"0-1" test chaosu

The goal of this thesis is to research the 0-1 test for chaos, its application in Matlab, and testing on suitable models. Elementary tools of the dynamical systems analysis are introduced, that are later used in the main results part of the thesis. The 0-1 test for chaos is introduced in detail, defined, and implemented in Matlab. The application is then performed on two one-dimensional discrete models where the first one is in explicit and the second one in implicit form. In both cases, simulations in dependence of the state parameter were done and main results are given - the 0-1 test for chaos, phase, and bifurcation diagrams.Hlavním cílem bakalářské práce je studium 0-1 testu chaosu, jeho implementace v Matlabu a následné testování na vhodných modelech. V práci jsou zavedeny základní nástroje analýzy dynamických systémů, které jsou později použity v části hlavních výsledků. 0-1 test chaosu je podrobně uveden, řádně definován a implementován v Matlabu. Aplikace je provedena na dvou jednodimenzionálních diskrétních modelech z nichž jeden je v explicitním a druhý v implicitním tvaru. V obou případech byly provedeny simulace v závislosti na stavovém parametru a hlavní výsledky byly demonstrovány formou 0-1 testu chaosu, fázových a bifurkačních diagramů.470 - Katedra aplikované matematikyvýborn

Discrete-Time Chaotic-Map Truly Random Number Generators: Design, Implementation, and Variability Analysis of the Zigzag Map

In this paper, we introduce a novel discrete chaotic map named zigzag map that demonstrates excellent chaotic behaviors and can be utilized in Truly Random Number Generators (TRNGs). We comprehensively investigate the map and explore its critical chaotic characteristics and parameters. We further present two circuit implementations for the zigzag map based on the switched current technique as well as the current-mode affine interpolation of the breakpoints. In practice, implementation variations can deteriorate the quality of the output sequence as a result of variation of the chaotic map parameters. In order to quantify the impact of variations on the map performance, we model the variations using a combination of theoretical analysis and Monte-Carlo simulations on the circuits. We demonstrate that even in the presence of the map variations, a TRNG based on the zigzag map passes all of the NIST 800-22 statistical randomness tests using simple post processing of the output data.Comment: To appear in Analog Integrated Circuits and Signal Processing (ALOG

A simple method for detecting chaos in nature

Chaos, or exponential sensitivity to small perturbations, appears everywhere in nature. Moreover, chaos is predicted to play diverse functional roles in living systems. A method for detecting chaos from empirical measurements should therefore be a key component of the biologist's toolkit. But, classic chaos-detection tools are highly sensitive to measurement noise and break down for common edge cases, making it difficult to detect chaos in domains, like biology, where measurements are noisy. However, newer tools promise to overcome these limitations. Here, we combine several such tools into an automated processing pipeline, and show that our pipeline can detect the presence (or absence) of chaos in noisy recordings, even for difficult edge cases. As a first-pass application of our pipeline, we show that heart rate variability is not chaotic as some have proposed, and instead reflects a stochastic process in both health and disease. Our tool is easy-to-use and freely available

Entanglement, randomness and chaos

Entanglement is not only the most intriguing feature of quantum mechanics, but also a key resource in quantum information science. The entanglement content of random pure quantum states is almost maximal; such states find applications in various quantum information protocols. The preparation of a random state or, equivalently, the implementation of a random unitary operator, requires a number of elementary one- and two-qubit gates that is exponential in the number n_q of qubits, thus becoming rapidly unfeasible when increasing n_q. On the other hand, pseudo-random states approximating to the desired accuracy the entanglement properties of true random states may be generated efficiently, that is, polynomially in n_q. In particular, quantum chaotic maps are efficient generators of multipartite entanglement among the qubits, close to that expected for random states. This review discusses several aspects of the relationship between entanglement, randomness and chaos. In particular, I will focus on the following items: (i) the robustness of the entanglement generated by quantum chaotic maps when taking into account the unavoidable noise sources affecting a quantum computer; (ii) the detection of the entanglement of high-dimensional (mixtures of) random states, an issue also related to the question of the emergence of classicality in coarse grained quantum chaotic dynamics; (iii) the decoherence induced by the coupling of a system to a chaotic environment, that is, by the entanglement established between the system and the environment.Comment: Review paper, 40 pages, 7 figures, added reference

Deciphering a novel image cipher based on mixed transformed Logistic maps

Since John von Neumann suggested utilizing Logistic map as a random number generator in 1947, a great number of encryption schemes based on Logistic map and/or its variants have been proposed. This paper re-evaluates the security of an image cipher based on transformed logistic maps and proves that the image cipher can be deciphered efficiently under two different conditions: 1) two pairs of known plain-images and the corresponding cipher-images with computational complexity of $O(2^{18}+L)$; 2) two pairs of chosen plain-images and the corresponding cipher-images with computational complexity of $O(L)$, where $L$ is the number of pixels in the plain-image. In contrast, the required condition in the previous deciphering method is eighty-seven pairs of chosen plain-images and the corresponding cipher-images with computational complexity of $O(2^{7}+L)$. In addition, three other security flaws existing in most Logistic-map-based ciphers are also reported.Comment: 10 pages, 2 figure

Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems

We investigate a generalised version of the recently proposed ordinal partition time series to network transformation algorithm. Firstly we introduce a fixed time lag for the elements of each partition that is selected using techniques from traditional time delay embedding. The resulting partitions define regions in the embedding phase space that are mapped to nodes in the network space. Edges are allocated between nodes based on temporal succession thus creating a Markov chain representation of the time series. We then apply this new transformation algorithm to time series generated by the R\"ossler system and find that periodic dynamics translate to ring structures whereas chaotic time series translate to band or tube-like structures -- thereby indicating that our algorithm generates networks whose structure is sensitive to system dynamics. Furthermore we demonstrate that simple network measures including the mean out degree and variance of out degrees can track changes in the dynamical behaviour in a manner comparable to the largest Lyapunov exponent. We also apply the same analysis to experimental time series generated by a diode resonator circuit and show that the network size, mean shortest path length and network diameter are highly sensitive to the interior crisis captured in this particular data set