Irrespective of achieving certain success in comprehending Passive Dynamic Walking (PDW) phenomena from a viewpoint of the chaotic dynamics and bifurcation scenarios, a lot of questions still need to ...be answered. This paper provides an overview of the previous literature on the chaotic behavior of passive dynamic biped robots. A review of a broad spectrum of chaotic phenomena found in PDW in the past is presented for better understanding of the chaos detection and controlling methods. This paper also indicates that the bulk of literature on PDW robots is focused on locomotion on slope, but there is a thriving trend towards bipedal walking in more challenging environments.
•The chaos research in Passive Dynamic Walking (PDW) is covered in the reviewed literature.•An account of chaos control techniques in PDW bipeds is presented. This area certainly necessitates further investigation.•The need of new mathematical methods is emphasized so that PDW bipeds can be studied analytically.•Potential research directions have been identified.
Stream water temperature is considered both a dominant factor in determining the longitudinal distribution pattern of aquatic biota and as a general metabolic indicator for the water body, since so ...many biological processes are temperature dependent. Moreover, the plunging depth of stream water, its associated pollutant load, and its potential impact on lake/reservoir ecology is dependent on water temperature. Lack of detailed datasets and knowledge on physical processes of the stream system limits the use of a phenomenological model to estimate stream temperature. Rather, empirical models have been used as viable alternatives. In this study, an empirical model (artificial neural networks (ANN)), a statistical model (multiple regression analysis (MRA)), and the chaotic non-linear dynamic algorithms (CNDA) were examined to predict the stream water temperature from the available solar radiation and air temperature. Observed time series data were non-linear and non-Gaussian, thus the method of time delay was applied to form the new dataset that closely represent the inherent system dynamics. Phase-space reconstruction plots show that time lag equal to 0 and greater than 10 result in highly dependent (a well-defined attractor) and highly independent (no attractor at all) reconstructions, respectively, and, therefore, may not be appropriate to use. Delayed vector was found to be strongly correlated with the original vector when time lag is small (i.e. less than 3-day) and vice versa. Power spectrum analysis and autocorrelation function suggested that the time series data was chaotic and mutual information function indicates that optimum time lag was approximately 3-day. The chaotic non-linear dynamic algorithm and four-layer back propagation neural network (4BPNN) optimized by micro-genetic algorithms (μGA) showed that the prediction performance was optimum when data are presented to the model with 1-day and 3-day time lag, respectively. The prediction performance efficiency of MRA is higher for time lag greater than 3-day, however, the incremental performance efficiency rate significantly decreased after 3-day time lag. The prediction performance efficiency of μGA-4BPNN was found to be the highest among all algorithms considered in this study. Air temperature was found to be the most important variable in stream temperature forecasting; however, the prediction performance efficiency was somewhat higher if short wave radiation was included.
The dynamical system studied in previous papers of this series, namely a bound time-like geodesic motion in the exact static and axially symmetric space–time of an (originally) Schwarzschild black ...hole surrounded by a thin disc or ring, is considered to test whether the often employed ‘pseudo-Newtonian’ approach (resorting to Newtonian dynamics in gravitational potentials modified to mimic the black hole field) can reproduce phase-space properties observed in the relativistic treatment. By plotting Poincaré surfaces of section and using two recurrence methods for similar situations as in the relativistic case, we find similar tendencies in the evolution of the phase portrait with parameters (mainly with mass of the disc/ring and with energy of the orbiters), namely those characteristic to weakly non-integrable systems. More specifically, this is true for the Paczyński–Wiita and a newly suggested logarithmic potential, whereas the Nowak–Wagoner potential leads to a different picture. The potentials and the exact relativistic system clearly differ in delimitation of the phase-space domain accessible to a given set of particles, though this mainly affects the chaotic sea whereas not so much the occurrence and succession of discrete dynamical features (resonances). In the pseudo-Newtonian systems, the particular dynamical features generally occur for slightly smaller values of the perturbation parameters than in the relativistic system, so one may say that the pseudo-Newtonian systems are slightly more prone to instability. We also add remarks on numerics (a different code is used than in previous papers), on the resemblance of dependence of the dynamics on perturbing mass and on orbital energy, on the difference between the Newtonian and relativistic Bach–Weyl rings, and on the relation between Poincaré sections and orbital shapes within the meridional plane.
Physics-informed neural networks (PINNs) have recently emerged as an alternative way of numerically solving partial differential equations (PDEs) without the need of building elaborate grids, ...instead, using a straightforward implementation. In particular, in addition to the deep neural network (DNN) for the solution, an auxiliary DNN is considered that represents the residual of the PDE. The residual is then combined with the mismatch in the given data of the solution in order to formulate the loss function. This framework is effective but is lacking uncertainty quantification of the solution due to the inherent randomness in the data or due to the approximation limitations of the DNN architecture. Here, we propose a new method with the objective of endowing the DNN with uncertainty quantification for both sources of uncertainty, i.e., the parametric uncertainty and the approximation uncertainty. We first account for the parametric uncertainty when the parameter in the differential equation is represented as a stochastic process. Multiple DNNs are designed to learn the modal functions of the arbitrary polynomial chaos (aPC) expansion of its solution by using stochastic data from sparse sensors. We can then make predictions from new sensor measurements very efficiently with the trained DNNs. Moreover, we employ dropout to quantify the uncertainty of DNNs in approximating the modal functions. We then design an active learning strategy based on the dropout uncertainty to place new sensors in the domain in order to improve the predictions of DNNs. Several numerical tests are conducted for both the forward and the inverse problems to demonstrate the effectiveness of PINNs combined with uncertainty quantification. This NN-aPC new paradigm of physics-informed deep learning with uncertainty quantification can be readily applied to other types of stochastic PDEs in multi-dimensions.
•Hybridize tent map with deterministic finite state machine to enhance dynamical properties.•New map has high complexity without requiring external entropy source.•New image cipher that achieves ...confusion and diffusion simultaneously in one round.•Cipher has flexible key size depending on user requirement.•Cipher achieves the image authentication.
Image encryption protects visual information by transforming images into an incomprehensible form. Chaotic systems are used to design image ciphers due to properties such as ergodicity and initial condition sensitivity. A chaos-based cipher derives its security strength from its underlying digital chaotic map, thus a more complex map leads to higher security. This paper introduces an enhancement to a tent map’s chaotic properties by hybridizing it with a deterministic finite state machine. We denote the resulting digital one-dimensional chaotic system as TM-DFSM. Chaotic analyses indicate that the new chaotic system has higher nonlinearity, sensitivity to initial condition, and larger chaotic parameter range than other recently proposed one-dimensional chaotic systems. We then propose a new image encryption scheme based on TM-DFSM, capable of performing both confusion and diffusion operations in one pass while also having a flexible key space. The encryption operations are designed to achieve maximal confusion and diffusion properties. Changing a single bit of the plainimage or secret key will result in an entirely different cipherimage. The proposed cipher has been analyzed using histogram analysis, contrast analysis, local Shannon entropy, resistance against differential cryptanalysis, and key security. Performance comparison with other recent schemes also depicts the proposed cipher’s superiority.
In this paper, we study chaos for bounded operators on Banach spaces. First, it is proved that, for a bounded operator
T
defined on a Banach space, Li–Yorke chaos, Li–Yorke sensitivity, ...spatio-temporal chaos, and distributional chaos in a sequence are equivalent, and they are all strictly stronger than sensitivity. Next, we show that
T
is sensitive dependence iff
sup
{
‖
T
n
‖
:
n
∈
N
}
=
∞
. Finally, the following results are obtained: (1)
T
is chaotic iff
T
n
is chaotic for each
n
∈
N
. (2) The product operator
T
n
∗
=
∏
i
=
1
n
T
i
is chaotic iff
T
k
is chaotic for some
k
∈
{
1
,
2
,
…
,
n
}
.
H2 static and dynamic output-feedback control problems are investigated for linear time-invariant uncertain systems. The goal is to minimize the averaged H2 performance in the presence of nonlinear ...dependence on time-invariant probabilistic parametric uncertainties. By applying the polynomial chaos theory, the control synthesis problem is solved using a high-dimensional expanded system which characterizes stochastic state uncertainty propagation. Compared to existing polynomial chaos-based control designs, the proposed approach addresses the simultaneous presence of parametric uncertainties and white noises. The effect of truncation errors due to using finite-degree polynomial chaos expansions is captured by time-varying norm-bounded uncertainties, and is explicitly taken into account by adopting a guaranteed cost control approach. This feature avoids the use of high-degree polynomial chaos expansions to alleviate the destabilizing effect of expansion truncation errors, thus significantly reducing computational complexity. Moreover, rigorous analysis clarifies the condition under which the stability of the high-dimensional expanded system implies the internal mean square stability of the original system under control. Using iterations between synthesis and post-analysis, a bisection algorithm is proposed to find the smallest bounding parameter on the effect of expansion truncation errors such that robust closed-loop stability is guaranteed. A numerical example is used to illustrate the effectiveness of the proposed approach.
Chaos in a linear wave equation Corron, Ned J.
Chaos, Solitons & Fractals: X,
June 2019, 2019-06-00, 2019-06-01, Letnik:
2
Journal Article
Recenzirano
Odprti dostop
•A linear partial differential equation is shown to exhibit three properties often used to define chaotic dynamics.•The linear system is a wave equation with gain that models, for example, the ...optical gain medium in a laser.•This linear system suggests common definitions of chaos are being stressed to, and possibly beyond, their limits.•The model illuminates the significance of linear chaos and the rote application of a definition for chaos.
A linear partial differential equation is shown to exhibit three properties often used to define chaotic dynamics. The system comprises a one-dimensional wave equation with gain that operates on a semi-infinite line. A boundary condition enforces that the waves remain finite. It is shown that the resulting solution set is dense with periodic orbits, contains transitive orbits, and exhibits extreme sensitivity to initial conditions. Definitions of chaos are considered in light of such linear chaos.
This paper focuses on chaos analysis-based adaptive backstepping control of the microelectromechanical resonators. To better understand the deep-rooted operational mechanism, the bifurcation diagram, ...phase diagrams, and corresponding time histories are presented to analyze the nonlinear dynamics and chaotic behavior of the microelectromechanical resonators. Based on the potential function, it can be shown that the microelectromechanical resonators undergo homoclinic and heteroclinic orbits which relate closely to the appearance of chaos in the resonator response. To suppress chaos and vibration, an adaptive neural network-based backstepping scheme is developed to tune the random motion into regular motion without the need for more precise information of system model. The tangent barrier Lyapunov function is used to develop a control scheme for the microelectromechanical resonators capable of preventing output constraint violation. By using tracking differentiator in controller design, the "explosion of complexity" of backstepping and poor precision of the first-order filters is prevented. Meanwhile, to increase the robustness and adaptivity, an adaptive neural network is employed to approximate uncertain nonlinear item in the framework of backstepping. Finally, numerical simulations are conducted to validate the effectiveness and robustness of the proposed approach.
Due to the statistical uncertainty of loads and power sources found in smart grids, effective computational tools for probabilistic load flow analysis and planning are now becoming indispensable. In ...this paper, we describe a unified simulation framework that allows quantifying the probability distributions of a set of observation variables as well as evaluating their sensitivity to potential variations in the power demands. The proposed probabilistic technique relies on the generalized polynomial Chaos algorithm and on a regionwise aggregation/description of the time-varying load profiles. It is shown how detailed statistical distributions of some important figures of merit, which includes voltage unbalance factor in distribution networks, can be calculated with a two orders of magnitude acceleration compared to standard Monte Carlo analysis. In addition, it is highlighted how the associated sensitivity analysis is of guidance for the optimal allocation and planning of new loads.
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