Trajectories In State Space Research Articles

Is the Hénon map able to predict the interaction dynamics between the knee and hip joints emerged during sit-to-stand movement?

In this study, the performance of a two-dimensional Hénon map in predicting the interactive dynamics of the knee and hip joints emerging during a normative sit-to-stand movement was evaluated. The instantaneous values of the knee and hip joints were the model inputs, and the next values of the knee and hip joints were predicted by the Hénon map. The map predicted the desired relative behavior of the joints, showing synergetic coordination between the joints. The experimental data were recorded from four healthy participants and used to identify the Hénon map via a genetic algorithm. Model performance was quantitatively assessed by computing the calculated prediction error and analyzing the behavioral dynamics of the state spaces reconstructed via the captured kinematic data. According to the results, there was an obvious similarity between the dynamics of the state space trajectories of the identified model and those of the recorded data, not only in terms of stretching and folding dynamics, but also concerning generalized synchrony. The acceptable performance of the proposed modeling solution can also be demonstrated through these results.

Melting Ice With Your Mind: Representational Momentum for Physical States

When a log burns, it transforms from a block of wood into a pile of ash. Such state changes are among the most dramatic ways objects change, going beyond mere changes of position or orientation. How does the mind represent changes of state? A foundational result in visual cognition is that memory extrapolates the positions of moving objects—a distortion called representational momentum. Here, five experiments (N = 400 adults) exploited this phenomenon to investigate mental representations in state space. Participants who viewed objects undergoing state changes (e.g., ice melting, logs burning, or grapes shriveling) remembered them as more changed (e.g., more melted, burned, or shriveled) than they actually were. This pattern extended to several types of state changes, went beyond their low-level properties, and even adhered to their natural trajectories in state space. Thus, mental representations of objects actively incorporate how they change—not only in their relation to their environment, but also in their essential qualities.

Detecting chaos in lineage-trees: A deep learning approach

Many complex phenomena, from weather systems to heartbeat rhythm patterns, are effectively modeled as low-dimensional dynamical systems. Such systems may behave chaotically under certain conditions, and so the ability to detect chaos based on empirical measurement is an important step in understanding these processes. Classifying a system as chaotic entails estimating its largest Lyapunov exponent (LLE), which quantifies the average rate of convergence or divergence of initially close trajectories in state space. Estimating the largest Lyapunov exponent from observations of a process is especially challenging in low-dimensional systems affected by a small amount of dynamical noise, which are used to model many real-world processes, and in particular biological systems. We describe a method for accurately estimating the largest Lyapunov exponent from noisy data, based on training deep learning models on synthetically generated trajectories. Once the deep learning models are trained, they can be used to detect the LLE from empirical data without any assumption on the underlying dynamics. Our method naturally extends to different input topologies, allowing us to analyze tree-shaped data, characteristic of inheritance dynamics, where near-identical replication offers a unique and hitherto unstudied avenue to detect chaos. We also characterize the input information extracted by our models for their predictions, allowing for a deeper understanding into the different ways by which chaos can be analyzed in different topologies.

Open quantum dynamics with singularities: Master equations and degree of non-Markovianity

Master equations describing open quantum dynamics are typically first-order differential equations. When such dynamics brings the trajectories in state space of more than one initial state to the same point at finite instants in time, the generator of the corresponding master equation becomes singular while the dynamical map becomes noninvertible. The first-order, time-local, homogeneous master equations then fail to describe the dynamics beyond the singular point. Retaining time locality in the master equation necessitates a reformulation in terms of higher-order differential equations. We formulate a method to eliminate the divergent behavior of the generator by using a combination of higher-order derivatives of the generator with suitable weights and illustrate it with several examples. We also present a detailed study of the central spin model and we propose the average rate of information inflow in non-Markovian processes as a quantity that captures a different aspect of non-Markovian dynamics.

Decoding complex state space trajectories for neural computing.

In biological neural circuits as well as in bio-inspired information processing systems, trajectories in high-dimensional state-space encode the solutions to computational tasks performed by complex dynamical systems. Due to the high state-space dimensionality and the number of possible encoding trajectories rapidly growing with input signal dimension, decoding these trajectories constitutes a major challenge on its own, in particular, as exponentially growing (space or time) requirements for decoding would render the original computational paradigm inefficient. Here, we suggest an approach to overcome this problem. We propose an efficient decoding scheme for trajectories emerging in spiking neural circuits that exhibit linear scaling with input signal dimensionality. We focus on the dynamics near a sequence of unstable saddle states that naturally emerge in a range of physical systems and provide a novel paradigm for analog computing, for instance, in the form of heteroclinic computing. Identifying simple measures of coordinated activity (synchrony) that are commonly applicable to all trajectories representing the same percept, we design robust readouts whose sizes and time requirements increase only linearly with the system size. These results move the conceptual boundary so far hindering the implementation of heteroclinic computing in hardware and may also catalyze efficient decoding strategies in spiking neural networks in general.

A trajectory-based loss function to learn missing terms in bifurcating dynamical systems

Missing terms in dynamical systems are a challenging problem for modeling. Recent developments in the combination of machine learning and dynamical system theory open possibilities for a solution. We show how physics-informed differential equations and machine learning—combined in the Universal Differential Equation (UDE) framework by Rackauckas et al.—can be modified to discover missing terms in systems that undergo sudden fundamental changes in their dynamical behavior called bifurcations. With this we enable the application of the UDE approach to a wider class of problems which are common in many real world applications. The choice of the loss function, which compares the training data trajectory in state space and the current estimated solution trajectory of the UDE to optimize the solution, plays a crucial role within this approach. The Mean Square Error as loss function contains the risk of a reconstruction which completely misses the dynamical behavior of the training data. By contrast, our suggested trajectory-based loss function which optimizes two largely independent components, the length and angle of state space vectors of the training data, performs reliable well in examples of systems from neuroscience, chemistry and biology showing Saddle-Node, Pitchfork, Hopf and Period-doubling bifurcations.

Evolutional deep neural network.

The notion of an evolutional deep neural network (EDNN) is introduced for the solution of partial differential equations (PDE). The parameters of the network are trained to represent the initial state of the system only and are subsequently updated dynamically, without any further training, to provide an accurate prediction of the evolution of the PDE system. In this framework, the network parameters are treated as functions with respect to the appropriate coordinate and are numerically updated using the governing equations. By marching the neural network weights in the parameter space, EDNN can predict state-space trajectories that are indefinitely long, which is difficult for other neural network approaches. Boundary conditions of the PDEs are treated as hard constraints, are embedded into the neural network, and are therefore exactly satisfied throughout the entire solution trajectory. Several applications including the heat equation, the advection equation, the Burgers equation, the Kuramoto Sivashinsky equation, and the Navier-Stokes equations are solved to demonstrate the versatility and accuracy of EDNN. The application of EDNN to the incompressible Navier-Stokes equations embeds the divergence-free constraint into the network design so that the projection of the momentum equation to solenoidal space is implicitly achieved. The numerical results verify the accuracy of EDNN solutions relative to analytical and benchmark numerical solutions, both for the transient dynamics and statistics of the system.

Stochastic facilitation in heteroclinic communication channels.

Biological neural systems encode and transmit information as patterns of activity tracing complex trajectories in high-dimensional state spaces, inspiring alternative paradigms of information processing. Heteroclinic networks, naturally emerging in artificial neural systems, are networks of saddles in state space that provide a transparent approach to generate complex trajectories via controlled switches among interconnected saddles. External signals induce specific switching sequences, thus dynamically encoding inputs as trajectories. Recent works have focused either on computational aspects of heteroclinic networks, i.e., Heteroclinic Computing, or their stochastic properties under noise. Yet, how well such systems may transmit information remains an open question. Here, we investigate the information transmission properties of heteroclinic networks, studying them as communication channels. Choosing a tractable but representative system exhibiting a heteroclinic network, we investigate the mutual information rate (MIR) between input signals and the resulting sequences of states as the level of noise varies. Intriguingly, MIR does not decrease monotonically with increasing noise. Intermediate noise levels indeed maximize the information transmission capacity by promoting an increased yet controlled exploration of the underlying network of states. Complementing standard stochastic resonance, these results highlight the constructive effect of stochastic facilitation (i.e., noise-enhanced information transfer) on heteroclinic communication channels and possibly on more general dynamical systems exhibiting complex trajectories in state space.

Probabilistic Situation Assessment for Intelligent Vehicles with Uncertain Trajectory Distribution

Situation assessment is crucial for intelligent vehicles, enabling detection of potential risks to dynamic and complex traffic environments. In this paper, we propose a unified framework that tackles the coupling relationships between traffic participants and quantifies the possible range of vehicle trajectory generation and the expected losses caused by risk source attributes in the driving process. We first apply the state space trajectory planning scheme based on a sampling algorithm to generate the path candidates; each feasible path is designed through a parametric cubic spline. Then, to evaluate the risk range in the driving process, we quantify the interaction of traffic participants, and employ the principle of least action to calculate the cost of each feasible path when achieving the destination. The probability distribution map, namely the possible range of driving trajectories, can be obtained based on the path cost. Furthermore, the vehicle-to-vehicle interaction is calculated based on the equivalent force, which estimates the expected accident losses. Finally, the vehicle trajectory prediction and the expected loss are combined to output the probabilistic situation assessment of intelligent vehicles. The algorithm is implemented in different scenarios and applied to the trajectory planning process. Results demonstrate that, compared with the classical situation assessment metric, the developed method can determine and accurately identify the influence range of driving risk in real-time, predict a dangerous situation earlier, and ensure the vehicle avoids obstacles in advance.

Adaptive observer based-robust synchronization of switched fractional Rikitake systems with input nonlinearity

This paper mainly focuses on the robust synchronization of switched fractional-order improved Rikitake two-disk dynamo system which is a more general topology of the original Rikitake system. In particular, we first explore and design a simplified model very close to a real geophysical process which is described by an orbiting point, primarily using concepts from dynamical system theory. Dynamical properties, including symmetry, dissipation, stability of equilibria, Lyapunov exponents, and bifurcation, are analyzed on the basis of theoretical analysis and numerical simulation. Secondly, some less stringent conditions for adaptive control are derived when the system is subjected to external disturbances, parametric uncertainties and nonlinear input. The convergence of the state space trajectories with the overall robust asymptotic stability of the closed-loop system is obtained using the Lyapunov theory. To highlight our contribution, an illustrative example is presented to show the effectiveness and applicability of the proposed synchronization scheme. This proposal derives a systematic procedure for the design and control of a wide range of electromechanical systems. The results obtained in this work have not yet been reported in the literature to the best of the authors’ knowledge and thus deserve dissemination.

Ameliorated Frenet Trajectory Optimization Method Based on Artificial Emotion and Equilibrium Optimizer

An ameliorated Frenet trajectory optimization (AFTO) method based on artificial emotion (AE) and an equilibrium optimizer (EO) is proposed for the local trajectory planning of an unmanned ground vehicle (UGV). An artificial emotional potential field (AEPF) model is established to simulate AE. To realize a humanoid driving mode with emotional intelligence, AE is introduced into the Frenet trajectory optimization (FTO) method to determine the optimal trajectory. Based on the optimal discrete goal state of the FTO method, a first-sampling-then-optimization (FSTO) framework combining the FTO method with the EO is designed to obtain the optimal trajectory in a continuous goal state space. With different AEPF levels corresponding to different types of obstacles, simulation results show that the AEPF effectively adjusts the trajectory into different levels of safe distance between the UGV and obstacles corresponding to the humanoid driving mode. From the results of 30 independent experiments based on the AEPF, the FSTO framework in the AFTO method is effective for optimizing the trajectory of the FTO method at a lower cost. Moreover, the effectiveness of the proposed method for different types of roads is verified on a straight road and a curved road with obstacles in simulation. The improvement based on emotional intelligence and trajectory optimization in the AFTO method provides a humanoid driving mode for the UGV in the continuous goal state space.

QMH via Ontological Engineering Applied to Measurement of Knowledge, Truth, and Lies (‘An Element of the Meta Cycle of Life’)

In reference to S.A Nigosian: “Mantras were not viewed as the only means of expressing truth. However, thought (defined as internalized speech ) offers another aspect of truth”. He goes on to say: “If words and thoughts designated different aspect of truth, or reality, then there had to be an underlying unity behind all phenomena”. This paper naivete will attempt to clarify Nigosian's “unity” , His description above will be examined as an externally linear measurement of the internal human system elements of knowledge, truth, and lies . These elements are in themselves nonlinear functions of the internalized state-space variables which are moving in time. Whereas “measurement” may be considered as satisfying the problem of the interpretation of data via the statistical solution to the smoothing, filtering, and prediction problem, these methods are extremely susceptible to “bias creep”. For example, recent studies from the Rand Corporation produced a study entitled “Truth Decay”. Post-truth differs from traditional contesting and falsifying of facts by relegating facts and expert opinions to be of secondary importance relative to appeal to emotion. While this has been described as a contemporary problem, some observers have described it as a long-standing part of political life that was less notable before the advent of the Internet and related social changes. As such in order to quantify the metaphysics of Aristotle underlying this “Truth Decay” its internal dynamics which couple “knowledge, truth, and lies” visualized in 3 dimensions by their state-space trajectories are presented. In doing so naivete this paper attempts to clarify Nigosian's “unity” by the single external measurement of the internal state-space variables which by their design are not visible to the outside world and have been (for most of us) hard coded into our DNA

A Geometric Characterization of Population Coding in the Prefrontal Cortex and Hippocampus during a Paired-Associate Learning Task.

Large-scale neuronal recording techniques have enabled discoveries of population-level mechanisms for neural computation. However, it is not clear how these mechanisms form by trial-and-error learning. In this article, we present an initial effort to characterize the population activity in monkey prefrontal cortex (PFC) and hippocampus (HPC) during the learning phase of a paired-associate task. To analyze the population data, we introduce the normalized distance, a dimensionless metric that describes the encoding of cognitive variables from the geometrical relationship among neural trajectories in state space. It is found that PFC exhibits a more sustained encoding of the visual stimuli, whereas HPC only transiently encodes the identity of the associate stimuli. Surprisingly, after learning, the neural activity is not reorganized to reflect the task structure, raising the possibility that learning is accompanied by some "silent" mechanism that does not explicitly change the neural representations. We did find partial evidence on the learning-dependent changes for some of the task variables. This study shows the feasibility of using normalized distance as a metric to characterize and compare population-level encoding of task variables and suggests further directions to explore learning-dependent changes in the neural circuits.

Invariant states in inclined layer convection. Part 1. Temporal transitions along dynamical connections between invariant states

Thermal convection in an inclined layer between two parallel walls kept at different fixed temperatures is studied for fixed Prandtl number Pr=1.07. Depending on the angle of inclination and the imposed temperature difference, the flow exhibits a large variety of self-organized spatio-temporal convection patterns. Close to onset, these patterns have been explained in terms of linear stability analysis of primary and secondary flow states. At larger temperature difference, far beyond onset, experiments and simulations show complex, dynamically evolving patterns that are not described by stability analysis and remain to be explained. Here we employ a dynamical systems approach. We construct stable and unstable exact invariant states, including equilibria and periodic orbits of the fully nonlinear three-dimensional Oberbeck-Boussinesq equations. These invariant states underlie the observed convection patterns beyond their onset. We identify state-space trajectories that, starting from the unstable laminar flow, follow a sequence of dynamical connections between unstable invariant states until the dynamics approaches a stable attractor. Together, the network of dynamically connected invariant states mediates temporal transitions between coexisting invariant states and thereby supports the observed complex time-dependent dynamics in inclined layer convection.

Fault estimation observer design for fractional-order financial system subject to time-delay

This article is devoted to resolving the issue of time-varying fault in the fractional-order financial system subject to constant delay. A fault-tolerant control is addressed to solve the faults which occur by unpredictable affairs in the financial system. Further, a set of adequate constraints in the form of Linear Matrix Inequalities are derived to ensure asymptotic stability of the considered system under the theorem of Lyapunov. More precisely, Lyapunov–Krasovskii candidates are employed independently with various state-space trajectories such as interest rate, investment demand and price index. Lastly, a numerical example is provided to support the constructed theoretical results.

Forced deterministic dynamics on a random energy landscape: Implications for the physics of amorphous solids.

The dynamics of supercooled liquids and plastically deformed amorphous solids is known to be dominated by the structure of their rough energy landscapes. Recent experiments and simulations on amorphous solids subjected to oscillatory shear at athermal conditions have shown that for small strain amplitudes these systems reach limit cycles of different periodicities after a transient. However, for larger strain amplitudes the transients become longer and for strain amplitudes exceeding a critical value the system reaches a diffusive steady state. This behavior cannot be explained using the current mean-field models of amorphous plasticity. Here we show that this phenomenology can be described and explained using a simple model of forced dynamics on a multidimensional random energy landscape. In this model, the existence of limit cycles can be ascribed to confinement of the dynamics to a small part of the energy landscape which leads to self-intersection of state-space trajectories and the transition to the diffusive regime for larger forcing amplitudes occurs when the forcing overcomes this confinement.

Forced vibrations of a spring–dashpot mechanism with dry friction and backlash

Mechanical systems like the Centre Buffer Couplers, which connect railway coaches, are spring–dashpot mechanisms characterized by presence of dry friction and backlash and are subjected to large dynamic forces. While systems with individual characteristics mentioned above, have been studied by earlier researchers, a system where all of the above physical features are present simultaneously, is analysed in this study. A single degree of freedom system with dry friction, backlash and spring–dashpot element is considered and subjected to harmonic excitation. The response is analysed through displacement, phase-space plots, orbit diagrams and Poincare maps using numerical simulations. Occurrence of bifurcations and chaos for different system parameters is numerically evaluated. The qualitative behaviour of systems with deteriorated conditions of low viscous damping, higher backlash and increased dry friction is established through numerical simulations and comparisons are made with that of a normal system. A transitional value of deteriorated condition is illustrated in terms of the ratio of viscous damping to dry friction, beyond which the response does not remain periodic. Additional verification is done through the Filippov method. Since, two discontinuities, i.e. the dry friction and backlash are simultaneously present in the system, the vector space gets divided into nine subspaces. Various possible transitions of state space trajectory between two adjacent subspaces have been analysed. To the knowledge of authors, such complex planar system with presence of more than one mutually perpendicular line discontinuities which intersect each other, has been analysed in various subspaces for the first time.

A nonlinear dynamical approach to analysis of emotions using EEG signals based on the Poincaré map function and recurrence plots.

Dynamic variations of electroencephalogram (EEG) contain significant information in the study of human emotional states. Transient time methods are well suited to evaluate short-term dynamic changes in brain activity. Human affective states, however, can be more appropriately analyzed using chaotic dynamical techniques, in which temporal variations are considered over longer durations. In this study, we have applied two different recurrence-based chaotic schemes, namely the Poincaré map function and recurrence plots (RPs), to analyze the long-term dynamics of EEG signals associated with state space (SS) trajectory of the time series. Both approaches determine the system dynamics based on the Poincaré recurrence theorem as well as the trajectory divergence producing two-dimensional (2D) characteristic plots. The performance of the methods is compared with regard to their ability to distinguish between levels of valence, arousal, dominance and liking using EEG data from the "dataset for emotion analysis using physiological" database. The differences between the levels of emotional feelings were investigated based on the analysis of variance (ANOVA) test and Spearman's statistics. The results obtained from the RP features distinguish between the emotional ratings with a higher level of statistical significance as compared with those produced by the Poincaré map function. The scheme based on RPs was particularly advantageous in identifying the levels of dominance. Out of the 32 EEG electrodes examined, the RP-based approach distinguished the dominance levels in 23 electrodes, while the approach based on the Poincaré map function was only able to discriminate dominance levels in five electrodes. Furthermore, based on nonlinear analysis, significant correlations were observed over a wider area of the cortex for all affective states as compared with that reported based on the analysis of EEG power bands.

Basis for finding exact coherent states.

One of the outstanding problems in the dynamical systems approach to turbulence is to find a sufficient number of invariant solutions to characterize the underlying dynamics of turbulence [Annu. Rev. Fluid Mech. 44, 203 (2012)10.1146/annurev-fluid-120710-101228]. As a practical matter, the solutions can be difficult to find. To improve this situation, we show how to find periodic orbits and equilibria in plane Couette flow by projecting pseudorecurrent segments of turbulent trajectories onto the left-singular vectors of the Navier-Stokes equations linearized about the relevant mean flow (resolvent modes). The projections are, subsequently, used to initiate Newton-Krylov-hookstep searches, and new (relative) periodic orbits and equilibria are discovered. We call the process project-then-search and validate the process by first applying it to previously known fixed point and periodic solutions. Along the way, we find new branches of equilibria, which include bifurcations from previously known branches, and new periodic orbits that closely shadow turbulent trajectories in state space.

Heterogeneous recurrence analysis of spatial data.

Nonlinear dynamical systems often generate significant amounts of observational data such as time series, as well as high-dimensional spatial data. To delineate recurrence dynamics in the spatial data, prior efforts either extended the recurrence plot, which is a widely used tool for time series, to a four-dimensional hyperspace or utilized the network approach for recurrence analysis. However, very little has been done to differentiate heterogeneous types of recurrences in the spatial data (e.g., recurrence variations of state transitions in the spatial domain). Therefore, we propose a novel heterogeneous recurrence approach for spatial data analysis. First, spatial data are traversed with the Hilbert Space-Filling Curve to transform the variations of recurrence patterns from the spatial domain to the state-space domain. Second, we design an Iterated Function System to derive the fractal representation for the state-space trajectory of spatial data. Such a fractal representation effectively captures self-similar behaviors of recurrence variations and multi-state transitions in the spatial data. Third, we develop the Heterogeneous Recurrence Quantification Analysis of spatial data. Experimental results in both simulation and real-world case studies show that the proposed approach yields superior performance in the extraction of salient features to characterize and quantify heterogeneous recurrence dynamics in spatial data.