Unstable Node Research Articles

Novel routes to chaos through torus breakdown in non-invertible maps

The paper describes a number of new scenarios for the transition to chaos through the formation and destruction of multilayered tori in non-invertible maps. By means of detailed, numerically calculated phase portraits we first describe how three- and five-layered tori arise through period-doubling and/or pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. We then describe several different mechanisms for the destruction of five-layered tori in a system of two linearly coupled logistic maps. One of these scenarios involves the destruction of the two intermediate layers of the five-layered torus through the transformation of two unstable node cycles into unstable focus cycles, followed by a saddle-node bifurcation that destroys the middle layer and a pair of simultaneous homoclinic bifurcations that produce two invariant closed curves with quasiperiodic dynamics along the sides of the chaotic set. Other scenarios involve different combinations of local and global bifurcations, including bifurcations that lead to various forms of homoclinic and heteroclinic tangles. We finally demonstrate that essentially the same scenarios can be observed both for a system of nonlinearly coupled logistic maps and for a couple of two-dimensional non-invertible maps that have previously been used to study the properties of invariant sets.

Bifurcation delay - the case of the sequence: Stable focus - unstable focus - unstable node

We consider a two dimensional family of real vector fields. We suppose that there exists a stationary point where the linearized vector field has successively a stable focus, an unstable focus and an unstable node. It is known that when the parameter moves slowly, a bifurcation delay appears due to the Hopf bifurcation. The main question investigated in this article is the continuation of the delay in the region of the unstable node. Another problem is to determine the input-output relation which characterizes all the possible delays.

Formation and destruction of multilayered tori in coupled map systems

The paper first illustrates how multilayered tori can arise through one or more pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. The paper hereafter describes three different scenarios by which a multilayered torus can be destructed. One scenario involves a saddle-node bifurcation in which the middle layer of a three-layered torus disappears in an abrupt transition to chaos while the outer-layer manifolds and their associated saddle and unstable-focus cycles continue to exist and to control the transient dynamics. In a second scenario, the unstable focus cycles of the intermediate layers in a five-layered torus turn into unstable nodes, and closed loop connections are established between the unstable nodes and the points of the stable resonance node on the torus. Finally, a third scenario describes a transition in which homoclinic bifurcations destroy first the outer layers and thereafter also the inner layer. The paper also illustrates how the formation and destruction of multilayered tori can occur in the cluster dynamics of an ensemble of globally coupled maps. This leads to three additional scenarios for the destruction of multilayered tori.

Extended quintessence with nonminimally coupled phantom scalar field

We investigate evolutional paths of an extended quintessence with a nonminimally coupled phantom scalar field $\ensuremath{\psi}$ to the Ricci curvature. The dynamical system methods are used to investigate typical regimes of dynamics at the late time. We demonstrate that there are two generic types of evolutional scenarios which approach the attractor (a focus or a node type critical point) in the phase space: the quasioscillatory and monotonic trajectories approach the attractor which represents the Friedmann-Robertson-Walker model with the cosmological constant. We demonstrate that the dynamical system admits an invariant two-dimensional submanifold and discuss that which cosmological scenario is realized depends on the behavior of the system on the phase plane $(\ensuremath{\psi},{\ensuremath{\psi}}^{\ensuremath{'}})$. We formulate simple conditions on the value of the coupling constant $\ensuremath{\xi}$ for which trajectories tend to the focus in the phase plane and hence damping oscillations around the mysterious value $w=\ensuremath{-}1$. We describe this condition in terms of slow-roll parameters calculated at the critical point. We discover that the generic trajectories in the focus-attractor scenario come from the unstable node. We also investigate the exact form of the parametrization of the equation of state parameter $w(z)$ (directly determined from dynamics) which assumes a different form for both scenarios.

Phantom field dynamics in loop quantum cosmology

We consider a dynamical system of phantom scalar field under exponential potential in the background of loop quantum cosmology. In our analysis, there is neither stable node nor repeller unstable node but only two saddle points, hence no big rip singularity. Physical solutions always possess potential energy greater than the magnitude of the negative kinetic energy. We found that the universe bounces after accelerating even in the domination of the phantom field. After bouncing, the universe finally enters the oscillatory regime.

The geometry of separation boundaries. II. Mathematical formalism

AbstractThis article is concerned with the mathematical proof that distillation boundaries correspond to locally longest residue curves. Perturbation theory and the implicit function theorem are used to provide an analytical relationship between neighboring trajectories while local congruence and a variety of limiting arguments are used to establish the relative distances between neighboring residue curves. The mathematical conditions on which this analysis is based include (1) The assumptions that trajectories are twice continuously differentiable and do not cross. (2) The comparison of neighboring trajectories that converge to the same stable node. (3) The existence and placement of a proper focal point. (4) Local congruence. While all of these assumptions and conditions are important, it is the existence and proper placement of a focal point and local congruence that are the keys to establishing relative distances between neighboring residue curves and, in turn, the main result that distillation boundaries are defined by locally longest trajectories from any unstable node to all reachable stable nodes. Wherever possible, geometric illustrations are used to clarify key concepts. © 2007 American Institute of Chemical Engineers AIChE J, 2007

Preservation of local dynamics when applying central difference methods: application to SIR model

We apply the central difference method (u t+1 − u t − 1)/(2Δt) = f(u t ) to an epidemic SIR model and show how the local stability of the equilibria is changed after applying the numerical method. The above central difference scheme can be used as a numerical method to produce a discrete-time model that possesses interesting local dynamics which appears inconsistent with the continuous model. Any fixed point of a differential equation will become an unstable saddle node after applying this method. Two other implicitly defined central difference methods are also discussed here. These two methods are more efficient for preserving the local stability of the fixed points for the continuous models. We apply conformal mapping theory in complex analysis to verify the local stability results.

Energy avalanches in the central plasma sheet

The central plasma sheet (CPS) is simulated as a 1D cellular automaton. The system is driven deterministically and globally by a spatially non‐uniform energy loading (convection). Each node (a flux tube) evolves until one of two local instability criteria is exceeded. The unstable node releases a small amount of energy to the ionosphere and another small amount is distributed to its neighboring nodes. The partition between the two modes of energy distribution is the only random factor in the model. The energy redistribution relaxes the node deterministically to a stable state. The simulation suggests that a central plasma sheet driven in the above manner is in a self‐organized critical state, with energy avalanches obeying a scale‐free distribution. The avalanches, however, co‐exist with quasi‐periodic intermittencies manifested in ring‐current injection, which is correlated with strong CPS avalanches, and tailward energy ejection, which shows no apparent correlation in this aspect.

Azeotropic behaviour of (benzene + cyclohexane + chlorobenzene) ternary mixture using chlorobenzene as entrainer at 101.3 kPa

In this paper, the azeotropic behaviour of the (benzene + cyclohexane + chlorobenzene) ternary mixture was experimentally investigated with the aim of enhancing the knowledge for the feasible use of chlorobenzene as an entrainer for the azeotropic distillation of the binary azeotrope. Such a study has not been reported in the literature to the best of the authors’ knowledge. (Vapour + liquid) equilibria data for (benzene + cyclohexane + chlorobenzene) at 101.3 kPa were obtained with a Othmer-type ebulliometer. Data were tested and considered thermodynamically consistent. The experimental results showed that this ternary mixture is completely miscible and exhibits an unique binary homogeneous azeotrope, an unstable node at the conditions studied, and the propitious topological characteristics (residual curve map and relative volatility) to be separated. Satisfactory results were obtained for the correlation of equilibrium compositions with the UNIQUAC activity coefficients model and also for prediction with the UNIFAC method. In both cases, low root mean square deviations of the vapour mole fraction and temperature were calculated. The capability of chlorobenzene as a modified distillation agent at atmospheric condition is discussed in terms of the thermodynamic topological analysis. A conceptual distillation scheme with reversed volatility is proposed to separate the azeotropic mixture. In order to reduce the operational cost requirements of the sequence of columns proposed, the range for optimal reflux and the ratio for feed flow conditions were studied.

Study on the feasibility of split crossing distillation compartment boundary

This paper proposes a graphical method for analyzing the feasibility of splits crossing distillation compartment boundaries. In this method, the univolatility curve, which reflects the inflexions of residue curves, and the volatility order diagram, are employed. The analysis shows that, for ternary system with one binary minimum azeotrope, splits can only cross the distillation compartment boundary in one direction. For a given feed composition, the feasible product compositions lie in a small region bounded by two straight lines: one passes through the stable (or unstable) node and feed point; the other is the transition line passing through the feed point and tangential to the feed residue curve. Aspen Plus simulation is used to check the feasibility of splits and product composition regions. The results are in good agreement with the analysis. The proposed method is simpler than the existing residue curve map based tools but does effectively predict feasibility with the same efficiency. It can only apply to ternary mixtures at present.

Geometry of Separation Boundaries: Systems with Reaction

Residue curves and residue curve maps have been extensively studied for over 100 years. Much of the literature in this field deals with systems that do not react. Chemical reactions can influence residue curve maps in some important ways. For example, it is known that reactions can lead to both the appearance and disappearance of stationary points (azeotropes), and that reactive azeotropes can exist even in systems that otherwise would be considered thermodynamically ideal. It follows that chemical reactions can influence the very existence of separation boundaries and, therefore, the design and synthesis of reactive separation processes. Lucia and Taylor (AIChE J. 2006, 52, 582) have shown that distillation boundaries in nonreacting ternary liquid mixtures can be defined as local maxima in the line integral from any unstable node to all reachable stable nodes. This paper shows how this geometric approach can be adapted for the rigorous determination of separation boundaries in reacting mixtures. Evidence...

Feasible products in complex batch reactive distillation

AbstractA new feasibility evaluation procedure for reactive mixtures is presented, where simple homogeneous reactive batch columns cannot produce pure products. Such columns are not feasible for producing pure products if no node products exist that are reachable by residue curves from the reaction equilibrium manifold. Three alternatives are presented. If an unstable node heterogenous azeotrope decants to an almost‐pure product, and that azeotrope is always reachable from the reaction equilibrium manifold, then a batch reactive rectifier can produce pure products with a decanter. If a homogeneous entrainer allows extractive section profiles to connect the reaction equilibrium manifold to an entrainer‐product binary edge, then a homogeneous batch reactive extractive distillation (BRED) column can produce a pure product. If these criteria are not met, then an entrainer that induces an unstable node heterogenous azeotrope between itself and one of the products should be used in a rectifier, a middle‐vessel column, or a BRED column. © 2006 American Institute of Chemical Engineers AIChE J, 2006

FRONTIER DIAGRAMS: PARTITION OF PHASE SPACE ACCORDING TO THE SIGNS OF EIGENVALUES OR SIGN PATTERNS OF THE CIRCUITS

Nonlinear dynamic systems can be considered in terms of feedback circuits (for short, circuits), which are circular interactions between variables. Each circuit can be identified without ambiguity from the Jacobian matrix of the system. Of special relevance are those circuits (or unions of disjoint circuits) that involve all the variables of the system. We call them "nuclei" because, in the same way as, in Biology, the cell nucleus contains essential genetic information, in nonlinear dynamics, nuclei are crucial elements in the genesis of steady states. Indeed, each nucleus taken alone can generate one or more steady states, whose nature is determined by the sign patterns of the nucleus. There can be up to two nuclei in 2D systems, six in 3D systems, … n! in nD systems. However, many interesting systems of high dimensionality have only two, sometimes even one, nucleus. This paper is based on an extensive exploitation of the Jacobian matrix of systems, in order to figure the global structure of phase space. In nonlinear systems, the value, and often the sign, of terms of the Jacobian matrix depend on the location in phase space. In contrast with a current usage, we consider this matrix, as well as its eigenvalues and eigenvectors, not only in close vicinity of steady states of the system, but also everywhere in phase space. Two distinct, but complementary approaches are used here. In Sec. 2, we define frontiers that partition phase space according to the signs of the eigenvalues (or of their real part if they are complex), and where required, to the slopes of the eigenvectors. From then on, the exact nature of any steady state that might be present in a domain can be identified on the sole basis of its location in that domain and, in addition, one has at least an idea of the possible number of steady states. In Sec. 3, we use a more qualitative approach based on the theory of circuits. Here, phase space is partitioned according to the sign patterns of elements of the Jacobian matrix, more specifically, of the nuclei. We feel that this approach is more generic than that described in Sec. 2. Indeed, it provides a global view of the structure of phase space, and thus permits to infer much of the dynamics of the system by a simple analysis of sign patterns within the Jacobian matrix. This approach also turned out to be extremely useful for synthesizing systems with preconceived properties. For a wide variety of systems, once the partition process has been achieved, each domain comprises at most one steady state. We found, however, a family of systems in which two steady states (typically, two stable or two unstable nodes) differ by neither of the criteria we use, and are thus not separated by our partition processes. This counter-example implies non-polynomial functions. It is essential to realize that the frontier diagrams permit a reduction of the dimensionality of the analysis, because only those variables that are involved in nonlinearities are relevant for the partition process. Whatever the number of variables of a system, its frontier diagram can be drawn in two dimensions whenever no more than two variables are involved in nonlinearities. Pre-existing conjectures concerning the necessary conditions for multistationarity are discussed in terms of the partition processes.

Theoretical and experimental study on residue curve maps of propyl acetate synthesis reaction

Residue curve maps (RCMs) of propyl acetate synthesis reaction in the batch reactive distillation process are studied. In order to adapt the model equations of residue curve maps to a practicable heating policy, the theoretical analysis and experimental measurements in this paper are carried out isothermally instead of the autonomous heat policy first introduced by Venimadhavan et al. (A.I.Ch.E. Journal 40 (1994) 1814–1824). The chemical equilibrium constant of this reaction is determined by experiments to be 20 within the temperature range 80–110 °C. Using this equilibrium constant, the RCMs predicted by simulation are in good agreement with the experimental measurements. The results show that there is an unstable node branch emerging from the propyl acetate–water edge, moving toward the chemical equilibrium surface with the increasing Damköhler number ( Da), and eventually reaching the quaternary reactive azeotrope when Da → ∞ . Residue curves are measured with initial compositions around the unstable node, and thus the results verify the existence of this reactive azeotrope. Further bifurcation analysis shows that different heat policies will influence the singular points and topology of kinetically controlled RCMs, but not the cases when Da = 0 or Da → ∞ .

Feasible products in batch reactive extractive distillation

AbstractA systematic feasibility evaluation method is presented for batch reactive distillation columns with extractive feed streams. In this batch configuration, one feasibility criterion is obtained for quaternary reactive systems by using residue curve maps and reaction equilibrium manifolds. If all reaction products are saddles and one of the binary azeotropes between the desired products and the other components is an unstable node, then a reactive rectifier with a side‐feed entrainer can produce pure products regardless of the type and number of azeotropes involved in the system. © 2004 American Institute of Chemical Engineers AIChE J, 50: 1484–1492, 2004

Feasibility of Continuous Reactive Distillation with Azeotropic Mixtures

By using phase and reaction equilibrium information, the feasibility criteria for single-feed and double-feed reactive distillation systems are derived: When the products are a stable node and an unstable node and a part of the reaction equilibrium curve lies in the same distillation region as the products, both reactive rectifying and reactive stripping sections can feasibly produce pure products in a single-feed column. If one of the products is a saddle and no azeotrope exists between the products, a single-feed column is still feasible, but only with large reaction equilibrium constants. If a saddle product forms a minimum-boiling azeotrope with another product, an entrainer is needed to break this azeotrope in a double-feed column. If both reaction products are saddles, the same double-feed column configuration with an entrainer can produce pure products.

Adaptive stabilization of unstable steady states of nonlinear dynamical systems

An adaptive dynamic state feedback controller for stabilizing and tracking unknown steady states of dynamical systems is considered. We prove that the steady state can never be stabilized if the system and controller as a whole has an odd number of real positive eigenvalues. For two-dimensional systems, this topological limitation states that only an unstable focus or node can be stabilized with a stable controller, and a stabilization of a saddle requires the presence of an unstable degree of freedom in a feedback loop. The use of the controller to stabilize saddles and unstable spirals is demonstrated numerically with several models: a pendulum driven with a constant torque, the Lorenz system, and an electrochemical Ni dissolution system.

Feasible products in batch reactive distillation

AbstractA systematic feasibility evaluation for batch reactive distillation systems is addressed. In each batch configuration, feasibility criteria are obtained for ternary reactive systems by using residual curve maps (RCMs) and reaction equilibrium manifolds. If one of the reaction products is an unstable node (UN) and is reachable from all distillation regions, a rectifier can be used for producing pure products. Even if this unstable node is only reachable from one specific distillation region, a rectifier can still be used when a part of the reaction equilibrium curve lies within the same distillation region as the product. Symmetric results are conserved for a batch stripper with one stable node (SN) product. Finally, pure products are produced in middle‐vessel columns when the UN and SN products share the same distillation region with a part of the reaction equilibrium curve. Although these criteria are obtained for ternary systems, they can be extended to multicomponent reactive systems.

Stabilizing and Tracking Unknown Steady States of Dynamical Systems

An adaptive dynamic state feedback controller for stabilizing and tracking unknown steady states of dynamical systems is proposed. We prove that the steady state can never be stabilized if the system and controller in sum have an odd number of real positive eigenvalues. For two-dimensional systems, this topological limitation states that only an unstable focus or node can be stabilized with a stable controller, and stabilization of a saddle requires the presence of an unstable degree of freedom in a feedback loop. The use of the controller to stabilize and track saddle points (as well as unstable foci) is demonstrated both numerically and experimentally with an electrochemical Ni dissolution system.

Stochastic vs. Deterministic Modeling of Intracellular Viral Kinetics

Within its host cell, a complex coupling of transcription, translation, genome replication, assembly, and virus release processes determines the growth rate of a virus. Mathematical models that account for these processes can provide insights into the understanding as to how the overall growth cycle depends on its constituent reactions. Deterministic models based on ordinary differential equations can capture essential relationships among virus constituents. However, an infection may be initiated by a single virus particle that delivers its genome, a single molecule of DNA or RNA, to its host cell. Under such conditions, a stochastic model that allows for inherent fluctuations in the levels of viral constituents may yield qualitatively different behavior. To compare modeling approaches, we developed a simple model of the intracellular kinetics of a generic virus, which could be implemented deterministically or stochastically. The model accounted for reactions that synthesized and depleted viral nucleic acids and structural proteins. Linear stability analysis of the deterministic model showed the existence of two nodes, one stable and one unstable. Individual stochastic simulation runs could access and remain at the unstable node. In addition, deterministic and averaged stochastic simulations yielded different transient kinetics and different steady-state levels of viral components, particularly for low multiplicities of infection (MOI), where few virus particles initiate the infection. Furthermore, a bimodal population distribution of viral components was observed for low MOI stochastic simulations. The existence of a low-level infected subpopulation of cells, which could act as a viral reservoir, suggested a potential mechanism of viral persistence.