Cancer, a disease of multicellular organisms, probably developed almost immediately following the transition from unicellular to metazoan life, about one billion years ago. Great efforts have been made to understand the carcinogenesis for many years. In this paper, We tried to explain the cancer based on “chaos”, “adaptation” and “information” with the context of new literature findings.

We work with reference to a well-known semiclassical model, in which quantum degrees of freedom interact with classical ones. We show that, in the classical limit, it is possible to represent classical results (e.g., classical chaos) by means a pure-state density matrix.

Type II diabetes (T2D) is a multifaceted metabolic disease influenced by various factors (genetic, epigenetic, environmental, and other). An alarming surge in T2D prevalence is estimated, reaching 700 million cases by 2045 worldwide. This review explores the complex interactions of multiple risk factors throughout life and offers a new perspective on T2D. Genetic and epigenetic influences are described in detail. Prenatal factors, such as exposure to undernutrition and maternal overfeeding, induce lasting epigenetic changes, affecting T2D susceptibility. Furthermore, birthweight is a crude proxy for perinatal programming, with low and high birthweights associated with increased T2D risk. Often overlooked, the early postnatal period also significantly contributes to perinatal programming. Neonatal over-nutrition and feeding choices exert lasting effects on DNA methylation and metabolic outcomes later in life. Stress during this period further exacerbates T2D susceptibility. In continuation, childhood behaviors which are influenced by prior periods contribute substantially to T2D risk. Adverse childhood experiences amplify the risk of many diseases, including T2D. Sleep hygiene throughout life plays a role in metabolic health, influencing insulin resistance and appetite regulation. In adulthood, obesity remains a prominent predictor of T2D, with chronic stress contributing to weight gain and unhealthy lifestyle. This review proposes a novel perspective, considering T2D pathophysiology as a nonlinear process governed by numerous factors and their interactions throughout life. The inherent unpredictability suggests a need for personalized interventions tailored to specific risk profiles, emphasizing the importance of understanding gene-environment interactions and embracing a holistic view of T2D pathogenesis.

The paper is dedicated to the numerical and experimental study of nonlinear oscillations exhibited by the Vilnius chaotic generator. The need for practical applicability of the mentioned generator in chaotic communication systems defined the necessity to study the operation of the circuit in a wide parameter range. The bifurcation maps and corresponding one-parameter diagrams, obtained and analysed utilising the Method of Complete Bifurcation Groups, reveal the complex nonlinear dynamics and regions of robust chaotic oscillations. Laboratory experiments allow the verification of the numerical results and identification of the most practically relevant phenomena.

The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender-Brody-Müller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a unique Hermitian Hamiltonian that maps to the zeros of the analytic continuation of the Riemann zeta function, and provide an analytical expression for the eigenvalues of the results. The holomorphicity of the resulting eigenvalues is examined. Moreover, a second quantization of the resulting Schrödinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the eigenvalues it is shown that the real part of every nontrivial zero of the Riemann zeta function converges at σ = 1/2.

This paper suggests a novel one-dimensional (1D) map to address the limitations of traditional chaotic 1D maps. The main challenge with traditional chaotic 1D maps is the limited control parameter range and the potential risk of collapsing as a result under the finite precision implementation. To overcome these limitations, the new 1D map hybridises the traditional logistic map with tent map, and a linear tent-like function. This hybridization results in a wider range of control parameters to produce chaotic behavior. The dynamic behavior of the new 1D map has been analyzed using well known numerical methods, including the bifurcation diagram and Lyapunov exponent. Both tests have shown their complex and diverse behavior. In addition, a novel image encryption scheme has been devised using the new function as its pseudo-random-number generator. The proposed encryption algorithm has been tested and found to be robust and secure, passing all statistical tests applied to the encrypted images. The results of this study demonstrate the effectiveness of the new 1D map for use in secure image cryptography applications, providing a more robust and secure alternative to traditional chaotic 1D maps. The proposed algorithm has demonstrated high performance in NPCR and UACI tests. It also has shown good results in the MSE and PSNR tests.

In this paper we have investigated the driven quintic Duffing equa- tion. Using some analysis technique, we are able to predict the number of limit cycles around the equilibrium and to develop a theoretical ap- proach to chaos transition in damped driven systems.

In the previous work in Ref.[1], we presented a new interaction scheme that describes the interaction of thermodynamic systems. Using this interaction scheme, we showed that the interactions of dark matter and dark energy are chaotic. We also predicted that the results could be generalized to all particle and thermodynamic systems. In addition, we gave physical definitions of the concepts of chaos and self-organization. More importantly, we proposed a new law of physics based on interacting systems. In this study, we will consider two thermodynamic systems interacting under different conditions to prove the universality of the physical law we propose. We will show that these systems also have chaotic interaction dynamics.

The following is the Collatz Conjecture: Suppose we begin with a positive number, multiply it by 3 and add 1 if it is odd, and divide it by 2 if it is even. Then continue doing this as long as you can. Will it matter where you start, whether you end up at the number 1? The Collatz conjecture has been studied for around 85 years. We transform the Collatz function from decimal to binary, then use the binary string's character to prove the Collatz conjecture. In addition, we use mathematics to give another interpretation to chaos, which is the ultimately periodic positive integer sequence.

What is a complex system? The definition of a complex system remains somewhat ambiguous. A complex system can be defined as a system consisting of many interacting parts exhibiting emergent behaviors. The emerging field of complexity science entails a change in the language of scientific research and thinking. As such, the general properties, tools and definitions pertaining to complex systems must be made accessible to multi-disciplinary systems scientists and thinkers. With this intention, this literary survey presents the development and glossary of essential concepts steering complex systems.

The perspective suggests an unorthodox interpretation of cancer dynamics in terms of strange attractors and fluid turbulence. The relevance of strange attractors in tumor growth-invasion dynamics and chemical pattern formation are reviewed. Chemical turbulence, fractals and chaos are discussed as the principal characteristics of tumor ecosystems and their complex cybernetics. To conclude, cancers are presented as complex systems exhibiting chaotic gene expression dynamics.

One of the oldest problems in physics is that of calculating the motion of N particles under a specified mutual force: the N-body problem. Much is known about this problem if the specified force is non-relativistic gravity, and considerable progress has been made by considering the problem in one spatial dimension. Here I review what is known about the relativistic gravitational N-body problem. Reduction to one spatial dimension has the feature that gravitational radiation is absent, thereby allowing for a clear comparison between the physics of one-dimensional relativistic and non-relativistic self-gravitating systems. After describing how to obtain a relativistic theory of gravity coupled to N point particles, I discuss in turn the 2-body, 3-body, 4-body, and N-body problems. Quite general exact solutions can be obtained for the 2-body problem, unlike the situation in general relativity in 3 spatial dimensions for which only highly specified solutions exist. The 3-body problem exhibits mild forms of chaos, and provides one of the first theoretical settings in which relativistic chaos can be studied. For N4 other interesting features emerge. Relativistic self-gravitating systems have a number of interesting problems awaiting further investigation, providing us with a new frontier for exploring relativistic many-body systems.

The paper focuses on the different dynamical systems for the prey-predator model for different cases which includes greater depth of the food chain and varying number of preys and predators. We simulate the different postulated models and provide a comparative study for the same.

We humans are natural-born engineers. As such, we model after machines not only isolated, naturally occurring systems, but also the basic laws of physics, sharing with machines a local-evolution-of-state `grammar'. However, previous work by the author casts doubt upon this mechanistic paradigm, suggesting that it is to blame for the stubbornness of many open problems in physics. Simple experiments are therefore proposed to identify `non-machines'. In one experiment, `non mechanistic correlations' in the spirit of Bell are sought in a pair of separated but previously coupled (macroscopic) chaotic systems. In another it is tested whether chaotic systems could fuzzily `remember their future' in the sense that a future binary perturbation applied to them could be inferred from their present behavior with probability>1/2. Chaotic systems are chosen as candidates because the long-time application of their (deterministic) mechanistic description is conceptually groundless: All scales are significant and mutually coupled in that regime, down to scales governed by QM, yet the quantum-classical transition remains inexplicable---if only due to the measurement problem. Similarly for non-deterministic chaotic systems, modeled as such purely for defying a short-time deterministic mechanistic description. Should a non mechanistic signature be found in such simple systems, the implications for science as a whole and life sciences in particular cannot be overstated.

Being able to forecast events is of great importance in many fields, from brain behavior to earthquakes or stock markets. Because each dynamical system has intrinsic features, different statistical tools have to be used for each system. Here we study the time series of the output intensity of a fiber laser with an ordinal patterns analysis, and we look for temporal correlations in order to statistically forecast the most intense events. We set two thresholds, a low one and a high one, to distinguish between low intensity versus high intensity events. We find that when the time series is performing events below the low threshold it shows some preferred temporal patterns before performing events above a high threshold.

We discuss the role of the opposing principles of order and disorder in physical and biological systems in determining stability, growth and evolution and bring forth the potential role of a cosmic ordering agency. We analyze its role in decreasing entropy by coarse-graining and hence in determining the initial low entropy state of the big bang universe. Since all physical and biological systems have either cycles of order and disorder alternating, or may have chaotic evolution with non-linear laws, the same is expected of the dynamics of the whole universe as well. The entropy of the initial state of the universe could be low because of the reduction of degrees of freedom (DoF) as one moves from physical encoding to neural encoding and then on to psychic encoding of information in a nested manner by coarse-graining. It is by such encoding that this cosmic agency enables the universe to pass through the big crunch phase and then rolls it out as the big bang universe from the initial state of low entropy.

In this paper, we consider the synchronization of chaotic oscillators interconnected via a second order dynamic coupling. The design of the proposed coupling is inspired in the ancient synchronization experiment of synchronized pendulum clocks, as described by the Dutch scientist Christiaan Huygens more than three centuries ago. It is demonstrated that the coupled chaotic systems may exhibit not only complete synchronization, but also mixed synchronization—some states synchronize in anti-phase whereas other states synchronize in-phase. In fact, a transition from complete synchronization to mixed synchronization or viceversa can be induced by just changing a single parameter in the coupling. Additionally, the stability of the synchronous solution is investigated by using the Master Stability Function approach and the largest transverse Lyapunov exponent. The Lorenz system is considered as particular application example. Finally, the performance of the proposed synchronization scheme is illustrated with computer simulations and validated by means of experiments using electronic circuits.

Introduction; to the best of the authors knowledge no past research has established how illness narrative master plots are expressed initially, and then if, and how they change longitudinally following musculoskeletal trauma. The aim of the current research was able to consider how specific master plots were expressed, interact and change across time following musculoskeletal trauma. Methods; a narrative analysis was undertaken that included individuals who had experienced a musculoskeletal traumatic injury. Individuals were included if they were an inpatient within 4 weeks of the first interview had mental capacity to participate and were able to communicate in English. Three interviews were undertaken (within 4 weeks of injury, 6 and 12 months). A 5-stage categorical form type narrative analysis was performed. Results; Twelve individuals (49.9±17.5 years; 7 male, 5 female) completed interviews at three time points following the trauma event (<4 weeks months, 6 months and 12 months). Three main narratives that appeared to work together to facilitate a positive accommodation of the trauma event into the individual’s life were presented. These included the resumption narrative, the activity narrative and quest narrative. Finally, less often regressive narratives were identified although these narratives were, at times, actively avoided. Discussion; The current results provide important consideration for how narratives are used within clinical practice, in particular the value of how these three narratives could be access and promoted.

During the progression of cancer cells, the degree of genome instability increases leading to genome chaos in populations of malignant cells. While normally chaos is associated with er-godicity, i.e., the state when the time averages of relevant parameters is equal to their phase space averages, the situation with cancer propagation is more complex. Chromothripsis, a catastrophic massive genomic rearrangement, is observed in many types of cancer leading to increased mu-tation rates. We present an entropic model of genome chaos and ergodicity and experimental evidence that increasing the degree of chaos beyond the non-ergodic threshold may lead to the self-destruction of the tumor cells. We study time and population averages of chromothripsis frequency in cloned rhabdomyosarcoma from rat stem cells. Clones with frequency above 10% result in cell apoptosis possibly due to mutations in the BCL2 gene. Potentially this can be used for suppressing cancer cells by shifting them into a non-ergodic proliferation regime.

Finite-time control theory has been widely used as a mathematical tool to design robust controllers. By manipulating the finite-time convergence proof of this theory, we developed a new control design appropriately tuned for the finite-time control of the chaotic logistics system. In our experimental setup, the logistic equation is programmed into a PIC microcontroller, and a part of the controller was conceived using analog electronics. Because the system to be controlled is in the discrete-time domain, and the finite-time stability proof is stated in the continuous-time representation, our finite-time control approach is a good example for designing control algorithms in both time domain schemes. Hence, our experimental results support our main contribution. Pulse Width Modulation (PWM) is the signal format used to translate digital signals into continuous-time fields.

The two main strategies used in this study are the binary representation and the decomposition of a natural number into many compound functions of odd function and even function. The Collatz conjecture regarding odd even numbers in number theory can be examined and discussed using them in this way: The sequence created by the finite iterations of the Collatz function becomes the ultimately periodic sequence if any natural number is the beginning value, proving the conjecture that has been held for 85 years.

A second order rational difference equation $$x_{n+1}=px_{n}+\frac{q}{x_{n-1}^2}$$ with the parameters $p$ and $q$ which lies in $(0,1)$, is studied. The dynamics of the equilibrium is characterized through the trichotomy of the parameter $p<\frac{1}{2}$, $p=\frac{1}{2}$ and $p>\frac{1}{2}$. It is found that there is no periodic solution of period $2$ and $3$ but there exists periodic solutions with only periodic solution $5$ and $10$ are achieved computationally.

Cancers remain the lead cause of disease-related, pediatric death in North America. The emerging field of complex systems has redefined cancer networks as a computational system with intractable algorithmic complexity. Herein, a tumor and its heterogeneous phenotypes are discussed as dynamical systems having multiple, strange attractors. Machine learning, network science and algorithmic information dynamics are discussed as current tools for cancer network reconstruction. Deep Learning architectures and computational fluid models are proposed for better forecasting gene expression patterns in cancer ecosystems. Cancer cell decision-making is investigated within the framework of complex systems and complexity theory.

Cancers are complex, adaptive ecosystems. They remain the leading cause of disease-related death among children in North America. As we approach computational oncology and Deep Learning Healthcare, our mathematical models of cancer dynamics must be revised. Recent findings support the perspective that cancer-microenvironment interactions consist of turbulent flows. As such, cancer pattern formation, protein-folding and metastatic invasion are discussed herein as processes driven by chemical turbulence within the framework of complex systems theory. Current state-of-the-art quantitative approaches used in reconstructing cancer stem cell networks are reviewed. To conclude, cancer stem cells are presented as strange attractors of the Waddington landscape.

Authentication systems based on biometrics characteristics and data represents one of the most important trend in the evolution of our world. In the near future, biometrics systems will be everywhere in the society, such as government, education, smart cities, banks etc. Due to its uniqueness characteristic, biometrics systems will become also vulnerable, privacy being one of the most important challenge. The classic cryptographic primitives are not sufficient to assure a strong level of secureness for privacy. The following work paper represents an effort to present the main cryptographic techniques and algorithms that can give us the possibility to raise a certain level of secureness for privacy. We will show their own challenges (strengths and weaknesses). We will demonstrate how we can use the most common and well-known techniques and algorithms in order to get a maximum efficiency and a high level in assuring the integrity of the biometrics data.

This article addresses the trajectory tracking between two non-identical systems with chaotic properties. We employ the Rossler chaotic and RCL-shunted Josephson junctions model in similar phase space to study trajectory tracking. In order to achieve the goal tracking, we afford two stages to approximate the target tracking. The first stage utilizes the active control technique to transfer the output signal from the RCLs-J system into the quasi-Rossler system. Then next, the RCLs-J system employs the proposed the iterative learning control scheme and the control signal from the drive system to trace the trajectory of Rossler system. The numerical results demonstrate the proposed method and the tracking system is asymptotically stable.

Recent studies have indicated that terahertz time-domain spectroscopy (THz-TDS) can stably and efficiently acquire output spectra using an affordable and compact multimode laser diode (MMLD) with delayed optical feedback as the light source. This research focused on a numerical analysis of the optimal conditions for employing an MMLD with delayed optical feedback (chaotic oscillating laser diode) in THz-TDS, utilizing multimode rate equations. The findings revealed that the intermittent chaotic output generated by the MMLD, characterized by concurrent picosecond pulse oscillations lasting several tens of picoseconds, proves to be highly effective for THz-TDS. By appropriately setting the amounts of injection current and optical feedback, and the delay time of optical feedback, intermittent chaotic oscillation can be attained within a considerably broad parameter range. Moreover, both the MMLD output spectrum and the THz-TDS output spectrum exhibit a consistently stable shape at the microsecond scale, demonstrating the attractor properties inherent in an MMLD with delayed optical feedback.

The occurrence of chaotic instability of oscillations at the output of a single-stage amplifier on a powerful bipolar microwave transistor when a biharmonic signal is amplified is considered. It has been established that at a high level of the input signal at the output of the amplifier, the signal contains a significant chaotic component. Cases were experimentally studied when the frequency distance between the partial frequency components of a biharmonic signal is close to the gain band and when it is much smaller than this band. It has been established that in the first case, the chaos in oscillations is associated with the emergence of a sequence of trains of a forced oscillatory process with arbitrary initial phases and durations. In the second case, the appearance of a chaotic pedestal is associated with the chaos in the passive underexcited mode of the nonlinear circuit formed by the capacitance of the p-n junction and the input matching circuits of the amplifier stage.

Fractals and multifractals are well-known trademarks of nonlinear dynamics and classical chaos. The goal of this work is to tentatively uncover the unforeseen path from multifractals and selfsimilarity to the framework of effective field theory (EFT). An intriguing finding is that the partition function of multifractal geometry includes a signature analogous to that of gravitational interaction. Our results also suggest that multifractal geometry may offer insights into the non-renormalizable interactions presumed to develop beyond the Standard Model scale.

Three body systems in which one of the bodies is ejected without escaping the binary system have previously been studied in various restricted forms. However, none of these studies dwells on the problem in a general setting. Thus, to study this phenomenon qualitatively, we try to expand this problem's scope to unequal mass systems and generalize them by considering various configurations of fixed initial points with precisely calculated initial velocities, some zero velocity models and some optimized models. We will see the use of terminology similar to the previous studies done in this domain but incorporate different analytical and evaluation methods.

In this paper the dynamics of spatially extended infected phytoplankton with the Holling type II functional response and logistically growing susceptible phytoplankton system is studied. The proposed model is an extension of temporal model available [6], in spatiotemporal domain. The reaction diffusion system exhibits spatiotemporal chaos in phytoplankton dynamics. This is particularly important for the spatially extended systems that are studied in this paper as they display a wide spectrum of ecologically relevant behavior, including chaos. In this system stability of the system is studied with respect to disease contact rate and the growth fraction of infected phytoplankton indirectly rejoin the susceptible phytoplankton population. The results of numerical experiments in one dimension and two dimensions in space as well as time series in temporal models are presented using MATLAB simulation. Moreover, the stability of the corresponding temporal model is studied analytically. Finally, the comparison of the three types of numerical experimentations are discussed in conclusion.

A new approach is presented to defining the amount of information, in which information is understood as the data about a finite set as a whole, whereas the average length of an integrative code of elements serves as a measure of information. We obtain a formula for the syntropy of reflection, that is, the information that two intersecting finite sets reflect about each other. The specificities of reflection of discrete systems through the combination of their parts are considered, and it is shown that the additive syntropy and entropy of reflection are measures of structural order and chaos. Three information laws have been established: the law of conservation of the sum of chaos and order; information law of reflection; the law of conservation and transformation of information. An assessment of the structural organization and the level of development of discrete systems is presented. It is shown that various measures of information are structural characteristics of integrative codes of elements of discrete systems. The conclusion is made that, from the informational-genetic positions, the synergetic theory of information is primary in relation to the Hartley-Shannon information theory. In the appendix we consider the asymmetry of the mutual reflection of finite sets and the proportionality of arbitrary quantities.

Nuclear fusion reactors are composed by several complex components whose behaviour may be not certain a priori. This uncertainty may have a significant importance in the evolution of fault transients in the machine, causing unexpected damages to its components. For this reason a suitable method for the uncertainty propagation during those transient is required. The Monte Carlo method would be the reference option, but it is, in most of the cases, not applicable due to the large amount of required simulations to be repeated. In this context the Polynomial Chaos Expansion has been considered as a valuable alternative. It allows to create a surrogate model of the original one in terms of orthogonal polynomials. Then, the uncertainty quantification is performed repeatedly calling this much simpler and faster model. Using the fast current discharge in the Divertor Tokamak Test Toroidal Field (DTT TF) coils as a reference scenario, this method has been applied: the uncertainty on the parameters of the Fast Discharge Unit (FDU) varistor disks is propagated to the simulated electrical and electro-magnetic relevant effects. Eventually two worst case scenarios have been analyzed from the thermal-hydraulic point of view with the 4C code, simulating a fast current discharge as a consequence of a coil quench. Eventually it has been demonstrated that the uncertainty on the inputs (varistor parameters) strongly propagates obtaining a wide range of possible scenarios in case of accidental transients. This result underlines the necessity to take into account and propagate all possible uncertainties in the design of a fusion reactor, according to the Best Estimate Plus Uncertainty approach.

Many probability-based uncertainty quantification (UQ) schemes require a large amount of sampled data to build credible probability density function (PDF) models for uncertain parameters. Unfortunately, the data collected in compressor blades of aero-engine are mostly limited due to expensive and time-consuming tests. In this paper, we develop a preconditioner-based data-driven polynomial chaos (PDDPC) method that can efficiently deal with uncertainty propagation of limited sampled data. The calculation accuracy of PDDPC method is closely related to the sample size of collected data. Therefore, the influence of sample size on PDDPC method is investigated using a nonlinear test function. Subsequently, we consider the real manufacturing errors of stagger angle for compressor blades. Under three different operating conditions, PDDPC method is applied to investigate the effect of stagger angle error on UQ results of multiple aerodynamic parameters of a two-dimensional compressor blade. The results show that as the sample size of measured data increases, UQ results of aerodynamic performance obtained by PDDPC method gradually converge. There exists a critical sample size that ensures accurate UQ analysis of compressor blades. The probability information contained in the machining error data is analyzed through Kullback-Leibler divergence, and the critical sample size is determined. The research results can make a valuable reference for the fast and cheap UQ analysis of compressor blades in practical engineering.

Chaotic systems are systems whose results are sensitive to the initial conditions. Chaotic systems generally have nonlinearities that can be difficult to model. One example of a chaotic system with nonlinearities is the Lorenz system. The Lorenz system can be used to model weather, wind disturbances, and electronic circuit design, among other applications. In this manuscript, the Lorenz system is modeled and various control methods are applied in an effort to dictate the system’s state and rate trajectories. The combination of the linear feedback and nonlinear feedforward controllers can show over an 80% improvement in state trajectory errors when compared to the baseline run. However, the improved state trajectory performance comes at a cost of as much as 1800% error in the rate trajectories.

In this study we used a non-autonomous Chua’s Circuit, and the fractional Lorenz chaos system together with a detection method from Extension theory to analyze the voltage signals. The measured bearing signals by acceleration sensor were introduced into the master and slave systems through a Chua’s Circuit. In a chaotic system minor differences can cause significant changes that generate dynamic errors, and extension matter-element models can be used to judge the bearing conditions. Extension theory can be used to establish classical and sectional domains using the dynamic errors of the fault conditions. The results obtained were compared with those from Discrete Fourier Transform analysis, Wavelet analysis and an integer order chaos system. The diagnostic ratio showed the fractional order master and slave chaos system calculations. The results show that the method presented in this paper is very suitable for monitoring the operational state of ball bearing system to be superior to the other methods. The diagnosis ratio was better and there were other significant advantages such as low cost and few.

This paper first discusses a fractional-order Liu system of order as low as 2.7 and shows its chaotic characteristics by carrying out numerical simulations such as Lyapunov exponents, bifurcation diagrams and phase portraits. Then, by using the topological horseshoe theory and computer-assisted proof, the existence of chaos in the system is verified theoretically. Finally, the fractional-order system is implemented on a Field Programmable Gate Array (FPGA) and the results obtained show that the fractional-order Liu system is indeed chaotic.

This research not only dedicated a less restrictive method of iteration-varying function for a learning control law to design a controller but also synchronize two nonlinear systems with free time-delay. In addition, the mathematical theory of system synchronization has proved rigorously and the theory verified through an example to demonstrate the behavior of each parameter in the theory. The design of a controller using the iterative learning control law is significant for robotic tracking. The controller in this research generates a feed-forward control input using the error dynamics among the drive-response systems. The error dynamics satisfies the Lyapunov function and the combination of output errors, which respectively represented relative estimated differences of the drive-response systems. The iterative learning control rule serves the function of a filter adding previous control error after the end of each iteration. The numerical example of a synchronous system is given a Lorenz system for driving and another with the iterative learning control law for response under different initial condition. The results verify and demonstrate the proposed mathematical theory. The simulation exhibits consistency in the behavior of each parameter to match mathematical theory.

As paradigm of complex behavior, Self-organized Criticality (SOC) reflects the ability of nonequilibrium dynamical systems to self-adjust into metastable states that are scale independent. The goal of this report is to tentatively show that the hierarchy of Standard Model masses and mixing angles follows from the universal scaling attributes of SOC.

We construct a microscopic model to study discrete randomness in bistable systems coupled to an environment comprising many degrees of freedom. A quartic double well is bilinearly coupled to a finite number N of harmonic oscillators. Solving the time-reversal invariant Hamiltonian equations of motion numerically, we show that for N=1, the system exhibits a transition with increasing coupling strength from integrable to chaotic motion, following the KAM scenario. Raising N to values of the order of 10 and higher, the dynamics crosses over to a quasi-relaxation, approaching either one of the stable equilibria at the two minima of the potential. We corroborate the irreversibility of this relaxation on other characteristic timescales of the system by recording the time dependences of autocorrelation, partial entropy, and the frequency of jumps between the wells as functions of N and other parameters. Preparing the central system in the unstable equilibrium at the top of the barrier and the bath in a random initial state drawn from a Gaussian distribution, symmetric under spatial reflection, we demonstrate that the decision whether to relax into the left or the right well is determined reproducibly by residual asymmetries in the initial positions and momenta of the bath oscillators. This result reconciles the randomness and spontaneous symmetry breaking of the asymptotic state with the conservation of entropy under canonical transformations and the manifest symmetry of potential and initial condition of the bistable system.

Quantum chaos is presented as a paradigm of information processing by dynamical systems at the bottom of the range of phase-space scales. Starting with a brief review of classical chaos as entropy flow from micro- to macro-scales, I argue that quantum chaos came as an indispensable rectification, removing inconsistencies related to entropy in classical chaos: Bottom-up information currents require an inexhaustible entropy production and a diverging information density in phase space, reminiscent of Gibbs' paradox in Statistical Mechanics. It is shown how a mere discretization of the state space of classical models already entails phenomena similar to hallmarks of quantum chaos, and how the unitary time evolution in a closed system directly implies the “quantum death” of classical chaos. As complementary evidence, I discuss quantum chaos under continuous measurement. Here, the two-way exchange of information with a macroscopic apparatus opens an inexhaustible source of entropy and lifts the limitations implied by unitary quantum dynamics in closed systems. The infiltration of fresh entropy restores permanent chaotic dynamics in observed quantum systems. Could other instances of stochasticity in quantum mechanics be interpreted in a similar guise? Where observed quantum systems generate randomness, that is, produce entropy without discernible source, could it have infiltrated from the macroscopic meter? This speculation is worked out for the case of spin measurement.

A probabilistic description is essential for understanding the dynamics of stochastic systems far from equilibrium. To compare different Probability Density Functions (PDFs), it is extremely useful to quantify the difference among different PDFs by assigning an appropriate metric to probability such that the distance increases with the difference between the two PDFs. This metric structure then provides a key link between stochastic processes and geometry. For a non-equilibrium process, we define an infinitesimal distance at any time by comparing two PDFs at times infinitesimally apart and sum these distances in time. The total distance along the trajectory of the system quantifies the total number of different states that the system undergoes in time and is called the information length. By using this concept, we investigate classical and quantum systems and demonstrate the utility of the information length as a unique Lagrangian diagnostic to quantify the information change as a system continuously evolves in time and to map out attractor structure. We further elucidate quantum effects (uncertainty relation) and the dual role of the width of PDF in quantum systems.

Our earlier work has tentatively shown that the hierarchy of fermion masses and mixing angles follows from the universal behavior of nonlinear dynamics. In this brief sequel we survey a similar scenario in which the Cabibbo angle arises from the nonlinear dynamics of charged-current interactions.

Industrial dynamical systems often exhibit multi-scale response due to material heterogeneities, operation conditions and complex environmental loadings. In such problems, it is the case that the smallest length-scale of the systems dynamics controls the numerical resolution required to effectively resolve the embedded physics. In practice however, high numerical resolutions is only required in a confined region of the system where fast dynamics or localized material variability are exhibited, whereas a coarser discretization can be sufficient in the rest majority of the system. To this end, a unified computational scheme with uniform spatio-temporal resolutions for uncertainty quantification can be very computationally demanding. Partitioning the complex dynamical system into smaller easier-to-solve problems based of the localized dynamics and material variability can reduce the overall computational cost. However, identifying the region of interest for high-resolution and intensive uncertainty quantification can be a problem dependent. The region of interest can be specified based on the localization features of the solution, user interest, and correlation length of the random material properties. For problems where a region of interest is not evident, Bayesian inference can provide a feasible solution. In this work, we employ a Bayesian framework to update our prior knowledge on the localized region of interest using measurements and system response. To address the computational cost of the Bayesian inference, we construct a Gaussian process surrogate for the forward model. Once, the localized region of interest is identified, we use polynomial chaos expansion to propagate the localization uncertainty. We demonstrate our framework through numerical experiments on a three-dimensional elastodynamic problem

Atherosclerosis (ATH) and Coronary Artery Disease (CAD) are chronic inflammatory diseases with an important genetic background which derive from the cumulative effect of multiple common risk alleles, most of them located in genomic non-coding regions. These complex diseases behave as non-linear dynamical systems that show a high dependence on their initial conditions, so that long-term predictions of disease progression are unreliable. One likely possibility is that the non-linear nature of ATH could be dependent on non-linear correlations in the structure of the human genome. In this review we show how Chaos theory analysis highlighted genomic regions that shared specific structural constraints that could have a role in ATH progression. These regions were shown to be enriched in repetitive sequences of the Alu family, genomic parasites which colonized the human genome, which show a particular secondary structure and have been involved in the regulation of gene expression. We also review the impact of Alu elements on the mechanisms that regulate gene expression, especially highlighting the molecular mechanisms by which the Alu elements could alter the inflammatory homeostasis. We devise especial attention to their relationship with the lncRNA ANRIL, the strongest risk factor for ATH, their role as miRNA sponges, and their ability to interfere with the regulatory circuitry of the NF-kB response. We aim to characterize ATH as a non-linear dynamic system in which small initial alterations in the expression of a number of repetitive elements are somehow amplified to reach phenotypic significance.

The Standard Model (SM) fails to account for either the triplication of fermion families or chiral symmetry breaking in the electroweak sector. Here we show that both phenomena arise from the approach to chaos of quantum theory near the Fermi scale.

In this article we study the presence of chaos in supersymmetric(SUSY) quantum mechanics. For that purpose we present a form of 4-point out of time order correlator(OTOC) for SUSY quantum mechanical systems using tensor product formalism. We calculate the 4-point OTOC for SUSY 1D harmonic oscillator and find that the OTOC is exactly equal to that of 1D bosonic harmonic oscillator system. In similar manner using the eigenstate representation of supersymmetric systems we calculate the generalized higher order out of time order correlator. The higher order OTOC is a more sensitive measure of chaos than the usual 4-point correlator used in literature. Finally, we calculate the generalized 2N-point OTOC for SUSY 1D harmonic oscillator.

The renewable energy source plays a major role in the grid side power production. The stability analysis is very essential in the renewable energy converters. In this paper the bifurcation is analyzed in ZETA converter and Continuous input and output(CIO) power Buck Boost converter. The ZETA converter gives positive step down and step up output voltage and the CIO power converter gives the negative step up and step down output voltage. These converters are used in the DC micro grid with renewable energy as the source. The current mode control technique is applied to analyze the bifurcation behavior and the reference current is taken as the bifurcation parameter. When the reference current is varied, both the converters loses its stability and it enters into chaotic region through period doubling bifurcation. The simulation results are presented to study the performance behavior of both the converters. The stability region of both the converters are determined by deriving the Monodromy matrix approach.

Chaos theory is widely used in the design of Pseudorandom Number Generators (PRNG), which produce number sequences with statistically uniform distributions and random appearance. However, certain methods for implementing chaotic maps can lead to dynamic degradation of the generated number sequences. To solve such problem, we develop a method for generating pseudo-random number sequences based on multiple one-dimensional chaotic maps. In particular, we introduce a Bernoulli chaotic map that utilizes function transformations and constraints on its control parameter, covering complementary regions of the phase space. This approach enables the generation of chaotic number sequences with a wide coverage of phase space, increasing uncertainty in the number sequence generation process. Moreover, by incorporating a scaling factor and a sinusoidal function, we develop a robust chaotic map, referred to as the Sine-Multiple Modified Bernoulli Chaotic Map (SM-MBCM), which ensures a high level of randomness, validated through statistical mechanics analysis tools. Using the SM-MBCM, we propose a Chaotic PRNG (CPRNG) and evaluate its quality through correlation coefficient analysis, key sensitivity tests, statistical and entropy analysis, key space evaluation, linear complexity analysis, and performance tests. Furthermore, we present an FPGA-based implementation scheme that leverages equivalent MBCM variants to optimize the electronic implementation process. Finally, we compare the proposed system with existing designs in terms of throughput and key space.

Reptilians or reptoids are staple characters of Hollywood cinema, science fiction literature, ufology and conspiracy theories. Despite its popularity, the reptoid has been dismissed by the scientific community as a figment of the imagination; born out of a speculation, paranoia and wanton anthropocentricism. In this paper, however, the Life Sequence Conjecture is advanced as a supportive framework for the notion of reptilian extraterrestrials. This paper is not an argument in support of the existence of extraterrestrials. Rather, it proposes that if extraterrestrials do exist, then a reptilian morphology is not untenable.

The out-of-time-ordered correlation (OTOC) function is an important new probe in quantum field theory which is treated as a significant measure of random quantum correlations. In this paper, with the slogan "Cosmology meets Condensed Matter Physics" we demonstrate a formalism using which for the first time we compute the Cosmological OTOC during the stochastic particle production during inflation and reheating following canonical quantization technique. In this computation, two dynamical time scales are involved, out of them at one time scale the cosmological perturbation variable and for the other the canonically conjugate momentum is defined, which is the strict requirement to define time scale separated quantum operators for OTOC and perfectly consistent with the general definition of OTOC. Most importantly, using the present formalism not only one can study the quantum correlation during stochastic inflation and reheating, but also study quantum correlation for any random events in Cosmology. Next, using the late time exponential decay of cosmological OTOC with respect to the dynamical time scale of our universe which is associated with the canonically conjugate momentum operator in this formalism we study the phenomena of quantum chaos by computing the expression for {\it Lyapunov spectrum}. Further, using the well known Maldacena Shenker Stanford (MSS) bound, on Lyapunov exponent, λ≤2π/β, we propose a lower bound on the equilibrium temperature, T=1/β, at the very late time scale of the universe. On the other hand, with respect to the other time scale with which the perturbation variable is associated, we find decreasing but not exponentially decaying behaviour, which quantifies the random correlation at out-of-equilibrium. Finally, we have studied the classical limit of the OTOC to check the consistency with the large time limiting behaviour.

In the context of the COVID-19 epidemic, and on the basis of the Theory of Dynamical Systems, we propose a simple model for the expansion of contagious diseases, with a particular focus on viral respiratory tracts. The infection develops through contacts between contagious and exposed people, with a rate proportional to contact duration and turnover, inversely proportional to the efficiency of protection measures, and balanced by the average immunological response. The obvious initial exponential increase is readily hindered by the size reduction of the exposed population. The system converges towards a stable attractor whose value is expressed in terms of the ratio C/D of contamination vs decay factors. Decreasing this ratio below a critical value leads to a tipping point beyond which the epidemic is over. By contrast, significant values of C/D may bring the system through a bifurcating hierarchy of stable cycles up to a chaotic behaviour.

Anticancer regimens for Hodgkin lymphoma (HL) patients include highly genotoxic drugs that have been very successful in killing tumor cells and providing a 90% disease-free survival at five years. However, these treatments do not have a specific cell target, damaging both cancerous and normal cells. Thus, HL survivors have a high risk of developing new primary cancers, both hematologic and solid tumors, that have been related to treatment. Several studies have shown that after-treatment, HL patients and survivors present persistent chromosomal instability, including non-clonal chromosomal aberrations. The frequency and type of chromosomal abnormalities appear to depend on the type of therapy and the cell type examined. For example, MOPP chemotherapy affects hematopoietic and germ stem cells leading to long-term genotoxic effects and azoospermia, while ABVD chemotherapy affects transiently sperm cells, with most of the patients showing recovery of spermatogenesis. Both regimens have long-term effects in somatic cells, presenting non-clonal chromosomal aberrations and genomic chaos in a fraction of non-cancerous cells. This is a source of karyotypic heterogeneity that could eventually generate a more stable population acquiring clonal chromosomal aberrations and leading towards the development of a new cancer.

In this work, an analytical and numerical analysis of the transition to chaos in five nonlinear systems of ordinary and partial differential equations, which are models of the autocatalytic chemical processes and the numbers of interacting populations, is carried out. It is analytically and numerically shown that in all considered systems of equations, further complication of the dynamics of solutions and the transition to chemical and biological turbulence is carried out in full accordance with the universal Feigenbaum-Sharkovsky-Magnitskii bifurcation theory through subharmonic and homoclinic cascades of bifurcations of stable limit cycles. In this case, irregular (chaotic) attractors in all cases are exclusively singular attractors in the sense of the FShM theory.

In physics we construct idealized mathematical models in order to explain various phenomena which we observe or create in our laboratories. In this article, I recall how sophisticated mathematical models evolved from the concept of a number created thousands years ago and I discuss some challenges and open questions in quantum foundations and in the standard model. We liberated nuclear energy, landed on the Moon and built ‘quantum computers’. Encouraged by these successes many believe, that when we reconcile the general relativity with the quantum theory then we will have the correct theory of everything. Perhaps, we should be much more humble. Our perceptions of reality are biased by our senses and by our brain bending them to meet our priors and expectations. Our abstract mathematical models describe only in an approximate way different layers of physical reality. To describe the motion of a meteorite we can use a concept of a material point but the point-like approximation breaks completely, when the meteorite hits the Earth. Similarly, thermodynamic, chemical, molecular, atomic, nuclear, elementary particle layers of physical reality are described using specific abstract mathematical models and approximations. In my opinion, the theory of everything does not exist.

This work is a top-level summary of several contributions published in the last three decades. It makes the case that complex dynamics of nonlinear systems lies at the heart of foundational physics.

Spatial-structured longitudinal light beams are optical fields sculpted in three-dimensional (3D) space by diffractive optical elements. These beams have been recently suggested for use in im-proving several imaging capabilities, such as 3D imaging, enhancing image resolution, engineering the depth of field, and sectioning 3D scenes. All these imaging tasks are performed using coded aperture correlation holography systems. Each system designed for a specific application is characterized by a point spread function of a different spatial-structured longitudinal light beam. This article reviews the topic of applying certain structured light beams for optical imaging.

Swift and diligent resilience response is mandatory in sustainable geo-distributed ecosystems. The real-time geo-spatial resilience requires agility in millions of parallel and distributed data processing tasks on data acquired from regional condition monitoring(RCM) systems. These tasks include expiditous resolution of complex sustainability conflict sets, promptly anomalies characterization in chaos sets, and resilience response uniqueness. This work is an archetype of a paragon geo-resilience support system(GRSS) for regional sustainability using a novel melioration in DADO production machine. The proposed expert system capitalized the synergic strengths of RETE, TREAT, LEAPS and GATOR networks was designed and implemented as a synthetic DADO machine(SDM). The generic architecture of DADO machine was improved in this work by enrichment in rule set conditions and solution set for regional scale resilience rule set conditions. The condition left-hand side(LHS) X equal to the solution set 2X for right-hand side (RHS) was the goal achieved by working memory(WM) optimization and conflict resolution strategy(CRS) in alpha and beta networks rules. A round-trip time of 80.2 seconds for first event response set using 1492 segment size and sequence number 360,000 with maximum packets at a single geospatial structure was 21 packets/sec was a noticeable landmark in this work. LEAPS and Concurrent-read algorithm for GATOR cluster networks in the proposed synthetic DADO machine architecture was the overall implementation that enabled urban scale resilient system practically possible on physical SHM deployment.

The present paper reviews evidence for ecological evolution of Montiaceae. Montiaceae (Portulacineae) comprise a family of ca. 275 species and ca. 25 subspecific taxa of flowering plants distributed mainly in extreme western America, with additional endemism elsewhere, including other continents and islands. They have diversified repeatedly across steep ecological gradients. Based on narrative analysis, I argue that phylogenetic transitions from annual to perennial life history have been more frequent than suggested by computational phylogenetic reconstructions. I suggest that a reported phylogenetic correlation between the evolution of life history and temperature niche is coincidental and not causal. I demonstrate how statistical phylogenetic comparative analysis (PhCA) missed evidence for marked moisture niche diversification among Montiaceae. I discount PhCA evidence for the relation between Montiaceae genome duplication and ecological diversification. Based on the present analysis of Montiaceae evolution, I criticize the premise of the prevalent statistical approach to PhCA, which tests Darwinian deterministic hypotheses against stochastic evolutionary null models. I discuss theoretical/empirical evidence that evolution is neither stochastic, nor Darwinistically-determined, but idiosyncratic. Idiosyncraticity describes the outcome of a stochastically perturbed nonlinear chaos-like process. The Principle of Evolutionary Idiosyncraticity (PEI) is based on the evolutionary theory of Natural Drift, which maintains that determinism in evolution is a property of the organism and not, as maintained by the theory of Natural Selection, its traits or its milieu. This determinism is characteristic of chaotic functions, which are absolutely determinate, generate self-similarity, but remain absolutely unpredictable. PEI explains precisely observations that evolution proceeds not linearly, but chaotically, producing both quasi-linear fractal-like patterns and non-linear jumps. PEI has ramifications for all areas of macroevolutionary research. In particular, it demonstrates both the fallacy and futility of the statistical PhCA approach that interprets evolutionary causes in terms of evolutionary correlations. However, statistical methods of PhCA can be applied heuristically and fruitfully to reveal idiosyncraticity and discover evolutionary novelty. This, in turn, is demonstrated by the emergence of statistical anomalies in evolutionary analyses of Montiaceae.

Consciousness is a special type of interaction between subjects exchanging by lingua quanta (phonemes). A set of lingua quanta composes a thesaurus placed on the edge of chaos. Its library is a memory, modification of which is due to tuning of memristive neural elements scattered in the brain volume. The memristive neural model takes into consideration two types of neurons - excitatory and inhibitory and the current leakage, all at the body temperature, T = 309 K. At such temperatures only heavy ions like hydrogen ions - protons can pass robustly through the water medium of the brain. The robust ion transport is proton water wires supported by the Grotthuss mechanism. Final aims of the ions are the gap junctions (electric synapses) linking nearest neurons. Following these observations, a model of excitable nervous tissue is constructed. Namely, a difference mapping on the basis of sigmoid curves can reproduce chaotic modes of neural activity proved by positive values of the Lyapunov exponent. The edge of chaos is placed near the bifurcation boundary dividing the chaos and the periodic convulsive activity typical at the epileptic discharges. In this region the self-sustained spiral waves occur. An intermittent activity of the competing excitatory and inhibitory neurons is observed on the edge of chaos. The intermittent electrical activity of neural tissues is demonstrated by records both from different literature issues and records made by the author together with DR. A. Dudkin on slices of CA1 field of the hippocampus.

This paper contains a comparative review of existing Zitterbewegung models. It attempts to synthesize them by offering a toy model containing an extra hypothesis of electromagnetic self-oscillation at the Fermi scale (10−15m). Inspired by Bell violation experiments, Zitterbewegung models and hydrodynamic walker studies, we revisit standing waves in quantum mechanics, echoing de Broglie-Bohm pilot wave theories. Starting with the electromagnetic self-oscillation hypothesis at the Fermi scale we propose an emergent and deductive hierarchical electron structure across multiple scales. The model spans Fermi, Compton (10−13m), Bohr (10−11m) and finally de Broglie (&gt;10−10m). At the Fermi scale, the hypothesis is one of "source-free chiral light", known as the "zilch", emerging as a self-oscillating electromagnetic field at the Schwinger limit. This self-interaction of retarded potentials due to real relativistic delays is a generic self-oscillation of Liénard-Wiechert potentials. This primary Fermi oscillation accounts for charge and mass at the Fermi scale. It generates secondary stable "auto-orbits" dynamics at the Compton scale, manifesting as observable spin and other quantum properties of the usual Dirac zitterbewegung models. These zitter models account for known electron characteristics, including mass, charge, spin and g/2 anomalous factors up to 9 digits precision. We explain the de Broglie wavelength through Doppler-shifted Compton-scale standing waves, in the tradition of de Broglie. The framework extends to electron-positron pair production at the Schwinger limit, described as a result of natural light focusing in non-homogeneous media, utilizing René Thom’s catastrophe theory. This offers a new perspective on baryogenesis and matter emergence from electromagnetic fields. We dub this toy model the "sub-standard" model of the electron.

The Van der Pol oscillator is a chaotic non-linear system. Small perturbations in initial conditions may result in wildly different trajectories. Because of its chaotic nature, controlling, or forcing, the behavior of a Van der Pol oscillator is difficult to achieve through traditional adaptive control methods. Connecting two Van der Pol oscillators together where the output of one oscillator, the driver, drives the behavior of its partner, the responder, is a proven technique for controlling the Van der Pol oscillator. Deterministic AI (DAI) is an adaptive feedback control method that leverages the known physics of the Van der Pol system to learn optimal system parameters for the forcing function. We assessed the performance of DAI employing three different online parameter estimation algorithms. Our evaluation criteria include mean absolute error (MAE) between the target trajectory and the response oscillator trajectory over time. RLS with exponential forgetting (RLS-EF) had the lowest MAE overall, with a 2.46% reduction in error. However, another method was notable. Least Mean Squares with normalized gradient adaptation (LMS-NG) had worse initial error in the first 10% of the simulation, but after that point had consistently better performance. We found that over the last 90% of the simulation, DAI with LMS-NG had a 48.7% reduction in MAE compared to feedforward alone.

This study delves into the intricate interplay between cosmology and gravity in dilaton-based models, enhancing our understanding. Exploring boundary theories coupled to gravity, we derive a four-dimensional effective theory from the cosmological constant. Dilaton gravity is then scrutinized, revealing elegant solutions and insights into brane embedding. The analysis extends to probe scalar fields, employing the WKB approximation in both conformal and non-conformal cases. In summary, this work contributes to a nuanced understanding of the complex relationship between cosmology and gravity in dilaton-based frameworks.