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This research explores the incorporation of variable order (VO) fractional calculus into the hyperchaotic Lorenz system and studies various chaotic features and attractors. Initially, we propose a variable fractional order hyperchaotic Lorenz system and numerically solve it. The solutions are obtained for multiple choices of control parameters, and these results serve as reference solutions for exploring chaos with the artificial intelligence tool radial basis function network (RBFN). We rebuild phase spaces and trajectories of system states to exhibit chaotic behavior at various levels. To further assess the sensitivity of chaotic attractors, Lyapunov exponents are calculated. The efficacy of the designed computational RBFN is validated through the RMSE and extensive error analysis. The proposed research on AI capabilities aims to introduce an innovative methodology for modeling and analyzing hyperchaotic dynamical systems with variable orders. [ABSTRACT FROM AUTHOR]

As a nonlinear dynamic system, the dynamic equation of triboelectric nanogenerator system is not only difficult to accurately establish, but also difficult to solve. The output signals of triboelectric nanogenerator system depends on the initial conditions of the system, which reflects the dynamic behavior of the system. Currently, there is little research on using the output signals to study the dynamic behavior of triboelectric nanogenerator system. Therefore, a dynamic simulation model of a freestanding triboelectric nanogenerator (F-TENG) system was established using COMSOL software. The mathematical model of the F-TENG system was fitted. Using chaos theory and its phase-space reconstruction technology, the phase-space trajectories of the F-TENG system were reconstructed through voltage signals under external factors, and the dynamic behavior of the F-TENG system was discussed and studied. Finally, validation experiments were carried out. The research shows that with the variation of internal factors and operating frequency, the mathematical model of the F-TENG system remains basically unchanged, and the variation law of output characteristics also remains basically unchanged. The F-TENG system is in periodic motion; with the variation of external factors such as resistance and friction layer spacing, the mathematical model of the F-TENG system will evolve in a complex direction, and the output characteristics will gradually distort. The F-TENG system will evolve from periodic motion to chaotic motion. The research results not only reveal the dynamic behavior and enrich the nonlinear dynamic theory of triboelectric nanogenerator system, but also provide theoretical guidance for increasing the working bandwidth and stability of triboelectric nanogenerator system. [ABSTRACT FROM AUTHOR]

Effectively identifying and characterizing the various dynamics present in complex and chaotic systems is fundamental for chaos control, chaos classification, and behavior-transition forecasting, among others. It is a complicated task that becomes increasingly difficult as systems involve more dimensions and parameters. Here, we extend methods inspired in ordinal patterns to analyze 2D iterative maps to unveil underlying approximate symmetries of their dynamics. We distinguish different families of chaos within the systems, find similarities among chaotic maps, identify approximate temporal and dynamical symmetries, and anticipate sharp transitions in dynamics. We show how this methodology displays the evolution of the spatial correlations in a dynamical system as the control parameter varies. We prove the power of these techniques, which involve simple quantifiers as well as combinations of them, in extracting relevant information from the complex dynamics of 2D systems, where other techniques are less informative or more computationally demanding. [ABSTRACT FROM AUTHOR]

The influence of orientation on the control of chaotic behaviors of the system in a porous medium is studied in this article based on the theory of dynamical systems. A three-dimensional system model is obtained using the spectral method. The results showed that chaos can be suppressed in the case of a low or moderate Prandtl number by choosing appropriate tilt angles. [ABSTRACT FROM AUTHOR]

Complex systems, as interwoven miscellaneous interacting entities that emerge and evolve through self-organization in a myriad of spiraling contexts, exhibit subtleties on global scale besides steering the way to understand complexity which has been under evolutionary processes with unfolding cumulative nature wherein order is viewed as the unifying framework. Indicating the striking feature of non-separability in components, a complex system cannot be understood in terms of the individual isolated constituents' properties per se, it can rather be comprehended as a way to multilevel approach systems behavior with systems whose emergent behavior and pattern transcend the characteristics of ubiquitous units composing the system itself. This observation specifies a change of scientific paradigm, presenting that a reductionist perspective does not by any means imply a constructionist view; and in that vein, complex systems science, associated with multiscale problems, is regarded as ascendancy of emergence over reductionism and level of mechanistic insight evolving into complex system. While evolvability being related to the species and humans owing their existence to their ancestors' capability with regards to adapting, emerging and evolving besides the relation between complexity of models, designs, visualization and optimality, a horizon that can take into account the subtleties making their own means of solutions applicable is to be entailed by complexity. Such views attach their germane importance to the future science of complexity which may probably be best regarded as a minimal history congruent with observable variations, namely the most parallelizable or symmetric process which can turn random inputs into regular outputs. Interestingly enough, chaos and nonlinear systems come into this picture as cousins of complexity which with tons of its components are involved in a hectic interaction with one another in a nonlinear fashion amongst the other related systems and fields. Relation, in mathematics, is a way of connecting two or more things, which is to say numbers, sets or other mathematical objects, and it is a relation that describes the way the things are interrelated to facilitate making sense of complex mathematical systems. Accordingly, mathematical modeling and scientific computing are proven principal tools toward the solution of problems arising in complex systems' exploration with sound, stimulating and innovative aspects attributed to data science as a tailored-made discipline to enable making sense out of voluminous (-big) data. Regarding the computation of the complexity of any mathematical model, conducting the analyses over the run time is related to the sort of data determined and employed along with the methods. This enables the possibility of examining the data applied in the study, which is dependent on the capacity of the computer at work. Besides these, varying capacities of the computers have impact on the results; nevertheless, the application of the method on the code step by step must be taken into consideration. In this sense, the definition of complexity evaluated over different data lends a broader applicability range with more realism and convenience since the process is dependent on concrete mathematical foundations. All of these indicate that the methods need to be investigated based on their mathematical foundation together with the methods. In that way, it can become foreseeable what level of complexity will emerge for any data desired to be employed. With relation to fractals, fractal theory and analysis are geared toward assessing the fractal characteristics of data, several methods being at stake to assign fractal dimensions to the datasets, and within that perspective, fractal analysis provides expansion of knowledge regarding the functions and structures of complex systems while acting as a potential means to evaluate the novel areas of research and to capture the roughness of objects, their nonlinearity, randomness, and so on. The idea of fractional-order integration and differentiation as well as the inverse relationship between them lends fractional calculus applications in various fields spanning across science, medicine and engineering, amongst the others. The approach of fractional calculus, within mathematics-informed frameworks employed to enable reliable comprehension into complex processes which encompass an array of temporal and spatial scales notably provides the novel applicable models through fractional-order calculus to optimization methods. Computational science and modeling, notwithstanding, are oriented toward the simulation and investigation of complex systems through the use of computers by making use of domains ranging from mathematics to physics as well as computer science. A computational model consisting of numerous variables that characterize the system under consideration allows the performing of many simulated experiments via computerized means. Furthermore, Artificial Intelligence (AI) techniques whether combined or not with fractal, fractional analysis as well as mathematical models have enabled various applications including the prediction of mechanisms ranging extensively from living organisms to other interactions across incredible spectra besides providing solutions to real-world complex problems both on local and global scale. While enabling model accuracy maximization, AI can also ensure the minimization of functions such as computational burden. Relatedly, level of complexity, often employed in computer science for decision-making and problem-solving processes, aims to evaluate the difficulty of algorithms, and by so doing, it helps to determine the number of required resources and time for task completion. Computational (-algorithmic) complexity, referring to the measure of the amount of computing resources (memory and storage) which a specific algorithm consumes when it is run, essentially signifies the complexity of an algorithm, yielding an approximate sense of the volume of computing resources and seeking to prove the input data with different values and sizes. Computational complexity, with search algorithms and solution landscapes, eventually points toward reductions vis à vis universality to explore varying degrees of problems with different ranges of predictability. Taken together, this line of sophisticated and computer-assisted proof approach can fulfill the requirements of accuracy, interpretability, predictability and reliance on mathematical sciences with the assistance of AI and machine learning being at the plinth of and at the intersection with different domains among many other related points in line with the concurrent technical analyses, computing processes, computational foundations and mathematical modeling. Consequently, as distinctive from the other ones, our special issue series provides a novel direction for stimulating, refreshing and innovative interdisciplinary, multidisciplinary and transdisciplinary understanding and research in model-based, data-driven modes to be able to obtain feasible accurate solutions, designed simulations, optimization processes, among many more. Hence, we address the theoretical reflections on how all these processes are modeled, merging all together the advanced methods, mathematical analyses, computational technologies, quantum means elaborating and exhibiting the implications of applicable approaches in real-world systems and other related domains. [ABSTRACT FROM AUTHOR]

Pseudo‐random number generators (PRNGs) are the cornerstone of various fields including computer science, cryptography, and scientific simulation. To generate high‐quality and unpredictable pseudo‐random sequences, many works draw attention to the use of chaos theory and dynamical systems. However, most of them focus on continuous or low‐dimensional chaotic systems with strong temporal correlations or insufficient nonlinear dynamics. Thus, in this study, a new PRNG based on a four‐dimensional discrete system is proposed to solve the aforementioned flaws. The PRNG has been implemented on Xilinx Artix‐7 xc7a100tfgg484‐2, which can generate pseudo‐random sequences at a high speed of 12811.53 Mb/s without additional post‐processing. All these sequences have successfully passed the NIST SP800.22 standard test. Both throughput and randomness quality of the proposed PRNG outperform the state‐of‐the‐art. [ABSTRACT FROM AUTHOR]

Further exploration into the influence of a memristor on the behavior of chaotic systems deserves attention. When constructing memristor chaotic systems, it is commonly believed that increasing the number of memristors will lead to better system performance. This paper proposes a class of chaotic maps with different discrete memristors, achieved through internal perturbation based on the Sine map. The I-V curve of the discrete memristor has a symmetrical structure. The dynamic characteristics of the designed system are analyzed using the chaotic attractor phase diagram, Lyapunov exponent (LE) spectrum, and bifurcation diagram. Numerical simulations demonstrate that internal perturbations of discrete memristors enhance the Sine map's chaotic characteristics, expand the chaos range, and improve the ergodicity and LE value. Moreover, the type of discrete memristors has a significant impact on the dynamic characteristics of the system, while the number of discrete memristors has little influence. Therefore, in this paper, a direction for the design of a discrete memristor chaotic system is provided. Finally, a discrete memristor chaotic map with a simple structure and better performance is selected. Based on this, a pseudo-random sequence generator is designed, and the generated sequence passes the National Institute of Standards and Technology (NIST) test. [ABSTRACT FROM AUTHOR]

Theoretically, an eight-term chaos system is presented. The effect of changing the initial conditions values on behavior Chen system was studied. The basic dynamical properties of system are analyzed like time series, attractor, FFT spectrum, and bifurcation. Where the system appears steady state behavior at initial condition xi, yi, zi equal (0, 0, 0) respectively and it convert to quasi-chaotic at xi,yi,zi equal (-0.1, 0.5,-0.6). Finally, the system become hyper chaotic at xi,yi, zi equal(- 0.5, 0.5,-0.6 ) that can used it in many applications like secure communication. [ABSTRACT FROM AUTHOR]

Background: Conservative chaos is one of the most interesting objects in modern nonlinear dynamics and chaos theory. Purpose: In this paper, we are focused on a simple symmetric 2–4–6 polynomial oscillatory potential under external periodic excitation. Methods: The Poincaré sections, bifurcation diagrams, and the Lyapunov characteristic exponents for such a dynamical system are obtained and analyzed together with the fixed points. Results: Our numerical results show that the motion of a particle in this potential has rich nonlinear dynamics. Because of two local instabilities, such a system exhibits conservative chaos and can be considered as a simple and efficient source of the chaotic signal. Conclusion: The obtained results show that this system is not only of fundamental interest for the theory of nonlinear oscillations but also can be considered as a promising source of chaos in the field of secure communications and encryption. [ABSTRACT FROM AUTHOR]

Bifurcation analysis and dynamical system studies are carried out to find the stability regime and chaotic scenario in electron acoustic waves in relativistic degenerate plasma. We have obtained the quantum hydrodynamic model and obtain the Korteweg–de Vries equation describing the nature and characteristics of solitary structures. The amplitude modulated envelop soliton formation due to external perturbations has been studied by analyzing the nonlinear Schrodinger equation. Further to study the stability factors and the parametric range for such stability, the dynamical system is studied and bifurcation analysis has been carried out. The chaotic behavior of the system is studied through largest Lyapunov exponent (LLE). This work will find application in theoretically predicting the stable modes in many solar plasma and stellar plasma applications and in laser plasma in future. [ABSTRACT FROM AUTHOR]

The Lyapunov exponent is the most-well-known measure for quantifying chaos in a dynamical system. However, its computation for any time series without information regarding a dynamical system is challenging because the Jacobian matrix of the map generating the dynamical system is required. The entropic chaos degree measures the chaos of a dynamical system as an information quantity in the framework of Information Dynamics and can be directly computed for any time series even if the dynamical system is unknown. A recent study introduced the extended entropic chaos degree, which attained the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. Moreover, an improved calculation formula for the extended entropic chaos degree was recently proposed to obtain appropriate numerical computation results for multidimensional chaotic maps. This study shows that all Lyapunov exponents of a chaotic map can be estimated to calculate the extended entropic chaos degree and proposes a computational algorithm for the extended entropic chaos degree; furthermore, this computational algorithm was applied to one and two-dimensional chaotic maps. The results indicate that the extended entropic chaos degree may be a viable alternative to the Lyapunov exponent for both one and two-dimensional chaotic dynamics. [ABSTRACT FROM AUTHOR]

In this paper we describe some basic notions of topological dynamical systems for maps of type f: X × X - X named general systems. This is proved that every uniformly expansive general system has the shadowing property and every uniformly contractive general system has the (asymptotic) average shadowing and shadowing properties. In the rest, Devaney chaos for general systems is considered. Also, we show that topological transitivity and density of periodic points of a general systems imply topological ergodicity. We also obtain some results on the topological mixing and sensitivity for general systems. [ABSTRACT FROM AUTHOR]

In the applied research of nonlinear system, the low degree of chaos in the dynamical system leads to the limitation of using the chaos method to solve some practical problems. In this paper, we use the product trigonometric function and ternary polynomial to build a dynamical system, which has strong chaotic characteristics. The dynamical system is constructed by two product trigonometric functions and a ternary linear equation, and its chaotic properties are verified by bifurcation diagrams, Lyapunov exponents, fractal dimensions, etc. The system has many parameters and large parameter intervals and is not prone to cycles. The conditions for the non-divergence of this system are given by mathematical derivation, and it is found that the linear part of the system can be replaced by an arbitrary ternary polynomial system and still not diverge, and the bifurcation diagram is drawn to verify it. Finally, the chaotic sequence is distributed more uniformly in the value domain space by adding the modulo operation. Then, the bit matrix of multiple images is directly permuted by the above system, and the experiment confirms that the histogram, information entropy, and pixel correlation of its encrypted images are satisfactory, as well as a very large key space. [ABSTRACT FROM AUTHOR]

Chua's groundbreaking discovery in 1971 introduced the memristor as the fourth essential electrical component, joining resistance, capacitance, and inductance. Over the past few years, numerous discrete memristor models have been developed following its formal proposal, expanding our understanding of this subject. However, there is limited discussion about multistable discrete memristor models. This research combines a comprehensive methodology for generating memristive maps using the modulo function and an exponential memristance, resulting in the development of a multistable discrete memristor model. Nonlinear techniques, such as bifurcation diagrams, Lyapunov spectrum diagrams, and phase portraits were utilized to analyze the dynamic characteristics of the system. The findings uncovered a range of captivating phenomena associated with chaos theory, including regular orbits, chaotic orbits, the mechanism of period doubling leading to chaos, crisis phenomena, and the presence of multistability. • Diagrams resembling bifurcation plots with two parameters were used to illustrate the dynamic behavior of the system. • Variations in initial conditions brought to light the presence of coexisting attractors. • Modified general discrete memristor model presented multistability under different initial conditions. • The behavior of the memristor as an active element becomes apparent when its internal state assumes smaller values. [ABSTRACT FROM AUTHOR]

Does complexity make multilingualism special? Since there is no unequivocal notion of complexity on which researchers agree, several characteristics that have been considered crucial for complexity are brought to bear on multilingualism. While multilingualism is fairly complex in some senses, for instance, because it requires that many variables be studied, it is less clear whether multilingualism becomes special in this way. The most salient possible way in which multilingualism might be special due to its complexity is that qualitatively new features emerge as we move from mono‐ or bilingualism to multilingualism. While research in mathematics and physics has shown examples in which novel features emerge at a certain level, related results are not easily transferred to multilingualism. This is shown by analyzing the application of dynamic systems theory to multilingualism by Herdina and Jessner. It is ultimately a matter of empirical research whether there are suitable novel features that make multilingualism special. [ABSTRACT FROM AUTHOR]

Copyright of Ingeniería (0121-750X) is the property of Ingenieria and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We investigate the ability of simple diagnostics based on Lagrangian descriptor (LD) computations of initially nearby orbits to detect chaos in conservative dynamical systems with phase space dimensionality higher than two. In particular, we consider the recently introduced methods of the difference and the ratio of the LDs of neighboring orbits, as well as a quantity related to the finite-difference second spatial derivative of the LDs, and use them to determine the chaotic or regular nature of ensembles of orbits of a prototypical area-preserving map model, the 4-dimensional symplectic standard map. We compare the characterization obtained by these LDs-based diagnostics against that achieved by the Smaller Alignment Index method of chaos detection, by recording the percentage agreement P A between the two classifications. We also study the influence of the final number of orbit iterations T , the order n of the indices, as well as the distance σ of neighboring orbits on the performance of these methods, and find appropriate T , n and σ values which allow the efficient use of the three indices as short time and computationally cheap chaos diagnostics achieving P A ≳ 90 %. Our findings clearly indicate the capability of LDs to efficiently identify chaos in systems whose phase space is difficult to visualize, due to its high dimensionality, without knowing the variational equations (tangent map) of continuous (discrete) time systems needed by traditional chaos indicators. • Presentation of chaos diagnostics based on computations of Lagrangian descriptors. • Identification of chaos in high dimensional symplectic maps. • Comparison of various efficient chaos detection techniques. • Identifying chaos without using variational equations or tangent maps. [ABSTRACT FROM AUTHOR]

A new 4-D dynamical system exhibiting chaos is introduced in this work. The proposed nonlinear plant with chaos has an unstable rest point and a line of rest points. Thus, the new nonlinear plant exhibits hidden attractors. A detailed dynamic analysis of the new nonlinear plant using bifurcation diagrams is described. Synchronization result of the new nonlinear plant with itself is achieved using Integral Sliding Mode Control (ISMC). Finally, a circuit model using MultiSim of the new 4-D nonlinear plant with chaos is carried out for practical use. [ABSTRACT FROM AUTHOR]

Chaos is an ubiquitous and fundamental phenomenon with a wide range of features pointing to a similar phenomenology. Although apparently distinct, it is natural to ask if all these features emerge from a unifying principle. Recently, it was realized that all continuous-time stochastic dynamical systems (DSs) — the most relevant in physics because natural DSs are always subject to noise and time is continuous — possess a topological supersymmetry (TS). It was then suggested that its spontaneous breakdown could be interpreted as the stochastic generalization of deterministic chaos. This conclusion stems from the fact that such phenomenon encompasses features that are traditionally associated with chaotic dynamics such as non-integrability, positive topological entropy, sensitivity to initial conditions and the Poincaré–Bendixson theorem. Here, we strengthen and complete this picture by showing that the remaining hallmarks of deterministic chaos — topological transitivity/mixing and dense periodic orbits — while being consistent with the TS breaking, do not actually admit a stochastic generalization. This is a major limitation, since all physical systems are always noisy to some extent. We, therefore, conclude that spontaneous TS breaking can be considered as the most general definition of continuous-time dynamical chaos. Contrary to the common perception and semantics of the word "chaos," this phenomenon should then be truly interpreted as the low-symmetry, or ordered phase of the DSs that manifest it. We also argue that the concept of chaos in low-dimensional, discrete-time DSs that do not obey the Poincaré–Bendixson theorem, is related to the explicit TS breaking. [ABSTRACT FROM AUTHOR]

Abstract We consider a two-dimensional dynamical system modeling the immune response to a target population. We perform a detailed linear stability analysis to investigate possible stability switches induced by the existence of characteristic delay times of the dynamical processes, mainly a delay in the replication of the target population and delays in the stimulated and cooperative immune responses. We show that stability switches of stable periodic cycle solutions can be induced by enforcing appropriated time delays in the target reproduction as well as in the cooperative immune response. The analytical results are supported by the numerical solution of the corresponding set of delayed differential equations, showing bifurcations and quasi-chaotic behavior. We discuss the possible relevance of the present results in tailoring therapeutic strategies of viral infections exhibiting periodic viral load oscillations. Highlights • Time delays in the immune response can lead to stability switches of the stationary state. • Delays in the virus replication and cooperative immune response are shown to induce bifurcations and chaos. • A chronic oscillatory state can be switched to a chronic stable state by properly tuning the response times. [ABSTRACT FROM AUTHOR]

Highlights • Advance security scheme is proposed via dynamical system (chaotic map). • The penetrating behavior of chaotic map yields significant diffusion. • The proposed.security scheme is secure against standard cryptanalysis. • Both diffusion and confusion is attained by means of three stages of the scheme. Abstract Chaotic maps contribute a vital part in modern cryptography. The algorithms constructed using chaotic maps are considered more effectual and secure. In this work, a novel technique for image encryption utilizing 3D mixed chaotic map is presented. In the proposed scheme, we used 3D mixed chaotic map for the transformation of pixels position by permuting then in row, column-wise and mixing, aim to fulfill the requirements of secure image transfer. We analyzed the proposed scheme with existing cryptanalysis and compare their results with the results of some other algorithms from literature. The proposed scheme is more efficacious than other algorithms and the outcomes of security analysis, statistical analysis and differential analysis validate that the anticipated scheme is exceptional for secure image transmission. [ABSTRACT FROM AUTHOR]

Modelling and control applications of dynamical systems in chaos theory arising in several areas are investigated and new control techniques are designed in the chaos literature. We propose a new complex finance chaotic model with states as the interest rate, investment demand, and price index. In this work, after studying the dynamical properties of the new finance model, an electronic chaotic circuit of the model is realized in Multisim. Based on passive control theory, we derive a new controller for globally synchronizing state trajectories of the new finance models. Using the new finance chaotic model developed in this work, a new voice encryption algorithm design is presented. With the proposed algorithm, voice encryption application is performed, and results are described. [ABSTRACT FROM AUTHOR]

Many definitions of chaos have appeared in the last decades and there is a concerned question if they are equivalent in some more specific spaces. Our focus will be on distributional chaos, first defined in 1994 and later subdivided into three major types (and even more subtypes). These versions of chaos are equivalent on a closed interval, but distinct in more complicated spaces. Since dendrites have much in common with the interval, we explore whether or not we can distinguish these kinds of chaos already on dendrites. At the end of the paper we will also briefly look at their correlation with other types of chaos. [ABSTRACT FROM AUTHOR]

In this paper, we present an analytical, numerical, and experimental description of the period addition phenomenon in a dynamical system arising from a boost converter controlled by ZAD strategy. The ideal model is presented, and compared with a model that includes parasitic resistances. We show the presence of chaos in the two systems, which is controlled by TDAS. In particular, numerical simulation shows the chaos control zone is greater in the system with internal resistances. [ABSTRACT FROM AUTHOR]

Extracting governing equations from dynamic data is an essential task in model selection and parameter estimation. The form of the governing equation is rarely known a priori; however, based on the sparsity-of-effect principle one may assume that the number of candidate functions needed to represent the dynamics is very small. In this work, we leverage the sparse structure of the governing equations along with recent results from random sampling theory to develop methods for selecting dynamical systems from undersampled data. In particular, we detail three sampling strategies that lead to the exact recovery of first-order dynamical systems when we are given fewer samples than unknowns. The first method makes no assumptions on the behavior of the data, and requires a certain number of random initial samples. The second method utilizes the structure of the governing equation to limit the number of random initializations needed. The third method leverages chaotic behavior in the data to construct a nearly deterministic sampling strategy. Using results from compressive sensing, we show that the strategies lead to exact recovery, which is stable to the sparse structure of the governing equations and robust to noise in the estimation of the velocity. Computational results validate each of the sampling strategies and highlight potential applications. [ABSTRACT FROM AUTHOR]

We examine the importance of chaotic attractors when modelling non-metastatic tumour growth using a model in which cells come in three types: host, immune, and tumour. The relationships between these cell populations are derived from the law of mass action, assuming that a conjugate is formed in the interaction between immune and tumour cells. A nonlinearity in the production of immune cells, based on previous analyses, is introduced and explained. Using previously chosen model parameters, the maximal Lyapunov exponent is calculated numerically as 0.0218, demonstrating the existence of chaotic behaviour. Under the variation of one particular nonlinear rate constant, four distinct types of attractor are observed. Of biological importance, chaotic behaviour is shown to lead to a significantly higher maximum tumour size when compared to non-chaotic behaviour. Counterintuitively, increasing the parameter associated with the killing of tumour cells by immune cells is demonstrated to increase the maximum tumour size when this parameter is below the threshold at which the equilibrium is zero tumour cells. [ABSTRACT FROM AUTHOR]

We study chaotic properties of uniformly convergent nonautonomous dynamical systems. We show that, contrary to the autonomous systems on the compact interval, positivity of topological sequence entropy and occurrence of Li-Yorke chaos are not equivalent, more precisely, neither of the two possible implications is true. [ABSTRACT FROM AUTHOR]

This paper is devoted to study some chaotic properties of iterated function systems (IFSs). Specially, a new notion named thick chaotic IFSs is introduced. The relationship between thick chaos and another properties of some notions in dynamical systems are studied. [ABSTRACT FROM AUTHOR]

This paper focuses on chaos induced by snap-back repellers in non-autonomous discrete systems. A new concept of snap-back repeller for non-autonomous discrete systems is introduced and several new criteria of chaos induced by snap-back repellers in non-autonomous discrete systems are established. In addition, it is proved that a regular and nondegenerate snap-back repeller in non-autonomous discrete systems implies chaos in the (strong) sense of Li-Yorke. Two illustrative examples are proved. [ABSTRACT FROM AUTHOR]

Non-linearity is essential for occurrence of chaos in dynamical system. The size of phase space and formation of attractors are much dependent on the setting of nonlinear function and parameters. In this paper, a three-variable dynamical system is controlled by different nonlinear function thus a class of chaotic system is presented, the Hamilton function is calculated to find the statistical dynamical property of the improved dynamical systems composed of hidden attractors. The standard dynamical analysis is confirmed in numerical studies, and the dependence of attractors and Hamilton energy on non-linearity selection is discussed. It is found that lower average Hamilton energy can be detected when intensity of nonlinear function is enhanced. It indicates that non-linearity can decrease the energy cost triggering for dynamical behaviors. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear three-degrees-of-freedom dynamical system consisting of a variable-length pendulum mass attached by a massless spring to the forced slider is investigated. Numerical solution is preceded by application of Euler-Lagrange equation. Various techniques like time histories, phase planes, Poincaré maps and resonance plots are used to observe and identify the system responses. The results show that the variable-length spring pendulum suspended from the periodically forced slider can exhibit quasi-periodic, and in a resonance state, even chaotic motions. It was concluded that near the resonance the influence of coupling of bodies on the system dynamics can lead to unpredictable dynamical behavior. [ABSTRACT FROM AUTHOR]

Over the past few decades, interval arithmetic has been attracting widespread interest from the scientific community. With the expansion of computing power, scientific computing is encountering a noteworthy shift from floating-point arithmetic toward increased use of interval arithmetic. Notwithstanding the significant reliability of interval arithmetic, this paper presents a theoretical inconsistency in a simulation of dynamical systems using a well-known implementation of arithmetic interval. We have observed that two natural interval extensions present an empty intersection during a finite time range, which is contrary to the fundamental theorem of interval analysis. We have proposed a procedure to at least partially overcome this problem, based on the union of the two generated pseudo-orbits. This paper also shows a successful case of interval arithmetic application in the reduction of interval width size on the simulation of discrete map. The implications of our findings on the reliability of scientific computing using interval arithmetic have been properly addressed using two numerical examples. [ABSTRACT FROM AUTHOR]

Recently, systems with chaos and the absence of equilibria have received a great deal of attention. In our work, a simple five-term system and its anti-synchronization are presented. It is special that the system has a hyperbolic sine nonlinearity and no equilibrium. Such a system generates chaotic behaviours, which are verified by phase portraits, positive Lyapunov exponent as well as an electronic circuit. Moreover, the system displays multistable characteristic when changing its initial conditions. By constructing an adaptive control, chaos anti-synchronization of the system with no equilibrium is obtained and illustrated via a numerical example. [ABSTRACT FROM AUTHOR]

Large scale networks may exhibit complicated dynamical behaviors, such as instability, bifurcation, and chaos. It is not easy to get which nodes or links need to be controlled for ensuring the desired dynamical behavior. This paper presents a detailed analysis on the dynamics and control of a delayed network by a pinning strategy. The network system is first mapped to a simple system by means of an orthogonal transformation. The control signals are then exerted only on a fraction of nodes in the transformed system. We analyze how to get the desired dynamics of the original controlled system (such as zeros solutions, periodic solutions, quasi-periodic solutions and chaos) via controlling the transformed system. It shows that the solutions of the original system can be obtained by the local dynamical behavior exhibiting in the transformed system. The results show that the proposed pinning control strategy is an effective approach for dynamics control. [ABSTRACT FROM AUTHOR]

This paper develops the Non-Intrusive Least Squares Shadowing (NILSS) method, which computes the sensitivity for long-time averaged objectives in chaotic dynamical systems. In NILSS, we represent a tangent solution by a linear combination of one inhomogeneous tangent solution and several homogeneous tangent solutions. Next, we solve a least squares problem using this representation; thus, the resulting solution can be used for computing sensitivities. NILSS is easy to implement with existing solvers. In addition, for chaotic systems with many degrees of freedom but few unstable modes, NILSS has a low computational cost. NILSS is applied to two chaotic PDE systems: the Lorenz 63 system and a CFD simulation of flow over a backward-facing step. In both cases, the sensitivities computed by NILSS reflect the trends in the long-time averaged objectives of dynamical systems. [ABSTRACT FROM AUTHOR]

Researches in natural neuron dynamics have shown that phase transition and chaos provide optimal behaviour for information processing. In artificial neural models that behavior is expected also to maximize the neuron learnability. By employing an open quantum systems approach, we propose a new quantum information extraction method in the quantum RAM node dynamics where complex values are iterated. We experimentally show bifurcation and chaos emergence by varying their parameters. [ABSTRACT FROM AUTHOR]

Copyright of Ingeniería (0121-750X) is the property of Ingenieria and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We study asymptotic properties of the dynamical system generated by the multidimensional Lasota equation. We give the conditions of its stability and chaos in the sense of Devaney in Orlicz spaces. We apply Matuszewska–Orlicz indices to a description of asymptotic behavior of the considered semigroup. We analyze also the conditions under which asymptotic behavior for more general form of the equation is provided. An example illustrates the criteria when the semigroups generated by the equations have not asymptotic behavior. [ABSTRACT FROM AUTHOR]

The following paper discusses the application of a multigrid-in-time scheme to Least Squares Shadowing (LSS), a novel sensitivity analysis method for chaotic dynamical systems. While traditional sensitivity analysis methods break down for chaotic dynamical systems, LSS is able to compute accurate gradients. Multigrid is used because LSS requires solving a very large Karush-Kuhn-Tucker system constructed from the solution of the dynamical system over the entire time interval of interest. Several different multigrid-in-time schemes are examined, and a number of factors were found to heavily influence the convergence rate of multigrid-in-time for LSS. These include the iterative method used for the smoother, how the coarse grid system is formed and how the least squares objective function at the center of LSS is weighted. [ABSTRACT FROM AUTHOR]

This paper studies chaos in the sense of Devaney for a class of two-dimensional time-varying generalized symbol systems of the form where for , is an integer, and are two well-defined functions. By introducing a more restrictive concept of chaos in the sense of Devaney, some sufficient conditions for this system to be completely chaotic in the sense of Devaney are derived. [ABSTRACT FROM AUTHOR]

Synchronization of chaotic systems which is defined based on the exponential stability for encrypts of signals is presented in this paper. An adaptive control scheme is proposed, and the convergence of the synchronized error is guaranteed. Masking and modulation methods are applied for encryption. To verify the effectiveness of the proposed schemes, numerical simulation was applied on a well-known system without linear term chaotic using MatLab software. The comparison of the proposed chaotic synchronization in the case of the synchronization rate and in the decryption precision of sent signal and image are shown. [ABSTRACT FROM AUTHOR]

We present analytic solutions for a chaotic dynamical system that do not have the regular timing characteristic of recently reported solvable chaotic systems. The dynamical system can be viewed as a first order filter with binary feedback. The feedback state may be switched only at instants defined by an external clock signal. Generalizing from a period one clock, we show analytic solutions for period two and higher period clocks. We show that even when the clock ‘ticks’ randomly the chaotic system has an analytic solution. These solutions can be visualized in a stroboscopic map whose complexity increases with the complexity of the clock. We provide both analytic results as well as experimental data from an electronic circuit implementation of the system. Our findings bridge the gap between the irregular timing of well known chaotic systems such as Lorenz and Rossler and the well regulated oscillations of recently reported solvable chaotic systems. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the geodesic motions of test particles in the intermediate vacuum between a monopolar core and an exterior shell of dipoles, quadrupoles and octopoles. The radii of the innermost stable circular orbits at the equatorial plane depend only on the quadrupoles. A given oblate quadrupolar leads to the existence of two innermost stable circular orbits, and their radii are larger than in the Schwarzschild spacetime. However, a given prolate quadrupolar corresponds to only one innermost stable circular orbit, and its radius is smaller than in the Schwarzschild spacetime. As to the general geodesic orbits, one of the recently developed extended phase space fourth order explicit symplectic-like methods is efficiently applicable to them although the Hamiltonian of the relativistic core-shell system is not separable. With the aid of both this fast integrator without secular growth in the energy errors and gauge invariant chaotic indicators, the effect of these shell multipoles on the geodesic dynamics of order and chaos is estimated numerically. [ABSTRACT FROM AUTHOR]