THE RÖSSLER EQUATIONS REVISITED IN TERMS OF FEEDBACK CIRCUITS

1999 ◽  
Vol 07 (02) ◽  
pp. 225-237 ◽  
Author(s):  
R. THOMAS
Keyword(s):  
Steady State ◽  
Chaotic Dynamics ◽  
Jacobian Matrix ◽  
Complex Dynamics ◽  
Deterministic Chaos ◽  
Logical Structure ◽  
Steady States ◽  
Global Behavior ◽  
Feedback Circuits ◽  
By Products

We would like to show that complex dynamics can be deciphered and at least partly understood in terms of its underlying logical structure, more specifically in terms of the feedback circuits built in the differential equations. This approach has permitted to build a number of new three- and four-variable systems displaying chaotic dynamics. Starting from the well-known Rössler equations for deterministic chaos, one asks how systems with the same types of steady states can be synthesized ab initio from appropriate feedback circuits (= appropriate logical structure). It is found that, granted an appropriate logical structure, the existence of a domain of chaotic dynamics is remarkably robust towards changes in the nature of the nonlinearity used, and towards those sign changes which respect the nature (positive vs negative) of the feedback circuits. Using logical arguments, it was also easy to find related systems with a single steady state. A variety of 3- and 4-d systems based on other combinations of feedback circuits and generating chaotic dynamics are described. The aim of this work is to contribute to a better understanding of the respective roles of feedback circuits and nonlinearity — both essential — in so-called "non trivial behavior", including deterministic chaos. Special emphasis is put on the interest of using the Jacobian matrix and its by-products (characteristic equation, eigenvalues, …) not only close to steady states (where linear stability analysis can be performed) but also elsewhere in phase space, where precious indications about the global behavior can be collected.

DETERMINISTIC CHAOS SEEN IN TERMS OF FEEDBACK CIRCUITS: ANALYSIS, SYNTHESIS, "LABYRINTH CHAOS"

1999 ◽  
Vol 09 (10) ◽  
pp. 1889-1905 ◽  
Author(s):  
R. THOMAS
Keyword(s):  
Dynamic Systems ◽  
Nonlinear Dynamic ◽  
Jacobian Matrix ◽  
Deterministic Chaos ◽  
Steady States ◽  
Nonlinear Dynamic Systems ◽  
Limit Case ◽  
Differential Systems ◽  
Feedback Circuits ◽  
Unstable Steady States

This paper aims to show how complex nonlinear dynamic systems can be classified, analyzed and synthesized in terms of feedback circuits. The Rössler equations for deterministic chaos are revisited and generalized in this perspective. It is shown that once a proper set of feedback circuits is present in the Jacobian matrix of the system, the chaotic character of trajectories is remarkably robust versus changes in the nature of the nonlinearities. "Labyrinth chaos", whereby simple differential systems generate large lattices of many unstable steady states embedded in a chaotic attractor, is constructed using this technique. In the limit case of a single three-element circuit without diagonal elements, one finds systems possessing an infinite lattice of unstable steady states between which trajectories percolate in a deterministic chaotic way.

NULLCLINES AND NULLCLINE INTERSECTIONS

2006 ◽  
Vol 16 (10) ◽  
pp. 3023-3033 ◽  
Author(s):  
RENÉ THOMAS
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Fixed Points ◽  
Ordinary Differential Equations ◽  
Chaotic Dynamics ◽  
Jacobian Matrix ◽  
Steady States ◽  
Sustained Oscillations ◽  
Sign Patterns

One purpose of this paper is to document the fact that, in dynamical systems described by ordinary differential equations, the trajectories can be organized not only around fixed points (steady states), but also around lines. In 2D, these lines are the nullclines themselves, in 3D, the intersections of the nullclines two by two, etc.We precise the concepts of "partial steady states" (i.e. steady states in a subsystem that consists of sections of phase space by planes normal to one of the axes) and of "partial multistationarity" (multistationarity in such a subsystem).Steady states, nullclines or their intersections are revisited in terms of circuits, defined from nonzero elements of the Jacobian matrix. It is shown how the mere examination of the Jacobian matrix and the sign patterns of its circuits can help interpreting (and often predicting) aspects of the dynamics of systems.The results reinforce the idea that chaotic dynamics requires both a positive circuit, to provide (if only partial) multistationarity, and a negative circuit, to provide sustained oscillations. As shown elsewhere, a single circuit may suffice if it is ambiguous (i.e. positive or negative depending on the location in phase space).The description in terms of circuits is by no means exclusive of the classical description. In many cases, a fruitful approach involves repeated feedback between the two viewpoints.

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

2005 ◽  
Vol 15 (10) ◽  
pp. 3051-3074 ◽  
Author(s):  
RENÉ THOMAS ◽  
MARCELLE KAUFMAN
Keyword(s):  
Steady State ◽  
Phase Space ◽  
Jacobian Matrix ◽  
Two Dimensions ◽  
Steady States ◽  
Exact Nature ◽  
Counter Example ◽  
Current Usage ◽  
Sign Patterns ◽  
Partition Process

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.

POVERTY TRAPS AND INFERIOR GOODS IN A DYNAMIC HECKSCHER–OHLIN MODEL

2012 ◽  
Vol 17 (6) ◽  
pp. 1227-1251 ◽  
Author(s):  
Eric W. Bond ◽  
Kazumichi Iwasa ◽  
Kazuo Nishimura
Keyword(s):  
Economic Theory ◽  
Steady State ◽  
World Economy ◽  
Steady States ◽  
The Other ◽  
Poverty Traps ◽  
Poverty Trap ◽  
Multiple Steady States ◽  
The World ◽  
State Capital

We extend the dynamic Heckscher–Ohlin model in Bond et al. [Economic Theory(48, 171–204, 2011)] and show that if the labor-intensive good is inferior, then there may exist multiple steady states in autarky and poverty traps can arise. Poverty traps for the world economy, in the form of Pareto-dominated steady states, are also shown to exist. We show that the opening of trade can have the effect of pulling the initially poorer country out of a poverty trap, with both countries having steady state capital stocks exceeding the autarky level. However, trade can also pull an initially richer country into a poverty trap. These possibilities are a sharp contrast with dynamic Heckscher–Ohlin models with normality in consumption, where the country with the larger (smaller) capital stock than the other will reach a steady state where the level of welfare is higher (lower) than in the autarkic steady state.

DETERMINISTIC CHAOS IN AN INTERPHASE LAYER OF A LIQUID–VAPOR SYSTEM

2004 ◽  
Vol 14 (10) ◽  
pp. 3671-3678
Author(s):  
G. P. BYSTRAI ◽  
S. I. IVANOVA ◽  
S. I. STUDENOK
Keyword(s):  
Phase Transitions ◽  
Chaotic Dynamics ◽  
Deterministic Chaos ◽  
Second Order ◽  
Homogeneous Layer ◽  
Time Interval ◽  
Evaporation And Condensation ◽  
Starting Conditions ◽  
Kolmogorov's Entropy ◽  
Dimensional Mapping

A second-order nonlinear differential equation with an aftereffect for the density of a thin homogeneous layer on a liquid and vapor interface is considered. The acts of evaporation and condensation of molecules, which are regarded as periodic "impacts", excite the layer. The mentioned NDE is integrated over a finite time interval to find a 2D (two-dimensional) mapping whose numerical solution describes the chaotic dynamics of density and pressure in time. The algorithms of constructing bifurcation diagrams, Lyapunov's exponents and Kolmogorov's entropy for systems with first-order, second-order phase transitions and Van der Waals' systems were elaborated. This approach allows to associate such concepts as phase transition, deterministic chaos and nonlinear processes. It also allows to answer a question whether deterministic chaos occurs in systems with phase transitions and how fast the information about starting conditions is lost within them.

scholarly journals Zero Eigenvalue Analysis for the Determination of Multiple Steady States in Reaction Networks

1998 ◽  
Vol 53 (3-4) ◽  
pp. 171-177
Author(s):  
Hsing-Ya Li
Keyword(s):  
Steady State ◽  
Rate Constants ◽  
Reaction Network ◽  
Steady States ◽  
Eigenvalue Analysis ◽  
Zero Eigenvalue ◽  
Positive Steady State ◽  
Positive Steady States ◽  
Positive Rate ◽  
System Of Inequalities

Abstract A chemical reaction network can admit multiple positive steady states if and only if there exists a positive steady state having a zero eigenvalue with its eigenvector in the stoichiometric subspace. A zero eigenvalue analysis is proposed which provides a necessary and sufficient condition to determine the possibility of the existence of such a steady state. The condition forms a system of inequalities and equations. If a set of solutions for the system is found, then the network under study is able to admit multiple positive steady states for some positive rate constants. Otherwise, the network can exhibit at most one steady state, no matter what positive rate constants the system might have. The construction of a zero-eigenvalue positive steady state and a set of positive rate constants is also presented. The analysis is demonstrated by two examples.

scholarly journals Thermal Robustness of Entanglement in a Dissipative Two-Dimensional Spin System in an Inhomogeneous Magnetic Field

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 1066
Author(s):  
Gehad Sadiek ◽  
Samaher Almalki
Keyword(s):  
Magnetic Field ◽  
Steady State ◽  
Nearest Neighbor ◽  
Quantum Phase Transitions ◽  
Inhomogeneous Magnetic Field ◽  
Steady States ◽  
Inhomogeneous Field ◽  
Spin States ◽  
Two Dimensional ◽  
Spin Lattice

Recently new novel magnetic phases were shown to exist in the asymptotic steady states of spin systems coupled to dissipative environments at zero temperature. Tuning the different system parameters led to quantum phase transitions among those states. We study, here, a finite two-dimensional Heisenberg triangular spin lattice coupled to a dissipative Markovian Lindblad environment at finite temperature. We show how applying an inhomogeneous magnetic field to the system at different degrees of anisotropy may significantly affect the spin states, and the entanglement properties and distribution among the spins in the asymptotic steady state of the system. In particular, applying an inhomogeneous field with an inward (growing) gradient toward the central spin is found to considerably enhance the nearest neighbor entanglement and its robustness against the thermal dissipative decay effect in the completely anisotropic (Ising) system, whereas the beyond nearest neighbor ones vanish entirely. The spins of the system in this case reach different steady states depending on their positions in the lattice. However, the inhomogeneity of the field shows no effect on the entanglement in the completely isotropic (XXX) system, which vanishes asymptotically under any system configuration and the spins relax to a separable (disentangled) steady state with all the spins reaching a common spin state. Interestingly, applying the same field to a partially anisotropic (XYZ) system does not just enhance the nearest neighbor entanglements and their thermal robustness but all the long-range ones as well, while the spins relax asymptotically to very distinguished spin states, which is a sign of a critical behavior taking place at this combination of system anisotropy and field inhomogeneity.

scholarly journals The unraveling of balanced complexes in metabolic networks

2021 ◽  
Author(s):  
Damoun Langary ◽  
Anika Kueken ◽  
Zoran Nikoloski
Keyword(s):  
Steady State ◽  
Large Scale ◽  
Metabolic Networks ◽  
Network Models ◽  
Steady States ◽  
Biochemical Networks ◽  
Computational Approaches ◽  
Metabolic Models ◽  
Large Scale Networks ◽  
General Conditions

Balanced complexes in biochemical networks are at core of several theoretical and computational approaches that make statements about the properties of the steady states supported by the network. Recent computational approaches have employed balanced complexes to reduce metabolic networks, while ensuring preservation of particular steady-state properties; however, the underlying factors leading to the formation of balanced complexes have not been studied, yet. Here, we present a number of factorizations providing insights in mechanisms that lead to the origins of the corresponding balanced complexes. The proposed factorizations enable us to categorize balanced complexes into four distinct classes, each with specific origins and characteristics. They also provide the means to efficiently determine if a balanced complex in large-scale networks belongs to a particular class from the categorization. The results are obtained under very general conditions and irrespective of the network kinetics, rendering them broadly applicable across variety of network models. Application of the categorization shows that all classes of balanced complexes are present in large-scale metabolic models across all kingdoms of life, therefore paving the way to study their relevance with respect to different properties of steady states supported by these networks.

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