Unsaturated porous media, characterized by the combined presence of several immiscible fluid phases in the pore space, are highly relevant systems in nature, because they control the fate of contaminants and the availability of nutrients in the subsoil. However, a full understanding of the mechanisms controlling solute mixing in such systems is still missing. In particular, the role of saturation in the development of chaotic solute mixing has remained unexplored. Using three-dimensional numerical simulations of flow and transport at the pore scale, built upon X-ray tomograms of a porous medium at different degrees of liquid (wetting)-phase saturation, we show the occurrence of chaotic dynamics in both the deformation of the solute plume, as characterized by computed chaos metrics (Lyapunov exponents), and the mixing of the injected solute. Our results show an enhancement of these chaotic dynamics at lower saturation and their occurrence even under diffusion-relevant conditions over the medium\'s length, also being strengthened by larger flow velocities. These findings highlight the dominant role of the pore-scale spatial heterogeneity of the system, enhanced by the presence of an immiscible phase (e.g., air), on the mixing efficiency. This represents a stepping stone for the assessment of mixing and reactions in unsaturated porous media.

Sea surface temperature (SST), with its complex and dynamic behavior, is a major driver of ocean-atmosphere interactions. The purpose of this study is to investigate the behavior of SST and its prediction using a chaotic approach. Average mutual information (AMI) and Cao methods were used to reconstruct the phase space. The Lyapunov exponent and correlation dimension were used to investigate chaos. The Lyapunov exponent index was used to predict SST with a 5-year average prediction horizon using the local prediction method between 2023 and 2027. The results showed a 3-month delay time for the Pacific and Antarctic Oceans, and a 2-month delay time for the Atlantic, Indian, and Arctic Oceans. The optimal embedding dimension for all oceans is between 6 and 7. Our analysis reveals that the dynamics of SST in all oceans exhibit varying degrees of chaos, as indicated by the correlation dimension. The local prediction method achieves relatively accurate short-term SST predictions due to the clustering of SST points around specific attractors in the phase space. However, in the long term, the accuracy of this method decreases as the points in the phase space of SST can spread randomly. The model performance ranking with a Percent Mean Relative Absolute Error shows that the Indian Ocean has the best performance compared to other oceans, while the Atlantic, Pacific, and Antarctic and Arctic Oceans are in the next ranks. This study contributes to understanding the dynamics of SST and has practical value for use in the development of climate models.

Accurate development of satellite maneuvers necessitates a broad orbital dynamical system and efficient nonlinear control techniques. For achieving the intended formation, a framework of a discrete fractional difference satellite model is constructed by the use of commensurate and non-commensurate orders for the control and synchronization of fractional-order chaotic satellite system. The efficacy of the suggested framework is evaluated employing a numerical simulation of the concerning dynamic systems of motion while taking into account multiple considerations such as Lyapunov exponent research, phase images and bifurcation schematics. With the aid of discrete nabla operators, we monitor the qualitative behavioural patterns of satellite systems in order to provide justification for the structure\'s chaos. We acquire the fixed points of the proposed trajectory. At each fixed point, we calculate the eigenvalue of the satellite system\'s Jacobian matrix and check for zones of instability. The outcomes exhibit a wide range of multifaceted behaviours resulting from the interaction with various fractional-orders in the offered system. Additionally, the sample entropy evaluation is employed in the research to determine complexities and endorse the existence of chaos. To maintain stability and synchronize the system, nonlinear controllers are additionally provided. The study highlights the technique\'s vulnerability to fractional-order factors, resulting in exclusive, changing trends and equilibrium frameworks. Because of its diverse and convoluted behaviour, the satellite chaotic model is an intriguing and crucial subject for research.

In many living organisms displaying circadian rhythms, the intake of energy often occurs in a periodic manner. Glycolysis is a prototypical biochemical reaction that exhibits a self-sustained oscillation under continuous injection of glucose. Here we study the effect of periodic injection of glucose on the glycolytic oscillation from a dynamical systems perspective. In particular, we employ Goldbeter\'s allosteric model of phosphofructokinase as a model system for glycolytic oscillations, and explore the effect of periodic substrate influx of varying frequencies and amplitudes by building the phase diagrams of Lyapunov exponents and oscillatory periods. When the frequency of driving is tuned around the harmonic and sub/super-harmonic conditions of the natural frequency, the system is entrained to a frequency-locked state, forming an entrainment band that broadens with an increasing amplitude of driving. On the other hand, if the amplitude is substantial, the system may transition, albeit infrequent, to a chaotic state which defies prediction of dynamical behaviour. Our study offers in-depth understandings into the controllability of glycolytic oscillation as well as explaining physical underpinnings that enable the synchronous oscillations among a dense population of cells.

Although various measures have been proposed to evaluate dynamic balance during walking, it is currently unclear which measures are most sensitive to dynamic balance. We aimed to investigate which dynamic balance measure is most sensitive to detecting differences in dynamic balance during walking across various gait parameters, including short- and long-term Lyapunov exponents (λs and λl), margin of stability (MOS), distance between the desired and measured centre of pressure (dCOP-mCOP) and whole-body angular momentum (WBAM). A total of 10 healthy young adults were asked to walk on a treadmill under three different conditions (normal walking, dual-task walking with a Stroop task as an unstable walking condition, and arm-restricted walking with arms restricted in front of the chest as another unstable walking condition) that were expected to have different dynamic balance properties. Overall, we found that λs of the centre of mass velocity, λs of the trunk velocity, λs of the hip joint angle, and the magnitude of the mediolateral dCOP-mCOP at heel contact can identify differences between tasks with a high sensitivity. Our findings provide new insights into the selection of sensitive dynamic balance measures during human walking.

Much attention have been devoted to control of chaos in nonlinear system in the last few decades and several control procedures have been derived to find the stability target in difference and differential equations. In this study, a novel hybrid chaos control procedure is derived which allows to stabilize the chaos in most accepted discrete chaotic equations of population growth models about the globally accepted stable equilibrium. Since the system depends on the parameters κ, α, and r, the chaos in the given system may be stabilized in different fixed points states of order p, when it is kicked with the parameter κ. From this point of view, the procedure is simple, flexible, and gives the advantage to take the numerous parameter values to reach the demanded stability in periodic states of order p. This hybrid approach to control makes it novel as compared to existing methods. Further, we provide the geometrical interpretation followed by a few examples, control curves, bifurcation plots, time-series plots, and Lyapunov exponent to illustrate our numerical results.

We present some stability criteria mainly for continuous-time Markovian switching systems, which can provide us with an insight into the mechanism that drives the subsystems to interact with each other so as to achieve stability. On the one hand, directly computing the Lyapunov exponent through the state-transition matrix of the system under consideration allows us to capture the interaction of subsystems by using the method developed in discrete-time setting. On the other hand, we can make sense of how the subsystems are orchestrated by the statistical property of switching signal through a family of coupled Lyapunov inequalities, which naturally covers the stability in the mean-square sense as a special case. Three examples are included to illustrate the theoretical results.

The purpose of this study was to identify how generalized fatigue along with hormonal changes throughout the menstrual cycle affects trunk variability and local dynamic stability during gait.
General fatigue was induced by an incremental test on a treadmill, and the menstrual cycle was divided into three phases: follicular, ovulatory, and luteal. Twenty-six healthy, young volunteers (aged 18 to 28 years) who did not use oral contraceptives or other hormonal drugs with a regular menstrual cycle participated in the study. They walked on the treadmill for 4 min at the preferred speed, before the incremental test, followed by four sets of 4 min alternating between walking, also at preferred speed, and resting. From trunk kinematic data, the following were extracted: the mean of the standard deviation along strides, as a measure of variability, and the maximum Lyapunov exponent, as a measure of local dynamic stability (LDS).
After the incremental test, variability increased, and LDS decreased. However, they showed a tendency to return to the initial value faster in women compared to previous results for men. In the follicular phase, which has less hormonal release, the volunteers had an almost complete recovery in LDS soon after the first rest interval, suggesting that female hormones can interfere with fatigue recovery. Nevertheless, concerning the LDS, it was significantly lower in the luteal phase than in the follicular phase.
Women that are not taking oral contraceptives should be aware that they are susceptible to increased gait instabilities in the pre-menstrual phase after strenuous activities.

Most existing chaotic systems have many drawbacks in engineering applications, such as the discontinuous range of chaotic parameters, weak chaotic properties, uneven chaotic sequence outputs, and dynamic degradation. Therefore, based on the above, this paper proposes a new method for the design of a three-dimensional chaotic map. One can obtain the desired number of positive Lyapunov exponents, and can also obtain the desired value of positive Lyapunov exponents. Simulation results show that the proposed system has complex chaotic behavior and high complexity. Finally, the method is implemented into an image encryption transmission scheme and experimental results show that the proposed image encryption scheme can resist brute force attacks, correlation attacks, and differential attacks, so it has a higher security.

The purpose of this study was to determine the test-retest repeatability of Blue Trident inertial measurement units (IMUs) and VICON Nexus kinematic modelling in analysing the Lyapunov Exponent (LyE) during a maximal effort 4000 m cycling bout in different body segments/joints. An additional aim was to determine if changes in the LyE existed across a trial. Twelve novice cyclists completed four sessions of cycling; one was a familiarisation session to determine a bike fit and become better accustomed to the time trial position and pacing of a 4000 m effort. IMUs were attached to the head, thorax, pelvis and left and right shanks to analyse segment accelerations, respectively, and reflective markers were attached to the participant to analyse neck, thorax, pelvis, hip, knee and ankle segment/joint angular kinematics, respectively. Both the IMU and VICON Nexus test-retest repeatability ranged from poor to excellent at the different sites. In each session, the head and thorax IMU acceleration LyE increased across the bout, whilst pelvic and shank acceleration remained consistent. Differences across sessions were evident in VICON Nexus segment/joint angular kinematics, but no consistent trend existed. The improved reliability and the ability to identify a consistent trend in performance, combined with their improved portability and reduced cost, advocate for the use of IMUs in analysing movement variability in cycling. However, additional research is required to determine the applicability of analysing movement variability during cycling.