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Extracting dynamical information from single‐cell transcriptomics is a novel task with the promise to advance our understanding of cell state transition and interactions between genes. Yet, theory‐oriented, bottom‐up approaches that consider differences among cell states are largely lacking. Here, we present spliceJAC, a method to quantify the multivariate mRNA splicing from single‐cell RNA sequencing (scRNA‐seq). spliceJAC utilizes the unspliced and spliced mRNA count matrices to constructs cell state‐specific gene–gene regulatory interactions and applies stability analysis to predict putative driver genes critical to the transitions between cell states. By applying spliceJAC to biological systems including pancreas endothelium development and epithelial–mesenchymal transition (EMT) in A549 lung cancer cells, we predict genes that serve specific signaling roles in different cell states, recover important differentially expressed genes in agreement with pre‐existing analysis, and predict new transition genes that are either exclusive or shared between different cell state transitions. Synopsis spliceJAC builds a multivariate mRNA splicing model from single‐cell transcriptome data to infer the context‐specific gene regulation and the key driver genes that guide the transition between cell states. spliceJAC constructs cell state‐specific gene regulatory networks and quantifies changes in signaling roles between cell states. spliceJAC employs stability analysis to identify driver genes that guide transitions between cell states. Context‐specific gene regulation and transition genes are identified using spliceJAC during pancreas endothelium development and epithelial–mesenchymal transition (EMT) in A549 lung cancer cells. Graphical Abstract spliceJAC builds a multivariate mRNA splicing model from single‐cell transcriptome data to infer the context‐specific gene regulation and the key driver genes that guide the transition between cell states.

In recent years, tremendous research efforts have been devoted to simple chaotic oscillators based on jerk equation that involves a third-time derivative of a single variable. In the present paper, we perform a systematic analysis of a simple autonomous jerk system with cubic nonlinearity. The system is a linear transformation of Model MO5 first introduced in Sprott (Elegant chaos: algebraically simple flow. World Scientific Publishing, Singapore, 2010 ) prior to the more detailed study by Louodop et al. (Nonlinear Dyn 78:597–607, 2014 ). The basic dynamical properties of the model are investigated including equilibria and stability, phase portraits, frequency spectra, bifurcation diagrams, and Lyapunov exponent plots. It is shown that the onset of chaos is achieved via the classical period-doubling and symmetry-restoring crisis scenarios. One of the key contributions of this work is the finding of a window in the parameter space in which the jerk system experiences the unusual and striking feature of multiple attractors (e.g. coexistence of four disconnected periodic and chaotic attractors). Basins of attraction of various coexisting attractors are computed showing complex basin boundaries. Among the very few cases of lower-dimensional systems (e.g. Newton–Leipnik system) capable of displaying such type of behaviour reported to date, the jerk system with cubic nonlinearity considered in this work represents the simplest and the most ‘elegant’ prototype. An appropriate electronic circuit describing the jerk system is designed and used for the investigations. Results of theoretical analyses are perfectly traced by laboratory experimental measurements.

Linear time-delay feedback method makes the stable system generate the infinite-dimensional hyper-chaos, which possesses more than one positive Lyapunov exponent, infinite-dimensional, a wider chaotic parameter range, and multi-attractors, including the single-scroll attractor, the double-scroll attractor, and the composite multi-scroll attractor. The infinite-dimensional hyper-chaotic multi-attractors Chen system generated by linear time-delay feedback (HCMACS) is used for image encryption. Firstly, the state time sequences of the infinite-dimensional HCMACS are preprocessed to achieve the ideal statistical property. Secondly, two random matrices generated by random number generators are used to expand the image size. Finally, the preprocessed chaotic sequences are used to confuse and diffuse the expanded digital image to obtain the encrypted image. The special feature of the proposed method is the theoretically infinite-dimensional secret key space. Results of analysis and computer simulation indicate that the encryption algorithm has good encryption performance, which can effectively resist various attacks.

The threat of global warming and the demand for reliable climate predictions pose a formidable challenge because the climate system is multiscale, high-dimensional and nonlinear. Spatiotemporal recurrences of the system hint to the presence of a low-dimensional manifold containing the high-dimensional climate trajectory that could make the problem more tractable. Here we argue that reproducing the geometrical and topological properties of the low-dimensional attractor should be a key target for models used in climate projections. In doing so, we propose a general data-driven framework to characterize the climate attractor and showcase it in the tropical Pacific Ocean using a reanalysis as observational proxy and two state-of-the-art models. The analysis spans four variables simultaneously over the periods 1979–2019 and 2060–2100. At each timet, the system can be uniquely described by a state space vector parametrized byNvariables and their spatial variability. The dynamics is confined on a manifold with dimension lower than the full state space that we characterize through manifold learning algorithms, both linear and nonlinear. Nonlinear algorithms describe the attractor through fewer components than linear ones by considering its curved geometry, allowing for visualizing the high-dimensional dynamics through low-dimensional projections. The local geometry and local stability of the high-dimensional, multivariable climate attractor are quantified through the local dimension and persistence metrics. Model biases that hamper climate predictability are identified and found to be similar in the multivariate attractor of the two models during the historical period while diverging under the warming scenario considered. Finally, the relationships between different subspaces (univariate fields), and therefore among climate variables, are evaluated. The proposed framework provides a comprehensive, physically based, test for assessing climate feedbacks and opens new avenues for improving their model representation.

In this study, we analyze the state space of a Boolean network modeling diphtheria pathogenesis, focusing on key genes such as Tox, Rep, INF1/INF2, TLR, AP1, IL6, and TNF. We introduce targeted perturbations to reveal how the network responds and converges to its attractors. Our approach utilizes semi-tensor product techniques and permutation methods to recast the Boolean dynamics into a linear algebraic scheme, enabling efficient identification of transient states, stable attractors, and Garden-of-Eden states. This work fills an important gap by clarifying how specific gene interactions drive the network toward non-pathogenic states. Our results show that altering regulatory relationships, particularly those between Rep, Tox, and interferon signals, significantly influences basin sizes and attractor stability, thereby enhancing our understanding of the network's resilience and informing potential therapeutic strategies.

We study the non-autonomously forced Burgers equation $$u_t(x,t) + u(x,t)u_x(x,t)-u_{xx}(x,t) = f(x,t)$$ on the space interval (0, 1) with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique H1 bounded trajectory of this equation defined for all t ∈ ℝ. Moreover we demonstrate that this trajectory attracts all trajectories both in pullback and forward sense. We also prove that for the Dirichlet case this attraction is exponential.

From the perspective of dynamic systems theory, stability and variability of biological signals are both understood as a functional adaptation to variable environmental conditions. In the present study, we examined whether this theoretical perspective is applicable to the pedalling movement in cycling. Non-linear measures were applied to analyse pedalling forces with varying levels of subjective load. Ten subjects completed a 13-sector virtual terrain profile of 15 km total length on a roller trainer with varying degrees of virtual terrain inclination (resistance). The test was repeated two times with different instructions on how to alter the bikes gearing. During the experiment, pedalling force and heart rate were measured. Force-time curves were sequenced into single cycles, linearly interpolated in the time domain, and z-score normalised. The established time series was transferred into a two-dimensional phase space with limit cycle properties given the applied 25% phase shift. Different representations of the phase space attractor were calculated within each sector and used as non-linear measures assessing pedalling forces. A contrast analysis showed that changes in pedalling load were strongly associated to changes in non-linear phase space attractor variables. For the subjects investigated in this study, this association was stronger than that between heart rate and resistance level. The results indicate systematic changes of the pedalling movement as an adaptive response to an externally determined increase in workload. Future research may utilise the findings from this study to investigate possible relationships between subjective measures of exhaustion, comfort, and discomfort with biomechanic characteristics of the pedalling movement and to evaluate connections with dynamic stability measures.

In this paper, a high dimensional double rotor model is proposed. We establish its dynamic equations, and simply it into a four-dimensional mapping form. The bifurcations of the double rotor mapping under different control parameters are investigated. The chaotic dynamic behavior of the model is controlled by improving the pole assignment method. With the linear control theory, a control parameter is selected and the period-1 is chosen as control target. When the mapping point wanders to the neighborhood of the periodic point, the control parameter is perturbed. The unstable period-1 orbit is controlled to be a stable periodic orbit. Numerical simulations are consistent with the theoretical analysis. The results of this research show that this chaos control method can be applied to the 4-dimensional model and can be realized.The research results indicate when the selected regulator poles are different, the control times are different.

In this paper, a three-layer Rao–Nakra sandwich beam is considered where the core viscoelastic layer is constrained by the purely elastic or piezoelectric outer layers. In the model, uniform bending motions of the overall laminate are coupled to the longitudinal motions of the outer layers, and the shear of the middle layer contributes to the overall motion. Together with nonlinear damping injection and nonlinear source terms, the existence and uniqueness of local and global weak solutions are obtained by the nonlinear semigroup theory and the theory of monotone operators. The global existence of potential well solutions and the uniform energy decay rates of such a solution, given as a solution to a certain nonlinear ODE, are shown are proved under certain assumptions of the parameters and by the Nehari manifold. Finally, the existence of a smooth global attractor with finite fractal dimension, which is characterized as an unstable manifold of the set of stationary solutions, and exponential attractors for the associated dynamical system are proved. The present paper extends the linear analysis of the stability of the Rao–Nakra sandwich beam to nonlinear analysis in the existing literature.

We present an approach to generate multiscroll attractors via destabilization of piecewise linear systems based on Hurwitz matrix in this paper. First we present some results about the abscissa of stability of characteristic polynomials from linear differential equations systems; that is, we consider Hurwitz polynomials. The starting point is the Gauss–Lucas theorem, we provide lower bounds for Hurwitz polynomials, and by successively decreasing the order of the derivative of the Hurwitz polynomial one obtains a sequence of lower bounds. The results are extended in a straightforward way to interval polynomials; then we apply the abscissa as a measure to destabilize Hurwitz polynomial for the generation of a family of multiscroll attractors based on a class of unstable dissipative systems (UDS) of affine linear type.