Complexity growth in integrable and chaotic models

The semiclassical (SC) initial value representation (IVR) for the time evolution operator e−iĤt/ℏ involves a phase space integral over the initial conditions of classical trajectories. It is shown in this paper how an IVR for the two time evolution operators in a typical quantum mechanical time correlation function, CAB(t)≡tr[ÂeiĤt/ℏB̂e−iĤt/ℏ], can be combined into one such phase space integral; i.e., time evolution from 0 to t and from t to 0 is combined into one overall SC-IVR propagation. This not only reduces the dimensionality of the phase space average, but the forward–backward (FB) nature of the net trajectory has a partial self-cancellation that reduces the oscillatory behavior of the integrand. Several applications of this FB-IVR to reactive flux correlation functions are presented to illustrate its possibilities.

Back to table of contents Previous article Next article LETTERFull AccessGlobal Versus Local Perspectives on SchizophreniaSareh ZendehrouhFatemeh BakouieShahriar Gharibzadeh,Sareh ZendehrouhSearch for more papers by this authorFatemeh BakouieSearch for more papers by this authorShahriar GharibzadehSearch for more papers by this author,Published Online:1 Apr 2009AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InEmail To the Editor: Recent neurobiological studies indicate that schizophrenia may be a neurodevelopmental and progressive disorder with multiple biochemical abnormalities involving dopamine, serotonin, glutamate, and gamma-aminobutyric acid secretion. In postmortem tissue studies, structural abnormalities and alterations in synaptic connectivity have been observed in the intracortical circuitry of the prefrontal dorsal cortex of schizophrenia patients. These morphological changes could be the sequelae of earlier environmental insults and genetic processes. There are probably multiple susceptibility genes, each of small effect, which act in conjunction with environmental factors as obstetric abnormalities, intrauterine infection, and abnormal nutrition. Candidate identified genes could influence neurodevelopment, synaptic plasticity, and neurotransmission. 1 The recent findings are consistent with notions that epigenetic factors play a major role in the disease process and epigenetic factors may continue to influence the expression of the affected genes in adulthood. 2 It is also demonstrated that schizophrenia is a dynamical disease—i.e., that important aspects of schizophrenia can be understood on the basis of concepts of the theory of nonlinear dynamical systems. 3 Chaotic dynamical systems are characterized by a lawful but delicate sensitivity to initial conditions. This leads to the observation that initially similar behaviors evolve into a striking divergence of behavioral patterns over time. Therefore, small differences in the input can result in an entirely different sequence of outputs. 4 We believe that genes act as initial conditions and environmental factors are the control parameters of a chaotic human brain system, which can force the system to special states with particular characteristics (e.g., schizophrenia). Since in chaotic dynamical systems, control parameters can direct the system to special states, environmental factors play a key role in aggravation or decline of schizophrenia symptoms. This complies with some reports that showed high relapse rate of schizophrenia in families that expressed high emotion, 5 which can be considered as a positive control factor. Based on our hypothesis, we propose that schizophrenia could be considered a chaotic model and the global features of the disease could be extracted instead of paying attention to local detailed features. In such a perspective, the manner of managing the disease will be changed. For example, we think that it is not necessary to include the effects of each part of the brain circuits and to correct any change in the amount of neurotransmitters; instead it is better to model the disease state as a chaotic system and recognize the interaction of different environmental factors, such as maternal health, birth complications, emotional states, familial relationships, etc., on it and try to minimize the pathological effects of them. We stress that it is not the compartments but the global interactions of different elements that play major roles in the disease and deserve attention. Modeling the behavior of schizophrenia on the basis of chaos theory to control the disease seems to be a good place to start.Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran

To explore the impact of unknown terms and parameters on chaotic characteristics in chaotic systems, this paper examines the effects of state variables and unknown parameters. The study focuses on different combinations of linear, nonlinear, and constant terms It primarily investigates the role of multi-order state variables and their application to chaotic system models of varying dimensions. Firstly, by simulating a three-dimensional chaotic system, the paper analyzes how different combinations of nonlinear terms and initial conditions affect the system's chaotic behavior. Secondly, it evaluates the chaotic characteristics of a four-dimensional system, combining nonlinear terms with unknown parameters, using tools such as Lyapunov index diagrams, sample entropy, and dynamic trajectory plots. Finally, the paper integrates the constructed chaotic system with chaotic mapping to develop a two-level key chaotic image encryption system, thoroughly assessing its security and resistance to interference.

The Dirac equation is formulated on (1+{ital D})-dimensional discrete Minkowski space-time. We find that the minimum number of components of the fermion field is 2{sup D{minus}1} in the massless case, which is smaller than 2{sup D+1} of the Kogut-Susskind fermion. There exist no extra poles in the fermion propagator. The action is not Hermitian but the quantization can be performed consistently. In the (1+3)-dimensional massless case the equation describes a single species of Dirac particle in the continuous space-time limit. In the (1+1)-dimensional massless case the equation is the same as the automaton equation by {close_quote}t Hooft and describes a chiral fermion. The time evolution operator is unitary and the norm is conserved. For interacting fermions with gauge fields the evolution operator is not unitary. But in the continuum limit the unitarity will be recovered. The consequences of loosening the unitarity condition on the time evolution operator are discussed. {copyright} {ital 1997} {ital The American Physical Society}

This paper explores the stabilization of the three-dimensional Fractional-Order (FO) chaotic Lorenz-84 atmosphere model through control strategies.Employing the Grnwald-Letnikov approximation for fractional integration, the chaotic system is simulated to understand its intricate dynamics.The main focus lies in utilizing state-space feedback control to adapt input signals based on system states, aiming to steer chaotic dynamics towards stability.Additionally, control gains optimization is conducted through Particle Swarm Optimization (PSO) to guarantee robust stabilization of the chaotic atmosphere model.

We study the time evolution of quantum systems with a time-dependent Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators. The invariant Hermitian operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. With the help of the invariant Hermitian operator we obtain not only the exact solutions of the Schr\"odinger equation but also the time-evolution operator. The adiabatic and nonadiabatic Berry phases are calculated with the exact solutions. \textcopyright{} 1996 The American Physical Society.

We explore the notion of the quantum auxiliary linear problem and the associated problem of quantum Backlund transformations (BT). In this context we systematically construct the analogue of the classical formula that provides the whole hierarchy of the time components of Lax pairs at the quantum level for both closed and open integrable lattice models. The generic time evolution operator formula is particularly interesting and novel at the quantum level when dealing with systems with open boundary conditions. In the same frame we show that the reflection K-matrix can also be viewed as a particular type of BT, fixed at the boundaries of the system. The q-oscillator (q-boson) model, a variant of the Ablowitz-Ladik model, is then employed as a paradigm to illustrate the method. Particular emphasis is given to the time part of the quantum BT as possible connections and applications to the problem of quantum quenches as well as the time evolution of local quantum impurities are evident. A discussion on the use of Bethe states as well as coherent states and the path integral formulation for the study of the time evolution is also presented.

We consider the time evolution in two spatial dimensions of a double vorticity layer consisting of two contiguous, infinite material fluid strips, each with uniform but generally differing vorticity, embedded in an otherwise infinite, irrotational, inviscid incompressible fluid. The potential application is to the wake dynamics formed by two boundary layers separating from a splitter plate. A thin-layer approximation is constructed where each layer thickness, measured normal to the common centre curve, is small in comparison with the local radius of curvature of the centre curve. The three-curve equations of contour dynamics that fully describe the double-layer dynamics are expanded in the small thickness parameter. At leading order, closed nonlinear initial-value evolution equations are obtained that describe the motion of the centre curve together with the time and spatial variation of each layer thickness. In the special case where the layer vorticities are equal, these equations reduce to the single-layer equation of Moore (Stud. Appl. Math., vol. 58, 1978, pp. 119–140). Analysis of the linear stability of the first-order equations to small-amplitude perturbations shows Kelvin–Helmholtz instability when the far-field fluid velocities on either side of the double layer are unequal. Equal velocities define a circulation-free double vorticity layer, for which solution of the initial-value problem using the Laplace transform reveals a double pole in transform space leading to linear algebraic growth in general, but there is a class of interesting initial conditions with no linear growth. This is shown to agree with the long-wavelength limit of the full linearized, three-curve stability equations.

In this study, a chaos-theoretic method is proposed to model the case of ferroresonance that can occur under nominal conditions in power systems, and the factors that determine the types of ferroresonance to occur are examined. In the ferroresonance chaotic system modeled in Matlab environment, the length of the transmission line and the breaker capacities in the circuit are fixed and its relationship with the transformer efficiency is investigated. In the proposed chaotic modeling, considering the situations that may occur in practical applications, the ferroresonance situations that occur when the single-phase remains open in the three-phase system are examined. In the study, ferroresonance, which occurs when one phase is open in a three-phase system, is analyzed by considering the situations that may happen during practical implementations. The similarity between the mathematical expressions obtained from the systems that create ferroresonance and Duffing oscillator is evaluated. In the chaotic system, fundamental ferroresonance, subharmonic ferroresonance, and chaotic ferroresonance situations are created depending on the transformer loss. Additionally, ferroresonance that occurs when the chaotic system is of fractional-order is analyzed, and it is observed that results of ferroresonance with different fractional-order values are not different. The results show that transformer loss is a significant element to determine the type of ferroresonance in power transformers. Also, when the chaotic system is operated in the fractional-order setting, the ferroresonance cases that occur are re-examined, and it is observed that the system can exit from the chaotic situation and prevent the formation of ferroresonance when fractional-order control is applied. According to the results, the fractional-order method can be used to control ferroresonance.

Time evolution of large classical systems.- Ergodic properties of infinite systems.- Time evolution and ergodic properties of harmonic systems.- The laser: A reversible quantum dynamical system with irreversible classical macroscopic motion.- What does it mean for a mechanical system to be isomorphic to the Bernoulli flow?.- The Geodesic flow on surfaces of negative curvature.- Lectures on the billiard.- Spectral invariants and smooth ergodic theory.- Nonlinear wave equations.- Integrable systems of nonlinear evolution equations.- Discrete and periodic illustrations of some aspects of the inverse method.- Finitely many mass points on the line under the influence of an exponential potential -- an integrable system.- On traveling wave solutions of nonlinear diffusion equations.- The existence of heteroclinic orbits, and applications.- Hadamard's generalization of hyperbolicity, with applications to the hopf bifurcation problem.- Hyperbolic sets and shift automorhpisms.- Triple collision in Newtonian gravitational systems.- Solutions of the collinear four body problem which become unbounded in finite time.- On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus.

Chaotic transverse-field Ising model with measurements exhibits interesting purification dynamics. Ensemble of non-unitary dynamics of a chaotic many-body system with measurements exhibits a purification phase transition. We numerically find that the law of the increase dynamics of the purity changes by projective measurements in the model. In order to study this behavior in detail, we construct the formalism of the tripartite mutual information (TMI) for non-unitary time evolution operator by using the state-channel map. The numerical result of the saturation value of the TMI indicates the existence of a measurement-induced phase transition. This implies the existence of two distinct phases, mixed phase and purified phase. Furthermore, the real-space spread of the TMI is investigated to explore spatial patterns of information spreading. Even in the purified phase, the spatial pattern of the light cone spread of quantum information is not deformed, but its density of information propagation is reduced on average by the projective measurements. We also find that this spatial pattern of the TMI distinguishes the chaotic and integrable regimes of the system.

This research aims to investigate the influence of model parameters and fractional order on a novel mathematical model with tangent hyperbolic memristor. This investigation conducted by applying Lyapunov exponents and bifurcation analysis. We utilize the Lyapunov exponent theory to understand and characterize these chaotic behaviors under fractional indices. The Lyapunov exponent, bifurcation, and phase diagrams have been depicted to explore the intricate dynamics of the chaotic system governed by the chaotic equation. A novel approach termed Atangana-Baleanu-Caputo (ABC) fractional derivative (FD) to generate phase portraits and gain insights into the system’s behavior. The random numbers generated by the chaotic system are employed to distort the image through an amalgamated image encryption (AIE) algorithm. Subsequently, the integrity of the scrambled image has been assessed using various image security evaluation methods to reinforce the notion that combining the chaotic system and image can constitute a valuable encryption key. Finally, the chaotic model circuit realization uses active and passive components, and the outcomes are compared with the numerical simulations.

Pinpoint authentication watermarking based on a chaotic system

A novel image encryption algorithm based on SPWLCM and DNA coding

We introduce and explore an interacting integrable cellular automaton, the Fredkin staircase, that lies outside the existing classification of such automata, and has a structure that seems to lie beyond that of any existing Bethe-solvable model. The Fredkin staircase has two families of ballistically propagating quasiparticles, each with infinitely many species. Despite the presence of ballistic quasiparticles, charge transport is diffusive in the dc limit, albeit with a highly non-Gaussian dynamic structure factor. Remarkably, this model exhibits persistent temporal oscillations of the current, leading to a delta-function singularity (Drude peak) in the ac conductivity at nonzero frequency. We analytically construct an extensive set of operators that anticommute with the time-evolution operator; the existence of these operators both demonstrates the integrability of the model and allows us to lower bound the weight of this finite-frequency singularity.