An implicit algorithm for validated enclosures of the solutions to variational equations for ODEs

We propose a new algorithm for computing validated bounds for the solutions to the first order variational equations associated to ODEs. These validated solutions are the kernel of numerics computer-assisted proofs in dynamical systems literature. The method uses a high-order Taylor method as a predictor step and an implicit method based on the Hermite-Obreshkov interpolation as a corrector step. The proposed algorithm is an improvement of the $C^1$-Lohner algorithm proposed by Zgliczy\'nski and it provides sharper bounds. As an application of the algorithm, we give a computer-assisted proof of the existence of an attractor set in the R\"ossler system, and we show that the attractor contains an invariant and uniformly hyperbolic subset on which the dynamics is chaotic, that is, conjugated to subshift of finite type with positive topological entropy.Comment: 33 pages, 11 figure

Connecting orbits for a singular nonautonomous real Ginzburg-Landau type equation

We propose a method for computation of stable and unstable sets associated to hyperbolic equilibria of nonautonomous ODEs and for computation of specific type of connecting orbits in nonautonomous singular ODEs. We apply the method to a certain a singular nonautonomous real Ginzburg-Landau type equation, which that arises from the problem of formation of spots in the Swift-Hohenberg equation.Comment: 36 pages, 6 figure

Computer assisted proof of the existence of homoclinic tangency for the Henon map and for the forced-damped pendulum

We present a topological method for the efficient computer assisted verification of the existence of the homoclinic tangency which unfolds generically in a one-parameter family of planar maps. The method has been applied to the Henon map and the forced damped pendulum ODE.Comment: 34 pages, 3 figure

Validated numerics for period-tupling and touch-and-go bifurcations of symmetric periodic orbits in reversible systems

We propose a general framework for computer-assisted verification of the presence of symmetry breaking, period-tupling and touch-and-go bifurcations of symmetric periodic orbits for reversible maps. The framework is then adopted to Poincar\'e maps in reversible autonomous Hamiltonian systems. In order to justify the applicability of the method, we study bifurcations of halo orbits in the Circular Restricted Three Body Problem. We give a computer-assisted proof of the existence of wide branches of halo orbits bifurcating from $L_{1,2,3}$-Lyapunov families and for wide range of mass parameter. For two physically relevant mass parameters we prove, that halo orbits undergo multiple period doubling, quadrupling and third-order touch-and-go bifurcations.Comment: 36 pages, 9 Figure

A geometric method for infinite-dimensional chaos : symbolic dynamics for the Kuramoto-Sivashinsky PDE on the line

We propose a general framework for proving that a compact, infinite-dimensional map has an invariant set on which the dynamics is semiconjugated to a subshift of finite type. The method is then applied to certain Poincaré map of the Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions and with parameter . We give a computer-assisted proof of the existence of symbolic dynamics and countable infinity of periodic orbits with arbitrary large periods

Optimizing a One DOF Robot Without a Mathematical Model Using a Genetic Algorithm

The purpose of the present study was to design a flight control system with no pre-determined mathematical model, but instead using a genetic algorithm to maintain the optimal altitude. The study is done through a quantitative empirical research method. In the process of conducting the research, we found that programming a genetic algorithm was cumbersome for novice users to implement. Due to this, we created and released an open source Python package called EasyGA. An initial population of 10 chromosomes and 5 generations were used during the trial. The throttle value of the device had an associated gene value of 1 second. When the trial of five generations was completed, the total increase percentage was 171 percent. Preliminary results showed that optimizing a one DOF device, in real-time, is possible without using a pre-determined mathematical model

Building an ensemble neural network optimized using uncertainty quantification for predicting metal oxide spectrograms from scanned metal oxide images.

Optical absorption spectroscopy is an important characterization of materials for applications such as solar energy generation. The purpose of the study is to build an ensemble neural network for predicting metal oxide spectrograms from images of metal oxide that have been scanned. With an ensemble network, several models are trained to produce a variety of predictions. By averaging these predictions, an even more accurate prediction can be made. Furthermore, uncertainty quantification will be applied by measuring the variance between the predictions, allowing more useful statistical analysis to be done such as producing confidence intervals to determine how accurate the results are. The study is done through a quantitative empirical research method. The research is a collaboration with Nevada National Security Site

C^{r}-Lohner algorithm

We present a Lohner type algorithm for the computation of rigorous bounds for the solutions of ordinary differential equations and its derivatives with respect to the initial conditions up to an arbitrary order