Attractor In Phase Space Research Articles

Large River Effect or Frozen Kinetics: How Complex Nonlinear Living Systems Solve Optimization Problems

In this paper, we introduce general idea of trajectories attraction in phase space, which is very common phenomenon for the processes in the Nature. We start from a rather general biological example of natural selection, where adaptation to the environmental conditions can be described as attraction of some population distribution in the phenotype space to a center of ecological niche. The niche is mathematically represented as the “survival coefficient” which in turn can be linked to a kind of energy potential. This link between biological and physical approaches may be very useful for solution of a wide range of biological problems. In particular, we discuss an evolution in complex potential with a lot of valleys in a multi-dimensional space accompanied by the so-called large river effect, which corresponds to an extremely slow evolution of some, normally close to final, stages of the adaptation. This effect is related to the practically important states of the “frozen kinetics” which accompanies extremely wide spectrum of phenomena and allows understanding different physical and biological processes.

The Effects of Graded Levels of Calorie Restriction XV: Phase Space Attractors Reveal Distinct Behavioral Phenotypes.

Calorie restriction (CR) has a positive impact on health and life span. Previous work, however, does not reveal the whole underlying mechanism of behavioral phenotypes under CR. We propose a new approach based on phase space reconstruction (PSR) to analyze the behavioral responses of mice to graded CR. This involved reconstructing high-dimensional attractors which topologically represent the intrinsic dynamics of mice based on low-dimensional time series of movement counts observed during the 90-day time course of restriction. PSR together with correlation dimensions (CD), Kolmogorov entropy (KE), and multifractal spectra builds a map from internal attractors to the phenotype of mice and reveals the mice with increasing CR levels undergo significant changes from a normal to a new state. Features of the attractors (CD and KE) were significantly associated with gene expression profiles in the hypothalamus of the same individuals.

Dynamics of Rubber Chaplygin Sphere under Periodic Control

This paper examines the motion of a balanced spherical robot under the action of periodically changing moments of inertia and gyrostatic momentum. The system of equations of motion is constructed using the model of the rolling of a rubber body (without slipping and twisting) and is nonconservative. It is shown that in the absence of gyrostatic momentum the equations of motion admit three invariant submanifolds corresponding to plane-parallel motion of the sphere with rotation about the minor, middle and major axes of inertia. The above-mentioned motions are quasi-periodic, and for the numerical estimate of their stability charts of the largest Lyapunov exponent and charts of stability are plotted versus the frequency and amplitude of the moments of inertia. It is shown that rotations about the minor and major axes of inertia can become unstable at sufficiently small amplitudes of the moments of inertia. In this case, the so-called “Arnol’d tongues” arise in the stability chart. Stabilization of the middle unstable axis of inertia turns out to be possible at sufficiently large amplitudes of the moments of inertia, when the middle axis of inertia becomes the minor axis for a part of a period. It is shown that the nonconservativeness of the system manifests itself in the occurrence of limit cycles, attracting tori and strange attractors in phase space. Numerical calculations show that strange attractors may arise through a cascade of period-doubling bifurcations or after a finite number of torus-doubling bifurcations.

Effect of Inclination and of Darcy Number on Bifurcations and Thermal Transfer in a Square Porous Cavity

In the present work a numerical study of natural convection in air in a square cavity filled with a porous media, was carried out in order to analyze the effects of the inclination angle and the Darcy number on the roads toward the chaos. The horizontal wall of the cavity is adiabatic while the vertical walls, composed of two part of the same size, are maintained at different temperatures. The dimensionless transfer equations, expressed in terms of stream function and vorticity had been solved using the Alternating Direction Implicit Method (ADI) and the GAUSS elimination method. The different bifurcations had been represented by attractor in phase space and by amplitude spectrum. We have noticed that for an inclination angle equal to 30°, the scenario of the roads towards the chaos is in agreement with the one of Feigen Baum. For the inclination angle equal to 60°, the scenario is that of Ruelle and Takens which is by quasi periodicity.

Nonlinear feature extraction and chaos analysis of flow field

A novel signal processing method named adaptive variational mode decomposition with the fractal (AFVMD), which is based on variational mode decomposition and fractal theory, is proposed in this paper for solving a problem that it is easy to misjudge the working conditions of the centrifugal compressor. The measured signal of a compressor is unstable, so a traditional method is used to analyze the nonlinear phenomenon of the stall flutter. Owing to the fact that the robustness of VMD method is strong and its combination with the fractal dimension can accurately describe self-similarity and fractal characteristics of a measured signal, the proposed AFVMD method can not only achieve noise reduction, but also extract nonlinear feature from a complex signal. Taking the dynamic pressure data of the impeller during the instability of a centrifugal compressor as an object to verify the effectiveness and superiority of the proposed AFVMD method, the results are obtained as follows. Firstly, compared with the wavelet noise reduction method, the proposed AFVMD method has both noise reduction and feature extraction functions, and the compressor pressure pulsation spectrum has more significant stall characteristics. Secondly, none of the traditional nonlinear analysis methods can reflect the stall process, so the chaotic phase space attractor is used to visualize the flow field changes. Due to the reasonable choice of the delay time and the embedding dimension, the physical information originally mixed in the signal is separated, so that the attractor phase diagram method has a better process of judging the flow stall than the frequency spectrum method. The results show that the proposed AFVMD method can judge the compressor about to enter into the deep surge earlier. Thirdly, In order to quantify the superiority of the proposed method, if the process of surging and the occurrence of deep wheezing can be predicted in advance, the largest Lyapunov exponent is used as an evaluation index. The above results show that the largest Lyapunov exponent of the proposed AFVMD is smallest for illustrating that the signal has more accurate flow field nonlinear information, which improves the predictability of the signal.

Layered chaos in mean-field and quantum many-body dynamics

We investigate the dimension of the phase-space attractor of a quantum chaotic many-body ratchet in the mean-field limit. Specifically, we explore a driven Bose-Einstein condensate in three distinct dynamical regimes---Rabi oscillations, chaos, and self-trapping regimes---and for each of them we calculate the correlation dimension. For the ground state of the ratchet formed by a system of field-free noninteracting particles, we find four distinct pockets of chaotic dynamics throughout these regimes. We show that a measurement of local density in each of the dynamical regimes has an attractor characterized by a higher fractal dimension, ${D}_{R}=2.59\ifmmode\pm\else\textpm\fi{}0.01$, ${D}_{C}=3.93\ifmmode\pm\else\textpm\fi{}0.04$, and ${D}_{S}=3.05\ifmmode\pm\else\textpm\fi{}0.05$, compared to the global measure of current, ${D}_{R}=2.07\ifmmode\pm\else\textpm\fi{}0.02$, ${D}_{C}=2.96\ifmmode\pm\else\textpm\fi{}0.05$, and ${D}_{S}=2.30\ifmmode\pm\else\textpm\fi{}0.02$. The deviation between local and global measurements of the attractor's dimension corresponds to an increase towards higher condensate depletion, which remains constant for long time scales in both Rabi and chaotic regimes. The depletion is found to scale polynomially with particle number $N$, namely, as ${N}^{\ensuremath{\beta}}$ with ${\ensuremath{\beta}}_{R}=0.51\ifmmode\pm\else\textpm\fi{}0.004$ and ${\ensuremath{\beta}}_{C}=0.18\ifmmode\pm\else\textpm\fi{}0.004$ for the two regimes. Thus, we find a strong deviation from the mean-field results, especially in the chaotic regime of the quantum ratchet. The ratchet also reveals quantum revivals in the Rabi and self-trapping regimes but not in the chaotic regime, with revival times scaling linearly in particle number for Rabi dynamics. Based on the obtained results, we outline pathways for the identification and characterization of emergent phenomena in driven many-body systems. This includes the identification of many-body localization from the many-body measures of the system, the influence of entanglement on the rate of the convergence to the mean-field limit, and the establishment of a polynomial scaling of the Ehrenfest time at which the mean-field description fails to describe the dynamics of the system.

Explaining Homeopathy with Quantum Electrodynamics.

Every living organism is an open system operating far from thermodynamic equilibrium and exchanging energy, matter and information with an external environment. These exchanges are performed through non-linear interactions of billions of different biological components, at different levels, from the quantum to the macro-dimensional. The concept of quantum coherence is an inherent property of living cells, used for long-range interactions such as synchronization of cell division processes. There is support from recent advances in quantum biology, which demonstrate that coherence, as a state of order of matter coupled with electromagnetic (EM) fields, is one of the key quantum phenomena supporting life dynamics. Coherent phenomena are well explained by quantum field theory (QFT), a well-established theoretical framework in quantum physics. Water is essential for life, being the medium used by living organisms to carry out various biochemical reactions and playing a fundamental role in coherent phenomena. Quantum electrodynamics (QED), which is the relativistic QFT of electrodynamics, deals with the interactions between EM fields and matter. QED provides theoretical models and experimental frameworks for the emergence and dynamics of coherent structures, even in living organisms. This article provides a model of multi-level coherence for living organisms in which fractal phase oscillations of water are able to link and regulate a biochemical reaction. A mathematical approach, based on the eigenfunctions of Laplace operator in hyper-structures, is explored as a valuable framework to simulate and explain the oneness dynamics of multi-level coherence in life. The preparation process of a homeopathic medicine is analyzed according to QED principles, thus providing a scientific explanation for the theoretical model of "information transfer" from the substance to the water solution. A subsequent step explores the action of a homeopathic medicine in a living organism according to QED principles and the phase-space attractor's dynamics. According to the developed model, all levels of a living organism-organelles, cells, tissues, organs, organ systems, whole organism-are characterized by their own specific wave functions, whose phases are perfectly orchestrated in a multi-level coherence oneness. When this multi-level coherence is broken, a disease emerges. An example shows how a homeopathic medicine can bring back a patient from a disease state to a healthy one. In particular, by adopting QED, it is argued that in the preparation of homeopathic medicines, the progressive dilution/succussion processes create the conditions for the emergence of coherence domains (CDs) in the aqueous solution. Those domains code the original substance information (in terms of phase oscillations) and therefore they can transfer said information (by phase resonance) to the multi-level coherent structures of the living organism. We encourage that QED principles and explanations become embodied in the fundamental teachings of the homeopathic method, thus providing the homeopath with a firm grounding in the practice of rational medicine. Systematic efforts in this direction should include multiple disciplines, such as quantum physics, quantum biology, conventional and homeopathic medicine and psychology.

Occasional uncoupling overcomes measure desynchronization.

Owing to the absence of the phase space attractors in the Hamiltonian dynamical systems, the concept of the identical synchronization between the dissipative systems is inapplicable to the Hamiltonian systems for which, thus, one defines a related generalized phenomenon known as the measure synchronization. A coupled pair of Hamiltonian systems-the full coupled system also being Hamiltonian-can possibly be in two types of measure synchronized states: quasiperiodic and chaotic. In this paper, we take representative systems belonging to each such class of the coupled systems and highlight that, as the coupling strengths are varied, there may exist intervals in the ranges of the coupling parameters at which the systems are measure desynchronized. Subsequently, we illustrate that as a coupled system evolves in time, occasionally switching off the coupling when the system is in the measure desynchronized state can bring the system back in measure synchrony. Furthermore, for the case of the occasional uncoupling being employed periodically and the corresponding time-period being small, we analytically find the values of the on-fraction of the time-period during which measure synchronization is effected on the corresponding desynchronized state.

Characteristic Fluctuations Around Stable Attractor Dynamics Extracted from Highly Nonstationary Electroencephalographic Recordings.

Since the discovery of electrical activity of the brain, electroencephalographic (EEG) recordings constitute one of the most popular techniques of brain research. However, EEG signals are highly nonstationary and one should expect that averages of the cross-correlation coefficient, which may take positive and negative values with equal probability, (almost) vanish when estimated over long data segments. Instead, we found that the average zero-lag cross-correlation matrix estimated with a running window over the whole night of sleep EEGs, or of resting state during eyes-open and eyes-closed conditions of healthy subjects shows a characteristic correlation pattern containing pronounced nonzero values. A similar correlation structure has already been encountered in scalp EEG signals containing focal onset seizures. Therefore, we conclude that this structure is independent of the physiological state. Because of its pronounced similarity across subjects, we believe that it depicts a generic feature of the brain dynamics. Namely, we interpret this pattern as a manifestation of a dynamical ground state of the brain activity, necessary to preserve an efficient operational mode, or, expressed in terms of dynamical system theory, we interpret it as a "shadow" of the evolution on (or close to) an attractor in phase space. Nonstationary dynamical aspects of higher cerebral processes should manifest in deviations from this stable pattern. We confirm this hypothesis through a correlation analysis of EEG recordings of 10 healthy subjects during night sleep, 20 recordings of 9 epilepsy patients, and 42 recordings of 21 healthy subjects in resting state during eyes-open and eyes-closed conditions. In particular, we show that the estimation of deviations from the stationary correlation structures provides a more significant differentiation of physiological states and more homogeneous results across subjects.

Steering multiattractors to overcome parameter inaccuracy and noise effects

Steering of attractors in multistable systems is used to increase the available parameter domains which lead to stable dynamics in nonlinear physical systems, reducing substantially undesirable effects of parametric inaccuracy and noise. The procedure proposed here uses time and/or space asymmetric perturbations to move independent multistable attractors in phase space. Applying this mechanism we increase around 230% the stable domains in H\'enon's map, roughly 85% in the ratchet current described by the Langevin equation and 60% in Chua's electronic circuit. The proposal is expected to have wide applications in generic nonlinear complex system presenting multistability, so that related experiments can increase robustness under parametric inaccuracy and noise.

A Novel Autonomous 5-D Hyperjerk RC Circuit with Hyperbolic Sine Function

A novel autonomous 5-D hyperjerk RC circuit with hyperbolic sine function is proposed in this paper. Compared to some existing 5-D systems like the 5-D Sprott B system, the 5-D Lorentz, and the Lorentz-like systems, the new system is the simplest 5-D system with complex dynamics reported to date. Its simplicity mainly relies on its nonlinear part which is synthetized using only two semiconductor diodes. The system displays only one equilibrium point and can exhibit both periodic and chaotic dynamical behavior. The complex dynamics of the system is investigated by means of bifurcation analysis. In particular, the striking phenomenon of multistability is revealed showing up to seven coexisting attractors in phase space depending solely on the system's initial state. To the best of author's knowledge, this rich dynamics has not yet been revealed in any 5-D dynamical system in general or particularly in any hyperjerk system. Pspice circuit simulations are performed to verify theoretical/numerical analysis.

A Semianalytical Approach for Nonlinear Dynamic System of Shallow Arches Using Higher Order Multistep Taylor Method

This study aimed at obtaining a semianalytical solution for nonlinear dynamic system of shallow arches. Taylor method was applied to find the analytical solution, and an investigation of their dynamic characteristic was carried out to verify the applicability of this methodology for the shallow arches under step or periodic excitation. A polynomial solution can be obtained from this multistep approach with respect to time, and direct buckling as well as indirect buckling of the shallow arches can be observed, also. The results indicated that the dynamic buckling load level was higher with higher shape factor. Additionally, a change of attractor in phase space was investigated. Coupling in symmetric mode as well as asymmetric mode was observed in case of indirect buckling, and a sensitive response was also manifested during sinusoidal and beating excitation. These results of applying multistep Taylor series for the investigation of displacement response and attractor change revealed that this analytical approach was valid in explaining the dynamic buckling behavior of shallow arches under direct and indirect snapping.

Open-loop control of periodic thermoacoustic oscillations: Experiments and low-order modelling in a synchronization framework

Open-loop forcing is known to be an effective strategy for controlling self-excited thermoacoustic oscillations, but the details of this synchronization process have yet to be comprehensively explored. In this study, we experimentally examine the synchronization dynamics of a laminar conical premixed flame in a tube combustor subjected to periodic acoustic forcing. We compare the response of this forced self-excited system with that of a forced Duffing–van der Pol oscillator, and find many similarities but also some differences. The similarities include (i) a torus-birth bifurcation from periodicity to quasiperiodicity at low forcing amplitudes, producing a stable ergodic T2 torus attractor in phase space; (ii) a transition from T2 quasiperiodicity to lock-in above a critical forcing amplitude, which increases linearly as the forcing frequency ff deviates from the natural frequency f1; (iii) two distinct routes to lock-in, one via a torus-death bifurcation if ff is far from f1 and one via a saddle-node bifurcation if ff is close to f1; and (iv) asynchronous quenching (AQ), which coincides with a torus-death bifurcation to lock-in and reduces the oscillation amplitude – by up to 90% in the combustor. There are, however, quantitative differences between the two systems, which pertain mainly to (i) the magnitude of the amplitude reduction achieved by AQ and (ii) the size of the AQ region in the ff/f1–forcing-amplitude plane.This study has three main contributions. First, it shows that studying open-loop control from a synchronization perspective can provide valuable insight into the optimal forcing conditions. Second, it shows that the optimal forcing condition for weakening thermoacoustic oscillations is that which causes the onset of lock-in via a torus-death bifurcation, as this is where AQ occurs. Third, it shows that the synchronization dynamics of a real combustor can be qualitatively modelled with a low-order universal oscillator. This suggests that it may be possible to develop and test new control strategies by analyzing the solutions to such an oscillator.

Nonlinear time-series analysis of thermoacoustic oscillations in a solid rocket motor

Recent applications of nonlinear time-series analysis have shown that thermoacoustic systems can exhibit self-excited behavior more complex than simple period-1 oscillations. Such behavior has been comprehensively explored in the Rijke tube and in gas turbines, but less so in rocket engines. In this experimental study, we examine the thermoacoustic oscillations of a full-scale solid rocket motor (SRM) by means of nonlinear time-series analysis, namely phase portraits, Poincaré maps and recurrence plots. In contrast to conventional linear tools based on the Fourier transform, these nonlinear tools can enable unambiguous identification of different dynamical states, such as period-k states, quasiperiodicity and chaos. In this study, these tools reveal limit-cycle dynamics not reported before in an SRM, such as transient mode switching between unstable attractors in phase space, with transitions between period-2 and period-3 states. In summary, this study constitutes the first experimental application of nonlinear time-series analysis to a full-scale SRM, demonstrating an alternative to conventional linear tools for the characterization of nonlinear thermoacoustic oscillations in rocket engines.

Resolving runaway electron distributions in space, time, and energy

Areas of agreement and disagreement with present-day models of runaway electron (RE) evolution are revealed by measuring MeV-level bremsstrahlung radiation from runaway electrons (REs) with a pinhole camera. Spatially resolved measurements localize the RE beam, reveal energy-dependent RE transport, and can be used to perform full two-dimensional (energy and pitch-angle) inversions of the RE phase-space distribution. Energy-resolved measurements find qualitative agreement with modeling on the role of collisional and synchrotron damping in modifying the RE distribution shape. Measurements are consistent with predictions of phase-space attractors that accumulate REs, with non-monotonic features observed in the distribution. Temporally resolved measurements find qualitative agreement with modeling on the impact of collisional and synchrotron damping in varying the RE growth and decay rate. Anomalous RE loss is observed and found to be largest at low energy. Possible roles for kinetic instability or spatial transport to resolve these anomalies are discussed.

Brilliant GeV gamma-ray flash from inverse Compton scattering in the QED regime

An all-optical scheme is proposed for studying laser plasma based incoherent photon emission from inverse Compton scattering in the quantum electrodynamic regime. A theoretical model is presented to explain the coupling effects among radiation reaction trapping, the self-generated magnetic field and the spiral attractor in phase space, which guarantees the transfer of energy and angular momentum from electromagnetic fields to particles. Taking advantage of a prospective ∼ 1023 W cm−2 laser facility, 3D particle-in-cell simulations show a gamma-ray flash with unprecedented multi-petawatt power and brightness of 1.7 × 1023 photons s−1 mm−2 mrad−2/0.1% bandwidth (at 1 GeV). These results bode well for new research directions in particle physics and laboratory astrophysics exploring laser plasma interactions.

From coupled networks of systems to networks of states in phase space

Dynamical systems on graphs can show a wide range of behaviours beyond simple synchronization - even simple globally coupled structures can exhibit attractors with intermittent and slow switching between patterns of synchrony. Such attractors, called heteroclinic networks, can be well described as networks in phase space and in this paper we review some results and examples of how these robust attractors can be characterised from the synchrony properties and how coupled systems can be designed to exhibit given but arbitrary network attractors in phase space.

Controlling intermediate dynamics in a family of quadratic maps.

The intermediate dynamics of composed one-dimensional maps is used to multiply attractors in phase space and create multiple independent bifurcation diagrams which can split apart. Results are shown for the composition of k-paradigmatic quadratic maps with distinct values of parameters generating k-independent bifurcation diagrams with corresponding k orbital points. For specific conditions, the basic mechanism for creating the shifted diagrams is the prohibition of period doubling bifurcations transformed in saddle-node bifurcations.

Proliferation of stability in phase and parameter spaces of nonlinear systems.

In this work, we show how the composition of maps allows us to multiply, enlarge, and move stable domains in phase and parameter spaces of discrete nonlinear systems. Using Hénon maps with distinct parameters, we generate many identical copies of isoperiodic stable structures (ISSs) in the parameter space and attractors in phase space. The equivalence of the identical ISSs is checked by the largest Lyapunov exponent analysis, and the multiplied basins of attraction become riddled. Our proliferation procedure should be applicable to any two-dimensional nonlinear system.

Stochastic inflation in phase space: is slow roll a stochastic attractor?

An appealing feature of inflationary cosmology is the presence of a phase-space attractor, ``slow roll'', which washes out the dependence on initial field velocities. We investigate the robustness of this property under backreaction from quantum fluctuations using the stochastic inflation formalism in the phase-space approach. A Hamiltonian formulation of stochastic inflation is presented, where it is shown that the coarse-graining procedure—where wavelengths smaller than the Hubble radius are integrated out—preserves the canonical structure of free fields. This means that different sets of canonical variables give rise to the same probability distribution which clarifies the literature with respect to this issue. The role played by the quantum-to-classical transition is also analysed and is shown to constrain the coarse-graining scale. In the case of free fields, we find that quantum diffusion is aligned in phase space with the slow-roll direction. This implies that the classical slow-roll attractor is immune to stochastic effects and thus generalises to a stochastic attractor regardless of initial conditions, with a relaxation time at least as short as in the classical system. For non-test fields or for test fields with non-linear self interactions however, quantum diffusion and the classical slow-roll flow are misaligned. We derive a condition on the coarse-graining scale so that observational corrections from this misalignment are negligible at leading order in slow roll.