In this paper we propose a model of a nonautonomous dynamical system whose operations show turbulent behavior. The appearance of the trajectory is similar to that of a simplified tornado. The electronic circuit implementation of this model contains simply two linear capacitors, a voltage-controlled switch (VCS), a linear inductor, two resistors, and a nonlinear negative resistor. Computer simul...

We explore the multifractal, self-similar organization of heteroclinic and homoclinic bifurcations of saddle singularities in the parameter space of the Shimizu–Morioka model that exhibits the Lorenz chaotic attractor. We show that complex transformations that underlie the transitions from the Lorenz attractor to wildly chaotic dynamics are intensified by Shilnikov saddle-foci. These transforma...

In 1963, meteorologist Edward Lorenz suggested an approximate mathematical model (the Lorenz system) for the Rayleigh-Bénard convection and discovered numerically a chaotic attractor in this model [1]. This discovery stimulated rapid development of the chaos theory, numerical methods for attractor investigation, and till now has received a great deal of attention from different fields [2–7]. Th...

In this assignment we will build a cortical network model that captures neuronal persistent activity, similar to the persistent activity recorded in the prefrontal cortex of animals performing a spatial working memory task. We will see how stable activity patterns emerge from the recurrent network dynamics and persist on time scales greatly exceeding biophysical time constants of constituent ne...

To study the effects of immigration of genes (possibly transgenic) into a natural population, a one-island selection-migration model with density-dependent regulation is used to track allele frequency and population size. The existence and uniqueness of a polymorphic genetic equilibrium is proved under a general assumption about dominance in fitnesses. Also, conditions are found which guarantee...

In this work the existence of a global attractor for the solution semiflow of the coupled two-cell Brusselator model equations is proved. A grouping estimation method and a new decomposition approach are introduced to deal with the challenge in proving the absorbing property and the asymptotic compactness of this type of fourvariable reaction-diffusion systems with cubic autocatalytic nonlinear...

We consider the critical behavior at the threshold of black hole formation for the five dimensional vacuum Einstein equations satisfying the cohomogeneity-two triaxial Bianchi IX ansatz. Exploiting a discrete symmetry present in this model we predict the existence of a codimension-two attractor. This prediction is confirmed numerically and the codimension-two attractor is identified as a discre...