We find conditions for existence of two cycles for a five-dimensional piecewise-linear dynamical system that models functioning of a circular gene network. Conditions for existence of a cycle were obtained by the authors earlier. The phase portrait of a system is divided into subdomains (or ...
An important engineering activity concerns the fracture mechanics study of cracked structures where prediction of safety is based upon the value of the stress intensity factor as calculated from a linear elastic analysis. The stress intensity factor is a measure of the amplitude of the dominant ...
The case\(m=1\), i.e. linear diffusion, is not completely covered in our theory, due to the lack of control on the compactness near the logarithmic singularity in the gradient flow approach. More precisely, our strategy does provide an approximating scheme, validated numerically in [10], b...
These are conditions which ensure that when all players’ strategies are implemented, then the differential equation (1) describing the evolution of the state admits a unique piecewise continuously differentiable solution for each initial conditionx0; see, e.g., Başar and Olsder (1999). 4. Wi...
Some techniques for proving the existence and uniqueness of limit cycles for smooth differential systems are extended to continuous piecewise linear differential systems with two and three zones and no symmetry. For planar systems with three linearity zones, the existence of two limit cycles surrounding...
The non-uniqueness of 𝒮 with respect to the boundary does not become a problem for the holonomy, as it does not depend on the surface, only on the variables on the loop. However, a problem appears if we consider the infinitesimal limit under the condition that the loop shrinks to a ...
The first is time-reversible, but nevertheless dissipative and periodic, the piecewise-linear compressible Baker Map. The fractal properties of that two-dimensional map are mirrored by an even simpler example, the one-dimensional random walk, confined to the unit interval. As a further puzzle the...
As demonstrated by the authors of [36], modeling the control with continuous variables allows some important guarantees in terms of uniqueness/optimality of the Nash Equilibria (NE) when applying game theory to these type of problems. However, thermostatically controlled loads, such as many other ...