We propose a reduced form set of two coupled continuous time equations linking the price of a representative asset and the price of a bond, the later quantifying the cost of borrowing. The feedbacks between asset prices and bonds are mediated by the
dependence of their fundamental values on past asset prices and bond themselves. The obtained nonlinear self-referencing price dynamics can induce, in a completely objective deterministic way, the appearance of periodically exploding bubbles ending in crashes. Technically, the periodically explosive bubbles arise due to the proximity of two types of bifurcations as a function of the two key control parameters $b$ and $g$, which represent, respectively, the sensitivity of the fundamental asset price on past asset and bond prices and of the fundamental bond price on past asset prices. One is a Hopf bifurcation, when a stable focus transforms into an unstable focus and a limit cycle appears. The other is a rather unusual bifurcation, when a stable node and a saddle merge together and disappear, while an unstable focus survives and a limit cycle develops. The lines, where the periodic bubbles arise, are analogous to the critical lines of phase transitions in statistical physics. The amplitude of bubbles and waiting times between them respectively diverge with the critical exponents $gamma = 1$ and $ u = 1/2$, as the critical lines are approached.
A trapped Bose-Einstein condensate, being strongly perturbed, exhibits several spatial structures. First, there appear quantum vortices. Increasing the amount of the injected energy leads to the formation of vortex tangles representing quantum vortex
turbulence. Continuing energy injection makes the system so strongly perturbed that vortices become destroyed and there develops another kind of spatial structures with essentially heterogeneous spatial density. These structures consist of high-density droplets, or grains, surrounded by the regions of low density. The droplets are randomly distributed in space, where they can move; however they live sufficiently long time to be treated as a type of metastable creatures. Such structures have been observed in nonequilibrium trapped Bose gases of $^{87}$Rb subject to the action of an oscillatory perturbation modulating the trapping potential. Perturbing the system even stronger transforms the droplet structure into wave turbulence, where Bose condensate is destroyed. Numerical simulations are in good agreement with experimental observations.
We show that there exists the inverse Kibble-Zurek scenario, when we start with an equilibrium system with broken symmetry and, by imposing perturbations, transform it to a strongly nonequilibrium symmetric state through the sequence of states with s
pontaneously arising topological defects. We demonstrate the inverse Kibble-Zurek scenario both experimentally, by perturbing the Bose-Einstein condensate of trapped $^{87}$Rb atoms, and also by accomplishing numerical simulations for the same setup as in the experiment, the experimental and numerical results being in good agreement with each other.
We present experimental observations and numerical simulations of nonequilibrium spatial structures in a trapped Bose-Einstein condensate subject to oscillatory perturbations. In experiment, first, there appear collective excitations, followed by qua
ntum vortices. Increasing the amount of the injected energy leads to the formation of vortex tangles representing quantum turbulence. We study what happens after the regime of quantum turbulence, with increasing further the amount of injected energy. In such a strongly nonequilibrium Bose-condensed system of trapped atoms, vortices become destroyed and there develops a new kind of spatial structure exhibiting essentially heterogeneous spatial density. The structure reminds fog consisting of high-density droplets, or grains, surrounded by the regions of low density. The grains are randomly distributed in space, where they move. They live sufficiently long time to be treated as a type of metastable objects. Such structures have been observed in nonequilibrium trapped Bose gases of $^{87}$Rb, subject to the action of alternating fields. Here we present experimental results and support them by numerical simulations. The granular, or fog structure is essentially different from the state of wave turbulence that develops after increasing further the amount of injected energy.
The method of self-similar factor approximants is completed by defining the approximants of odd orders, constructed from the power series with the largest term of an odd power. It is shown that the method provides good approximations for transcendent
al functions. In some cases, just a few terms in a power series make it possible to reconstruct a transcendental function exactly. Numerical convergence of the factor approximants is checked for several examples. A special attention is paid to the possibility of extrapolating the behavior of functions, with arguments tending to infinity, from the related asymptotic series at small arguments. Applications of the method are thoroughly illustrated by the examples of several functions, nonlinear differential equations, and anharmonic models.
