of calcium oscillations in the cilia of olfactory sensory neurons and show that it can have two limit cycles: a stable cycle appearing after a Bautin (generalized Hopf) bifurcation and an unstable cycle appearing after a subcritical Hopf bifurcation.
COBISS.SI-ID: 22250006
In this paper we generalize the notion of a persistent center to a persistent p : -q resonant center and find conditions for existence of a persistent p : -q resonant center for several p : -q resonant systems with quadratic nonlinearities. To prove the sufficiency of the obtained conditions we use either the Darboux theory of integrability or look for a formal first integral of the required form or we use the method based on the blow-up transformation
COBISS.SI-ID: 21997590