Symbolic Dynamics Techniques Research Articles

Fine-Grained Modeling of ROP Vulnerability Exploitation Process under Stack Overflow Based on Petri Nets

Software vulnerability discovery is currently a hot topic, and buffer overflow remains a prevalent security vulnerability. One of the key issues in vulnerability discovery and analysis is how to quickly analyze buffer overflow vulnerabilities and select critical exploitation paths. Existing modeling methods for vulnerability exploitation cannot accurately reflect the fine-grained execution process of stack overflow exploitation paths. This paper, based on the discussion of buffer overflow exploitation techniques, proposes a fine-grained modeling and analysis method based on Petri nets for the selection and execution of exploitation processes, specifically focusing on the return-oriented programming in stack overflow. Through qualitative analysis, we compared the simulated time of the software with the execution time of existing exploitation tools, achieving timeout-based simulation experiments. We validated the model’s effectiveness using symbolic execution and dynamic analysis techniques. The results indicate that this model performs well for vulnerable programs with Position Independent Executable (PIE) protection enabled and has an advantage in selecting exploitation paths, enabling timeout-based simulation. This method provides a reference for rapidly constructing exploitation implementations.

Markov partition in the attractor of Lozi maps

In this paper, we study iterations of two-dimensional maps, in particular iterations of Lozi maps in the region of the parameter space where it has a strange attractor. Using symbolic dynamics techniques for two-dimensional maps, based on the kneading theory of Milnor and Thurston and also in the symbolic dynamic formalism developed by Sousa Ramos, through the kneading sequence for the Lozi maps, we characterize the region in the parameter space that contains the kneading curves and present a method to define a Markov partition for the Lozi attractors. Consequently, the topological entropy for the Lozi map is computed.

Debugging Multithreaded Programs as if They Were Sequential

Debugging multithread programs is extremely difficult because the basic assumption that underlies sequential program debugging, that is, the program behavior is deterministic under a fixed input, is no longer valid due to the nondeterminism attributed to thread scheduling. It is because the programs behavior is non-deterministic due to the nondeterminism of parallel execution, which makes debugging or testing multithreaded programs extremely difficult. In this paper, we propose a proactive debugging method to restore this basic assumption so that programmers can debug multithreaded programs as if they were sequential. Our approach is based on the synergistic integration of symbolic analysis and dynamic analysis techniques. In particular, symbolic analysis investigates the program behavior under the same input to search whether there is a thread schedule would trigger a bug or a not-yet-explored path. Dynamic analysis is to execute these new paths with the guide of the generated thread schedules, thereby further guiding the symbolic analysis. The net effect of applying this feedback loop is a systematic and complete coverage of the program behavior under a fixed test input. We implement the proposed approach in a prototype tool called proactive-debugger. Our experiments show that proactive-debugger outperforms both ESBMC and Maple, which are two powerful and well-known testing tools for failure detection in multithreaded programs.

Symbolic dynamics techniques for complex systems: Application to share price dynamics

The symbolic dynamics technique is well known for low-dimensional dynamical systems and chaotic maps, and lies at the roots of the thermodynamic formalism of dynamical systems. Here we show that this technique can also be successfully applied to time series generated by complex systems of much higher dimensionality. Our main example is the investigation of share price returns in a coarse-grained way. A nontrivial spectrum of Rényi entropies is found. We study how the spectrum depends on the time scale of returns, the sector of stocks considered, as well as the number of symbols used for the symbolic description. Overall our analysis confirms that in the symbol space transition probabilities of observed share price returns depend on the entire history of previous symbols, thus emphasizing the need for a modelling based on non-Markovian stochastic processes. Our method allows for quantitative comparisons of entirely different complex systems, for example the statistics of symbol sequences generated by share price returns using 4 symbols can be compared with that of genomic sequences.

Epileptic Seizure Prediction Using a New Similarity Index for Chaotic Signals

Epileptic seizures are generated by abnormal activity of neurons. The prediction of epileptic seizures is an important issue in the field of neurology, since it may improve the quality of life of patients suffering from drug resistant epilepsy. In this study a new similarity index based on symbolic dynamic techniques which can be used for extracting behavior of chaotic time series is presented. Using Freiburg EEG dataset, it is found that the method is able to detect the behavioral changes of the neural activity prior to epileptic seizures, so it can be used for prediction of epileptic seizure. A sensitivity of 63.75% with 0.33 false positive rate (FPR) in all 21 patients and sensitivity of 96.66% with 0.33 FPR in eight patients were achieved using the proposed method. Moreover, the method was evaluated by applying on Logistic and Tent map with different parameters to demonstrate its robustness and ability in determining similarity between two time series with the same chaotic characterization.

Statistical Mechanics and Information-Theoretic Perspectives on Complexity in the Earth System

This review provides a summary of methods originated in (non-equilibrium) statistical mechanics and information theory, which have recently found successful applications to quantitatively studying complexity in various components of the complex system Earth. Specifically, we discuss two classes of methods: (i) entropies of different kinds (e.g., on the one hand classical Shannon and R´enyi entropies, as well as non-extensive Tsallis entropy based on symbolic dynamics techniques and, on the other hand, approximate entropy, sample entropy and fuzzy entropy); and (ii) measures of statistical interdependence and causality (e.g., mutual information and generalizations thereof, transfer entropy, momentary information transfer). We review a number of applications and case studies utilizing the above-mentioned methodological approaches for studying contemporary problems in some exemplary fields of the Earth sciences, highlighting the potentials of different techniques.

Strong and weak Allee effects and chaotic dynamics in Richards' growths

In this paper we define and investigate generalized Richards' growth models with strong and weak Allee effects and no Allee effect. We prove the transition from strong Allee effect to no Allee effect, passing through the weak Allee effect, depending on the implicit conditions, which involve the several parameters considered in the models. New classes of functions describing the existence or not of Allee effect are introduced, a new dynamical approach to Richards' populational growth equation is established. These families of generalized Richards' functions are proportional to the right hand side of the generalized Richards' growth models proposed. Subclasses of strong and weak Allee functions and functions with no Allee effect are characterized. The study of their bifurcation structure is presented in detail, this analysis is done based on the configurations of bifurcation curves and symbolic dynamics techniques. Generically, the dynamics of these functions are classified in the following types: extinction, semi-stability, stability, period doubling, chaos, chaotic semistability and essential extinction. We obtain conditions on the parameter plane for the existence of a weak Allee effect region related to the appearance of cusp points. To support our results, we present fold and flip bifurcations curves and numerical simulations of several bifurcation diagrams.

Dynamical analysis in growth models: Blumberg's equation

We present a new dynamical approach to the Blumberg's equation, a family of unimodal maps. These maps are proportional to $Beta(p,q)$ probability densities functions. Using the symmetry of the $Beta(p,q)$ distribution and symbolic dynamics techniques, a new concept of mirror symmetry is defined for this family of maps. The kneading theory is used to analyze the effect of such symmetry in the presented models. The main result proves that two mirror symmetric unimodal maps have the same topological entropy. Different population dynamics regimes are identified, when the intrinsic growth rate is modified: extinctions, stabilities, bifurcations, chaos and Allee effect. To illustrate our results, we present a numerical analysis, where are demonstrated: monotonicity of the topological entropy with the variation of the intrinsic growth rate, existence of isentropic sets in the parameters space and mirror symmetry.

Generalized Models from Beta(p, 2) Densities with Strong Allee Effect: Dynamical Approach

A dynamical approach to study the behaviour of generalized populational growth models from Beta(p, 2) densities, with strong Allee effect, is presented. The dynamical analysis of the respective unimodal maps is performed using symbolic dynamics techniques. The complexity of the correspondent discrete dynamical systems is measured in terms of topological entropy. Different populational dynamics regimes are obtained when the intrinsic growth rates are modified: extinction, bistability, chaotic semistability and essential extinction.

A discrete time neural network model with spiking neurons

We derive rigorous results describing the asymptotic dynamics of a discrete time model of spiking neurons introduced in Soula et al. (Neural Comput. 18, 1, 2006). Using symbolic dynamic techniques we show how the dynamics of membrane potential has a one to one correspondence with sequences of spikes patterns ("raster plots"). Moreover, though the dynamics is generically periodic, it has a weak form of initial conditions sensitivity due to the presence of a sharp threshold in the model definition. As a consequence, the model exhibits a dynamical regime indistinguishable from chaos in numerical experiments.

Analyticity of the SRB measure of a lattice of coupled Anosov diffeomorphisms of the torus

We consider the “thermodynamic limit” of a d-dimensional lattice of hyperbolic dynamical systems on the 2-torus, interacting via weak and nearest neighbor coupling. We prove that the SRB measure is analytic in the strength of the coupling. The proof is based on symbolic dynamics techniques that allow us to map the SRB measure into a Gibbs measure for a spin system on a (d+1)-dimensional lattice. This Gibbs measure can be studied by an extension (decimation) of the usual “cluster expansion” techniques.

Bowen–Franks Groups Associated with Linear Mod One Transformations

We consider those linear mod one transformations fβ,α on the unit interval, for which the respective kneading sequences are periodic. Then, we construct the corresponding Markov transition matrices Aβ,α. Using symbolic dynamics techniques, we prove that the Bowen–Franks groups BF(Aβ,α) can be directly computed from the number of symbols of the respective periodic words of the kneading sequences of fβ,α.

Symbolic dynamics of event-related brain potentials

We apply symbolic dynamics techniques such as word statistics and measures of complexity to nonstationary and noisy multivariate time series of electroencephalograms (EEG) in order to estimate event-related brain potentials (ERP). Their significance against surrogate data as well as between different experimental conditions is tested. These methods are validated by simulations using stochastic dynamical systems with time-dependent control parameters and compared with traditional ERP-analysis techniques. Continuous EEG data are cut into epochs according to stimuli events presented to the subjects. These ensembles of time series can be considered as ensembles of trajectories given by some dynamical systems. We employ a statistical mechanics approach motivated by the Frobenius-Perron equation and apply it to coarse-grained symbolic descriptions of the dynamics. We develop time-dependent measures of complexity founded on running cylinder sets and show that these quantities are able to distinguish simulated data obtained by different control parameters as well as experimental data between different experimental conditions. As a first finding, our approach restores the well-known ERP components and it reveals additionally qualitative changes in the EEG that cannot be detected by means of the traditional techniques. We criticize the prerequisites of the traditional approach to ERP analysis and propose to consider ERP instead in terms of dynamical system theory and information theory.

Some applications of symbolic dynamics techniques to toral skew endomorphisms

A class of dynamical systems of the 2-torus is considered. These systems have the form of a skew product between the Bernoulli endomorphism , , defined on the 1-torus and a translation on itself. Symbolic dynamics techniques allow one to single out wide classes of observables which show an exponential decay of correlations. For some observables the rate of correlation decay can be explicitly estimated.

Tranversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem

The restricted three-body problem is considered for values of the Jacobi constant C near the value C 2 associated to the Euler critical point L 2. A Lyapunov family of periodic orbits near L 2, the so-called family ( c), is born for C = C 2 and exists for values of C less than C 2. These periodic orbits are hyperbolic. The corresponding invariant manifolds meet transversally along homoclinic orbits. In this paper the variation of the transversality is analyzed as a function of the Jacobi constant C and of the mass parameter μ. Asymptotical expressions of the invariant manifolds for C ≲ C 2 and μ ≳ 0 are found. Several numerical experiments provide accurate information for the manifolds and a good agreement is found with the asymptotical expressions. Symbolic dynamic techniques are used to show the existence of a large class of motions. In particular the existence of orbits passing in a random way (in a given sense) from the region near one primary to the region near the other is proved.