CUPOLETS: Chaotic unstable periodic orbits theory and applications

Recent theoretical work suggests that periodic orbits of chaotic systems are a rich source of qualitative information about the dynamical system. The presence of unstable periodic orbits located densely on the attractor is a typical characteristic of chaotic systems. This abundance of unstable periodic orbits can be utilized in a wide variety of theoretical and practical applications [19]. In particular, chaotic communication techniques and methods of controlling chaos depend on this property of chaotic attractors [12, 13]. In the first part of this thesis, a control scheme for stabilizing the unstable periodic orbits of chaotic systems is presented and the properties of these orbits are investigated. The technique allows for creation of thousands of periodic orbits. These approximated chaotic unstable periodic orbits are called cupolets (C&barbelow;haotic U&barbelow;nstable P&barbelow;eriodic O&barbelow;rbit- lets). We show that these orbits can be passed through a phase transformation to a compact cupolet state that possesses a wavelet-like structure and can be used to construct adaptive bases. The cupolet transformation can be regarded as an alternative to Fourier and wavelet transformations. In fact, this new framework provides a continuum between Fourier and wavelet transformations and can be used in variety of applications such as data and music compression, as well as image and video processing. The key point in this method is that all of these different dynamical behaviors are easily accessible via small controls. This technique is implemented in order to produce cupolets which are essentially approximate periodic orbits of the chaotic system. The orbits are produced with small perturbations which in turn suggests that these orbits might not be very far away from true periodic orbits. The controls can be considered as external numerical errors that happen at some points along the computer generated orbits. This raises the question of shadowability of these orbits. It is very interesting to know if there exists a true orbit of the system with a slightly different initial condition that stays close to the computer generated orbit. This true orbit, if it exists, is called a shadow and the computer generated orbit is then said to be shadowable by a true orbit. We will present two general purpose shadowing theorems for periodic and nonperiodic orbits of ordinary differential equations. The theorems provide a way to establish the existence of true periodic and non-periodic orbits near the approximated ones. Both theorems are suitable for computations and the shadowing distances, i.e., the distance between the true orbits and approximated orbits are given by quantities computable form the vector field of the differential equation

The Relation between Approximation in Distribution and Shadowing in Molecular Dynamics

Molecular dynamics refers to the computer simulation of a material at the atomic level. An open problem in numerical analysis is to explain the apparent reliability of molecular dynamics simulations. The difficulty is that individual trajectories computed in molecular dynamics are accurate for only short time intervals, whereas apparently reliable information can be extracted from very long-time simulations. It has been conjectured that long molecular dynamics trajectories have low-dimensional statistical features that accurately approximate those of the original system. Another conjecture is that numerical trajectories satisfy the shadowing property: that they are close over long time intervals to exact trajectories but with different initial conditions. We prove that these two views are actually equivalent to each other, after we suitably modify the concept of shadowing. A key ingredient of our result is a general theorem that allows us to take random elements of a metric space that are close in distribution and embed them in the same probability space so that they are close in a strong sense. This result is similar to the Strassen-Dudley Theorem except that a mapping is provided between the two random elements. Our results on shadowing are motivated by molecular dynamics but apply to the approximation of any dynamical system when initial conditions are selected according to a probability measure.Comment: 21 pages, final version accepted in SIAM Dyn Sy

Numerical Calculation of Lyapunov Exponents for Three-Dimensional Systems of Ordinary Differential Equations

We consider two algorithms for the computation of Lyapunov exponents for systems of ordinary dierential equations: orbit separation and continuous Gram-Schmidt orthonormal-ization. We also consider two Runge-Kutta methods for the solution of ordinary dierential equations. These algorithms and methods are applied to four three-dimensional systems of ordinary dierential equations, and the results are discussed

Parameter shifts for nonautonomous systems in low dimension: Bifurcation- and Rate-induced tipping

We discuss the nonlinear phenomena of irreversible tipping for non-autonomous systems where time-varying inputs correspond to a smooth "parameter shift" from one asymptotic value to another. We express tipping in terms of pullback attraction and present some results on how nontrivial dynamics for non-autonomous systems can be deduced from analysis of the bifurcation diagram for an associated autonomous system where parameters are fixed. In particular, we show that there is a unique local pullback point attractor associated with each linearly stable equilibrium for the past limit. If there is a smooth stable branch of equilibria over the range of values of the parameter shift, the pullback attractor will remain close to (track) this branch for small enough rates, though larger rates may lead to rate-induced tipping. More generally, we show that one can track certain stable paths that go along several stable branches by pseudo-orbits of the system, for small enough rates. For these local pullback point attractors, we define notions of bifurcation-induced and irreversible rate-induced tipping of the non-autonomous system. In one-dimension, we give a number of sufficient conditions for the presence or absence of rate-induced tipping, and we discuss some applications of our results to give criteria for irreversible rate-induced tipping in a conceptual climate model example

Shadowing para Aproximar la Solución de problemas Parabólicos Semilineales para intervalos de Tiempo Grandes.

Se presenta un resultado de shadowing para un problema de evolucion parabolico no autonomo. Utilizando el metodo de Euler hacia atras se demuestra que bajo ciertas hipótesis de regularidad se puede aproximar usando shadowing la solucion de un problema de la forma                               u'(t) = A(t)u(t) + f(t);donde A(t) es el generador de un semigrupo analítico sobre un espacio de Banach

Development of a Novel Method for Biochemical Systems Simulation: Incorporation of Stochasticity in a Deterministic Framework

Heart disease, cancer, diabetes and other complex diseases account for more than half of human mortality in the United States. Other diseases such as AIDS, asthma, Parkinson’s disease, Alzheimer’s disease and cerebrovascular ailments such as stroke not only augment this mortality but also severely deteriorate the quality of human life experience. In spite of enormous financial support and global scientific effort over an extended period of time to combat the challenges posed by these ailments, we find ourselves short of sighting a cure or vaccine. It is widely believed that a major reason for this failure is the traditional reductionist approach adopted by the scientific community in the past. In recent times, however, the systems biology based research paradigm has gained significant favor in the research community especially in the field of complex diseases. One of the critical components of such a paradigm is computational systems biology which is largely driven by mathematical modeling and simulation of biochemical systems. The most common methods for simulating a biochemical system are either: a) continuous deterministic methods or b) discrete event stochastic methods. Although highly popular, none of them are suitable for simulating multi-scale models of biological systems that are ubiquitous in systems biology based research. In this work a novel method for simulating biochemical systems based on a deterministic solution is presented with a modification that also permits the incorporation of stochastic effects. This new method, through extensive validation, has been proven to possess the efficiency of a deterministic framework combined with the accuracy of a stochastic method. The new crossover method can not only handle the concentration and spatial gradients of multi-scale modeling but it does so in a computationally efficient manner. The development of such a method will undoubtedly aid the systems biology researchers by providing them with a tool to simulate multi-scale models of complex diseases

On shadowing methods for data assimilation

Combining orbits from a model of a (chaotic) dynamical system with measured data to arrive at an improved estimate of the state of a physical system is known as data assimilation. This thesis deals with various algorithms for data assimilation. These algorithms are based on shadowing. Shadowing is a concept from the theory of dynamical systems. When a dynamical system has the property that an exact orbit of the dynamical system is located in a neighborhood of each pseudo-orbit, then this exact orbit shadows the pseudo-orbit. Shadowing can be used to show that a numerical solution of a dynamical system is located in a neighborhood of an exact solution. Shadowing refinement is a numerical technique in which an improved approximation to an exact solution is found from a pseudo-orbit. It is possible to use a shadowing refinement technique for data assimilation. Starting from observations, Newton's method is applied to approximate a zero of a cost operator, where the cost operator assigns costs to deviations from model solutions. The algorithms of Chapter 2 are based on a numerical time-dependent split between stable and unstable directions. The algorithm uses time-dependent projections onto the unstable subspace determined by using Lyapunov exponents and Lyapunov vectors. A shadowing algorithm is used in the unstable subspace, while synchronization is used in the stable subspace. The method is further extended to include parameter estimation and to some cases where only partial observations are available. Chapter 3 discusses data assimilation for imperfect models. Through regularization according to the Levenberg-Marquardt method, imperfections in the model are considered. It also describes how the shadowing method compares, both analytically and numerically, with the weak constraint 4DVar method and shows that the shadowing method is consistent with the measurement error distribution, which is not the case for the weak constraint 4DVar method. This effect is particularly evident when there are fewer observations. Moreover, when there are few observations, they have a smaller impact on unobserved variables in the shadowing method than in the weak constraint 4DVar method. Chapter 4 extends the method of Chapter 2 to other cases of partial observations, in a similar way to Chapter 3. Local convergence to a solution manifold is proved and a lower bound on an algorithmic time step is provided. Numerical experiments with the Lorenz-'63 and Lorenz-'96 models show convergence of the algorithm and further show that the method compares favorably with the weak constraint 4DVar method and another shadowing method called pseudo-orbit data assimilation. Chapter 5 further develops the method of the previous chapters. The algorithm is extended to an ensemble of states for estimating uncertainties of the algorithm, based on the concept of indistinguishable states. The chapter also includes some proofs on uniqueness, accuracy and consistency of the algorithm. The algorithm is applied to an imperfect model to show how the unmodeled components of the model can be estimated using the data assimilation algorithm