Numerical controllability of fractional dynamical systems

In this paper, we provide a computational procedure for controlled state and steering control for linear and nonlinear fractional dynamical systems of order 1 < α ≤ 2 in finite-dimensional spaces using the Mittag-Leffler matrix function and the iterative technique. Some numerical examples are provided to illustrate the results.     

Existence and uniqueness results for a semilinear Black–Scholes type equation

This paper deals with the application of fixed point theorem to determine the source term of semilinear Black–Scholes type equation and thereby establish the existence and uniqueness of the solution. The proof mainly relies on the iteration method and the Schauder fixed point theorem.     

Existence and Uniqueness of Solutions of Predator-Prey Type Model with Mixed Boundary Conditions

In this work we consider the predator-prey model in ℝ³ with mixed boundary conditions on the Lipschitz boundary and prove the existence of solutions by Schauder’s fixed point theorem and uniqueness of solutions by Gronwall’s lemma.     

Classical topology and quantum states

Any two infinite-dimensional (separable) Hilbert spaces are unitarily isomorphic. The sets of all their self-adjoint operators are also therefore unitarily equivalent. Thus if all self-adjoint operators can be observed, and if there is no further major axiom in quantum physics than those formulated for example in Dirac’s ‘quantum mechanics’, then a quantum physicist would not be able to tell a torus from a hole in the ground. We argue that there are indeed such axioms involving observables with smooth time evolution: they contain commutative subalgebras from which the spatial slice of spacetime with its topology (and with further refinements of the axiom, its C ^K - and C ^--structures) can be reconstructed using Gel’fand-Naimark theory and its extensions. Classical topology is an attribute of only certain quantum observables for these axioms, the spatial slice emergent from quantum physics getting progressively less differentiable with increasingly higher excitations of energy and eventually altogether ceasing to exist. After formulating these axioms, we apply them to show the possibility of topology change and to discuss quantized fuzzy topologies. Fundamental issues concerning the role of time in quantum physics are also addressed.     

Expected escape times from attractor basins due to low intensity noise

In this paper, a numerical approach is described to estimate escape times from attractor basins when a dynamical system is subjected to noise or stochastic perturbations. Noise can affect nonlinear system response by driving solution trajectories to different attractors. The changes in physical behavior can be observed as amplitude and phase change of periodic oscillations, initiation or annihilation of chaotic motion, phase synchronization, and so on. Estimating probability of transitions from one attractor to another, and predicting escape times are essential for quantifying the effects of noise on the system response. In this paper, a numerical approach is outlined where probability transition maps are generated between grids. Then, these maps are iterated to find the probability distribution after long durations, wherein, a constant escape rate can be observed between basins. The constant escape rate is then used to estimate the average escape times. The approach is applicable to systems subjected to low-intensity stochastic disturbances and with long escape times, where Monte Carlo simulations are impractical. Escape times up to \(10^{13}\) periods are estimated without relying on computationally expensive computations.