Commentary to “pharmacokinetics from a dynamical systems point of view”

In their pioneering work BUSSE & HEIKES [2] looked at the classical Benard problem in a rotating frame. They observed rolls with a certain axial direction which seem to be stable, ie they remain unchanged for long periods of time, but suddenly the behavior changes: new rolls appear which are rotated with respect to the original rolls by approximately 60 degrees. GUCKENHEIMER & HOLMES [15] looked at the Busse-Heikes problem from a theoretical point of view. They derived a three dimensional ODE exhibiting heteroclinic cycles. Thereafter many papers have dealt with various aspects of cycles: existence, stability, bifurcations and structural stability under certain settings: [1, 3, 4, 5, 7, 8, 16, 17, 18, 20, 21, 22, 25, 23, 24, 26, 27, 28, 29]. Due to the fact that solutions which pass near steady states remain there for a long time heteroclinic cycles serve as a model for metastable behavior. We see such a metastable behavior if we look at the polarity reversals of the magnetic field of the Earth, see for example GHIL & CHILDRESS [13]. Since the origin of the magnetic field and the mechanism of its reversals are unknown we try to look at it from a dynamical systems point of view and ask ourselves whether there are heteroclinic cycles in this problem. Here, we only look at the fluid mechanical part and not at full MHD-equations. In this paper we emphasize the role of rotation and the question what happens to heteroclinic cycles in the presence of rotations. Of course this approach dictates to look at the non rotating case first and treat the rotating case as a perturbation of the non rotating one. It is not clear whether such a approach is reasonable for studying the Earth’ field but on the other hand numerical computations indicate that the region of validity of the results which are presented here exceeds the marginal speeds of rotation allowed by the usual perturbation methods.

Takagi-Sugeno (TS) fuzzy models have been the popular choice for multi-model based nonlinear system identification for controller design. However, the interpretation of the model from a dynamical systems point of view is still unclear. In this paper, we investigate the interpretation of the TS fuzzy models in terms of local dynamics of nonlinear systems in state space. The investigation is based on the comparison between the constant-affine recursive i/o equation and constant-affine state space representations. We show that the true local dynamics is not identifiable by TS fuzzy models by analysis and example. For the identification of constant-affine state space models, a subspace identification algorithm is proposed to identify the true local dynamics with zero initial conditions.

In the present paper, we study a new kind of p-adic measures for q + 1-state Potts model, called p-adic quasi Gibbs measure. For such a model, we derive a recursive relations with respect to boundary conditions. Note that we consider two mode of interactions: ferromagnetic and antiferromagnetic. In both cases, we investigate a phase transition phenomena from the associated dynamical system point of view. Namely, using the derived recursive relations we define a fractional p-adic dynamical system. In ferromagnetic case, we establish that if q is divisible by p, then such a dynamical system has two repelling and one attractive fixed points. We find basin of attraction of the fixed point. This allows us to describe all solutions of the nonlinear recursive equations. Moreover, in that case there exists the strong phase transition. If q is not divisible by p, then the fixed points are neutral, and this yields that the existence of the quasi phase transition. In antiferromagnetic case, there are two attractive fixed points, and we find basins of attraction of both fixed points, and describe solutions of the nonlinear recursive equation. In this case, we prove the existence of a quasi phase transition.

We investigate the effect and the impact of predator-prey interactions, diffusivity and chemotaxis on the ability of survival of multiple consumer levels in a predator-prey microbial food chain. We aim at answering the question of how many consumer levels can survive from a dynamical system point of view. To solve this standing issue on food-chain length, first we construct a chemotactic food chain model. A priori bounds of the steady state populations are obtained. Then under certain sufficient conditions combining the effect of conversion efficiency, diffusivity and chemotaxis parameters, we derive the co-survival of all consumer levels, thus obtaining the food chain length of our model. Numerical simulations not only confirm our theoretical results, but also demonstrate the impact of conversion efficiency, diffusivity and chemotaxis behavior on the survival and stability of various consumer levels.

Uncertainty arises in all stages of the visualization pipeline. However, the majority of flow visualization applications convey no uncertainty information to the user. In tools where uncertainty is conveyed, the focus is generally on data, such as error that stems from numerical methods used to generate a simulation or on uncertainty associated with mapping visualiza-tion primitives to data. Our work is aimed at another source of uncertainty - that associated with user-controlled input param-eters. The navigation and stability analysis of user-parameters has received increasing attention recently. This work presents an investigation of this topic for flow visualization, specifically for three-dimensional streamline and pathline seeding. From a dynamical systems point of view, seeding can be formulated as a predictability problem based on an initial condition. Small perturbations in the initial value may result in large changes in the streamline in regions of high unpredictability. Analyzing this predictability quantifies the perturbation a trajectory is subjugated to by the flow. In other words, some predictions are less certain than others as a function of initial conditions. We introduce novel techniques to visualize important user input parameters such as streamline and pathline seeding position in both space and time, seeding rake position and orientation, and inter-seed spacing. The implementation is based on a metric which quantifies similarity between stream and pathlines. This is important for Computational Fluid Dynamics (CFD) engineers as, even with the variety of seeding strategies available, manual seeding using a rake is ubiquitous. We present methods to quantify and visualize the effects that changes in user-controlled input parameters have on the resulting stream and pathlines. We also present various visualizations to help CFD scientists to intuitively and effectively navigate this parameter space. The reaction from a domain expert in fluid dynamics is also reported.

Flow invariant subspaces for lattice dynamical systems by F. Antoneli, A. P. S. Dias, M. Golubitsky, and Y. Wang Waves in extended systems: From low- to high-dimensional behavior by O. Batiste and E. Knobloch Mixed mode oscillations due to the generalized Canard phenomenon by M. Brons, M. Krupa, and M. Wechselberger Bioremediation of waste in a porous medium by C. Chen and J. Chadam Bifurcations of gyroscopic systems near a 0: 1 resonance by R. J. McDonald, N. S. Namachchivaya, and W. Nagata High dimensional data clustering from a dynamical systems point of view by J. Wu Computation of limit cycles-The second part of Hilbert's 16th problem by P. Yu.

The 120-year old so-called Painleve paradox involves the loss of determinism in models of planar rigid bodies in point contact with a rigid surface, subject to Coulomb-like dry friction. The phenomenon occurs due to coupling between normal and rotational degrees-of-freedom such that the effective normal force becomes attractive rather than repulsive. Despite a rich literature, the forward evolution problem remains unsolved other than in certain restricted cases in 2D with single contact points. Various practical consequences of the theory are revisited, including models for robotic manipulators, and the strange behaviour of chalk when pushed rather than dragged across a blackboard. Reviewing recent theory, a general formulation is proposed, including a Poisson or energetic impact law. The general problem in 2D with a single point of contact is discussed and cases or inconsistency or indeterminacy enumerated. Strategies to resolve the paradox via contact regularisation are discussed from a dynamical systems point of view. By passing to the infinite stiffness limit and allowing impact without collision, inconsistent and indeterminate cases are shown to be resolvable for all open sets of conditions. However, two unavoidable ambiguities that can be reached in finite time are discussed in detail, so called dynamic jam and reverse chatter. A partial review is given of 2D cases with two points of contact showing how a greater complexity of inconsistency and indeterminacy can arise. Extension to fully three-dimensional analysis is briefly considered and shown to lead to further possible singularities. In conclusion, the ubiquity of the Painleve paradox is highlighted and open problems are discussed.

Hysteresis bifurcation and application to delayed FitzHugh-Nagumo neural systems

We construct stable manifolds for a class of singular evolution equations including the steady Boltzmann equation, establishing in the process exponential decay of associated kinetic shock and boundary layers to their limiting equilibrium states. Our analysis is from a classical dynamical systems point of view, but with a number of interesting modifications to accomodate ill-posedness with respect to the Cauchy problem of the underlying evolution equation.

Predicting foreign exchange rates and stock market indices have been a well researched topic in the field of financial engineering. However, most methods suffer from serious drawback due to the inherent uncertainty in the data acquisition process. Here, we have analyzed the very nature of the time series data from a pure dynamic system point of view and explored the deterministic chaotic characteristic in it. In this research, the concept of chaos has been analyzed thoroughly and the relationships among chaos, stability and order have been explained with respect to the concept of time. A method of predicting time series data based on deterministic dynamically system has been presented in this monograph. The present research revolves around the concepts of embedding and fuzzy reconstruction. In this regard, the necessary and sufficient condition for this reconstruction of the state space of the dynamic system in a multi-dimensional Euclidean space has been substantiated in accordance to Theory of embedding. Finally, a fuzzy reconstruction method based on fuzzy multiple regression analysis method has been used to predict the foreign exchange rates with accuracy.

Predicting time series is of great importance in various scientific and engineering fields. However, in the context of limited and noisy data, accurately predicting the dynamics of all variables in a high‐dimensional system is a challenging task due to their nonlinearity and complex interactions. This study introduces the Delayformer framework for simultaneously predicting the dynamics of all variables by developing a novel multivariate spatiotemporal information (mvSTI) transformation that makes each observed variable into a delay‐embedded state (vector) and further cross‐learns those states from different variables. From a dynamical systems viewpoint, Delayformer predicts system states rather than individual variables, thus theoretically and computationally overcoming such nonlinearity and cross‐interaction problems. Specifically, it first utilizes a single shared Visual Transformer (ViT) encoder to cross‐represent dynamical states from observed variables in a delay‐embedded form and then employs distinct linear decoders for predicting next states, i.e., equivalently predicting all original variables in parallel. By leveraging the theoretical foundations of delay embedding theory and the representational capabilities of Transformers, Delayformer outperforms current state‐of‐the‐art methods in forecasting tasks on both synthetic and real‐world datasets. Furthermore, the potential of Delayformer as a foundational time‐series model is demonstrated through cross‐domain forecasting tasks, highlighting its broad applicability across various scenarios.

Equilibria with Many Nuclei for the Cahn–Hilliard Equation

Besides classical feed-forward neural networks such as multilayer perceptrons, also neural ordinary differential equations (neural ODEs) have gained particular interest in recent years. Neural ODEs can be interpreted as an infinite depth limit of feed-forward or residual neural networks. We study the input–output dynamics of finite and infinite depth neural networks with scalar output. In the finite-depth case, the input is a state associated with a finite number of nodes, which maps under multiple non-linear transformations to the state of one output node. In analogy, a neural ODE maps an affine linear transformation of the input to an affine linear transformation of its time- T map. We show that, depending on the specific structure of the network, the input–output map has different properties regarding the existence and regularity of critical points. These properties can be characterized via Morse functions, which are scalar functions where every critical point is non-degenerate. We prove that critical points cannot exist if the dimension of the hidden layer is monotonically decreasing or the dimension of the phase space is smaller than or equal to the input dimension. In the case that critical points exist, we classify their regularity depending on the specific architecture of the network. We show that, except for a Lebesgue measure zero set in the weight space, each critical point is non-degenerate if for finite-depth neural networks, the underlying graph has no bottleneck, and if for neural ODEs, the affine linear transformations used have full rank. For each type of architecture, the proven properties are comparable in the finite and infinite depth cases. The established theorems allow us to formulate results on universal embedding and universal approximation, i.e., on the exact and approximate representation of maps by neural networks and neural ODEs. Our dynamical systems viewpoint on the geometric structure of the input–output map provides a fundamental understanding of why certain architectures perform better than others.

A number of researches have been made for the (Euclidean) Taub-NUT metric, because the geodesic for this metric describes approximately the motion of well separated monopole-monopole interaction. From the viewpoint of dynamical systems, it is well known also that the Taub-NUT metric admits the Kepler-type symmetry, and hence provides a nontrivial generalization of the Kepler problem. More specifically speaking, because of a U(1) symmetry, the geodesic flow system as a Hamiltonian system for the Taub-NUT metric is reduced to a Hamiltonian system which admits a conserved Runge-Lenz-like vector in addition to the angular momentum vector, and thereby whose trajectories turn out to be conic sections. In particular, all the bounded trajectories of the reduced system are closed. The Taub-NUT metrics is generalized so that the reduced system may remain to have the property that all of bounded trajectories are closed. On the application of Bertrand's method to the reduced system, two types of systems are found: one is called the Kepler-type system and the other the harmonic oscillator-type system. Correspondingly, two types of metrics come out: the Kepler-type metric and the harmonic oscillator-type metric. Furthermore, the symmetry of the Kepler-type system and of the harmonic oscillator-type system are studied through forming accidental first integrals. Thus the generalization of the Taub-NUT metric accomplishes nontrivial generalizations of the Kepler problem and the harmonic oscillator.

Dynamics of relations associated with two core decompositions