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Stability and asymptotic behavior of dynamical systems defined by autonomous functional or partial differential equation and conditions for applying Liapunov theorem
Abstract Optimizing root system architecture offers a promising approach to developing stress tolerant cultivars in the face of climate change, as root systems are critical for water and nutrient uptake as well as mechanical stability. However, breeding for optimal root system architecture has been hindered by the difficulty in measuring root growth in the field. Here, we describe the RootTracker, a technology that employs impedance touch sensors to monitor in-field root growth over time. Configured in a cylindrical, window shutter-like fashion around a planted seed, 264 electrodes are individually charged multiple times over the course of an experiment. Signature changes in the measured capacitance and resistance readings indicate when a root has touched or grown close to an electrode. Using the RootTracker, we have measured root system dynamics of commercial maize (Zea mays) hybrids growing in both typical Midwest field conditions and under different irrigation regimes. We observed rapid responses of root growth to water deficits and found evidence for a “priming response” in which an early water deficit causes more and deeper roots to grow at later time periods. Genotypic variation among hybrid maize lines in their root growth in response to drought indicated a potential to breed for root systems adapted for different environments. Thus, the RootTracker is able to capture changes in root growth over time in response to environmental perturbations.
Parallel processing of real-time dynamic systems simulation on a multiprocessor system named OSCAR is presented. In the simulation of dynamic systems, generally, the same calculation are repeated every time step. However, we cannot apply to Do-all or the Do-across techniques for parallel processing of the simulation since there exist data dependencies from the end of an iteration to the beginning of the next iteration and furthermore data-input and data-output are required every sampling time period. Therefore, parallelism inside the calculation required for a single time step, or a large basic block which consists of arithmetic assignment statements, must be used. In the proposed method, near fine grain tasks, each of which consists of one or more floating point operations, are generated to extract the parallelism from the calculation and assigned to processors by using optimal static scheduling at compile time in order to reduce large run time overhead caused by the use of near fine grain tasks. The practicality of the scheme is demonstrated on OSCAR (Optimally SCheduled Advanced multiprocessoR) which has been developed to extract advantageous features of static scheduling algorithms to the maximum extent.
The results of a fixed-base simulator study of the effects of variable longitudinal control-system dynamics on pilot opinion are presented and compared with flight-test data. The control-system variables considered in this investigation included stick force per g, time constant, and dead-band, or stabilizer breakout force. In general, the fairly good correlation between flight and simulator results for two pilots demonstrates the validity of fixed-base simulator studies which are designed to complement and supplement flight studies and serve as a guide in control-system preliminary design. However, in the investigation of certain problem areas (e.g., sensitive control-system configurations associated with pilot- induced oscillations in flight), fixed-base simulator results did not predict the occurrence of an instability, although the pilots noted the system was extremely sensitive and unsatisfactory. If it is desired to predict pilot-induced-oscillation tendencies, tests in moving-base simulators may be required. It was found possible to represent the human pilot by a linear pilot analog for the tracking task assumed in the present study. The criterion used to adjust the pilot analog was the root-mean-square tracking error of one of the human pilots on the fixed-base simulator. Matching the tracking error of the pilot analog to that of the human pilot gave an approximation to the variation of human-pilot behavior over a range of control-system dynamics. Results of the pilot-analog study indicated that both for optimized control-system dynamics (for poor airplane dynamics) and for a region of good airplane dynamics, the pilot response characteristics are approximately the same.
Generalized dynamic systems defined by contingent equations
Stability problems in randomly excited dynamic systems, discussing partial differential equations governing evolution of conditional probabilities and expectations
Equilibria stability of linear discrete dynamic systems involving elastic, nonconservative, dissipative and gyroscopic forces, using Liapunov-type energy method
A computer simulation of the YF-12 aircraft motions and propulsion system dynamics is presented. The propulsion system was represented in sufficient detail so that interactions between aircraft motions and the propulsion system dynamics could be investigated. Six degree-of-freedom aircraft motions together with the three-axis stability augmentation system were represented. The mixed compression inlets and their controls were represented in the started mode for a range of flow conditions up to the inlet unstart boundary. Effects of inlet moving geometry on aircraft forces and movements as well as effects of aircraft motions on the inlet behavior were simulated. The engines, which are straight subjects, were represented in the afterburning mode, with effects of changes in aircraft flight conditions included. The simulation was capable of operating in real time.
This work attempts to test predictions based on the Dynamical Systems approach to Wall Layer Turbulence. We analyze the Dynamical Systems model for the nonlinear interaction mechanisms between the coherent structures and deduce qualitative behavior as expected. We then test for this behavior in data sets from D.N.S. The agreement is good, given the suboptimal conditions for the test. We discuss implications of this test and work to be done to deepen the understanding of control of turbulent boundary layers.
In this paper, a new systematic method is proposed to analyze the system frequency in power systems under uncertain variability (UV). UV gradually increases in power systems, which originates from the renewable energy generation, random loads, frequency measurement, and communication channels, exerting noteworthy impacts on dynamic system frequency. The UV is modeled as a stochastic process in this paper. Then, the stochastic differential equations are used to describe the dynamic system frequency response (SFR) of a power system under UV, which is based on the SFR model. To assess system frequency dynamics under UV, an index of intra-range probability is put forward. Based on stochastic analysis theory, an analytic method is proposed to analyze the system frequency under UV, by which the intra-range probability can be analytically solved. Compared with Monte Carlo simulation in a typical SFR case and the Iceland electricity transmission network, the proposed method shows almost the same results using much less computation resource. Finally, insights for improving the SFR under UV are provided. The proposed method can be used to quickly verify whether the system frequency could withstand the UV and stay in the security range.
Some progress has been made in a continuing effort to develop mathematical models of the behaviors of multi-agent systems known in biology, economics, and sociology (e.g., systems ranging from single or a few biomolecules to many interacting higher organisms). Living systems can be characterized by nonlinear evolution of probability distributions over different possible choices of the next steps in their motions. One of the main challenges in mathematical modeling of living systems is to distinguish between random walks of purely physical origin (for instance, Brownian motions) and those of biological origin. Following a line of reasoning from prior research, it has been assumed, in the present development, that a biological random walk can be represented by a nonlinear mathematical model that represents coupled mental and motor dynamics incorporating the psychological concept of reflection or self-image. The nonlinear dynamics impart the lifelike ability to behave in ways and to exhibit patterns that depart from thermodynamic equilibrium. Reflection or self-image has traditionally been recognized as a basic element of intelligence. The nonlinear mathematical models of the present development are denoted self-supervised dynamical systems. They include (1) equations of classical dynamics, including random components caused by uncertainties in initial conditions and by Langevin forces, coupled with (2) the corresponding Liouville or Fokker-Planck equations that describe the evolutions of probability densities that represent the uncertainties. The coupling is effected by fictitious information-based forces, denoted supervising forces, composed of probability densities and functionals thereof. The equations of classical mechanics represent motor dynamics that is, dynamics in the traditional sense, signifying Newton s equations of motion. The evolution of the probability densities represents mental dynamics or self-image. Then the interaction between the physical and metal aspects of a monad is implemented by feedback from mental to motor dynamics, as represented by the aforementioned fictitious forces. This feedback is what makes the evolution of probability densities nonlinear. The deviation from linear evolution can be characterized, in a sense, as an expression of free will. It has been demonstrated that probability densities can approach prescribed attractors while exhibiting such patterns as shock waves, solitons, and chaos in probability space. The concept of self-supervised dynamical systems has been considered for application to diverse phenomena, including information-based neural networks, cooperation, competition, deception, games, and control of chaos. In addition, a formal similarity between the mathematical structures of self-supervised dynamical systems and of quantum-mechanical systems has been investigated.
The Koopman operator has recently garnered much attention for its value in dynamical systems analysis and data-driven model discovery. However, its application has been hindered by the computational complexity of extended dynamic mode decomposition; this requires a combinatorially large basis set to adequately describe many nonlinear systems of interest, e.g. cyber-physical infrastructure systems, biological networks, social systems, and fluid dynamics. Often the dictionaries generated for these problems are manually curated, requiring domain-specific knowledge and painstaking tuning. In this paper we introduce a deep learning framework for learning Koopman operators of nonlinear dynamical systems. We show that this novel method automatically selects efficient deep dictionaries, outperforming state-of-the-art methods. We benchmark this method on partially observed nonlinear systems, including the glycolytic oscillator and show it is able to predict quantitatively 100 steps into the future, using only a single timepoint, and qualitative oscillatory behavior 400 steps into the future.
This work proposes a machine-learning framework for modeling the error incurred by approximate solutions to parameterized dynamical systems. In particular, we extend the machine-learning error models (MLEM) framework proposed in Ref. Freno and Carlberg (2019) to dynamical systems. The proposed Time-Series Machine-Learning Error Modeling (T-MLEM) method constructs a regression model that maps features – which comprise error indicators that are derived from standard a posteriori error-quantification techniques – to a random variable for the approximate-solution error at each time instance. The proposed framework considers a wide range of candidate features, regression methods, and additive noise models. We consider primarily recursive regression techniques developed for time-series modeling, including both classical time-series models (e.g., autoregressive models) and recurrent neural networks (RNNs), but also analyze standard non-recursive regression techniques (e.g., feed-forward neural networks) for comparative purposes. Finally, numerical experiments conducted on multiple benchmark problems illustrate that the long short-term memory (LSTM) neural network, which is a type of RNN, outperforms other methods and yields substantial improvements in error predictions over traditional approaches.
A classical reduced order model (ROM) for dynamical problems typically involves only the spatial reduction of a given problem. Recently, a novel space–time ROM for linear dynamical problems has been developed [Choi et al., Space–tume reduced order model for large-scale linear dynamical systems with application to Boltzmann transport problems, Journal of Computational Physics, 2020], which further reduces the problem size by introducing a temporal reduction in addition to a spatial reduction without much loss in accuracy. The authors show an order of a thousand speed-up with a relative error of less than 10−5 for a large-scale Boltzmann transport problem. In this work, we present for the first time the derivation of the space–time least-squares Petrov–Galerkin (LSPG) projection for linear dynamical systems and its corresponding block structures. Utilizing these block structures, we demonstrate the ease of construction of the space–time ROM method with two model problems: 2D diffusion and 2D convection diffusion, with and without a linear source term. For each problem, we demonstrate the entire process of generating the full order model (FOM) data, constructing the space–time ROM, and predicting the reduced-order solutions, all in less than 120 lines of Python code. We compare our LSPG method with the traditional Galerkin method and show that the space–time ROMs can achieve O(10−3) to O(10−4) relative errors for these problems. Depending on parameter–separability, online speed-ups may or may not be achieved. For the FOMs with parameter–separability, the space–time ROMs can achieve O(10) online speed-ups. Finally, we present an error analysis for the space–time LSPG projection and derive an error bound, which shows an improvement compared to traditional spatial Galerkin ROM methods.
Approximate model for simplification of linear dynamic system, neglecting effect of higher order time constants
This paper describes a constraint embedding approach for the handling of local closure constraints in multibody system dynamics. The approach uses spatial operator techniques to eliminate local-loop constraints from the system and effectively convert the system into tree-topology systems. This approach allows the direct derivation of recursive O(N) techniques for solving the system dynamics and avoiding the expensive steps that would otherwise be required for handling the closedchain dynamics. The approach is very effective for systems where the constraints are confined to small-subgraphs within the system topology. The paper provides background on the spatial operator O(N) algorithms, the extensions for handling embedded constraints, and concludes with some examples of such constraints.
Computational fluid dynamics (CFD) enables modeling, analysis, and visualization of phenomena that would usually be impossible or extremely expensive to measure experimentally.