Projective lag synchronization in chaotic systems

In this paper, a new projective lag synchronization is proposed, where a driven chaotic system synchronizes the past state of the driver up to a scaling factor α. An active control method is employed to design a controller to achieve the global synchronization of two identical chaotic systems. Based on Lyapunov stability theorem, a sufficient condition is then given for the asymptotical stability of the null solution of an error dynamics. The effectiveness of the proposed schemes is verified via numerical simulations.     

From phenomena of synchronization to exact synchronization and approximate synchronization for hyperbolic systems

In this survey paper, the synchronization will be initially studied for infinite dimensional dynamical systems of partial differential equations instead of finite dimensional systems of ordinary differential equations,and will be connected with the control theory via boundary controls in a finite time interval. More precisely,various kinds of exact boundary synchronization and approximate boundary synchronization will be introduced and realized by means of fewer boundary controls for a coupled system of wave equations with Dirichlet boundary controls. Moreover, as necessary conditions for various kinds of approximate boundary synchronization, criteria of Kalman's type are obtained. Finally, some prospects will be given.     

On the modelling of distributed outflow in one-dimensional models of arterial blood flow

Summary One-dimensional mathematical models of the flow of blood in arteries commonly make use of the concept of distributed outflow to simulate the loss of blood via side branches. It is shown that the usual approach leads to physically unrealistic results in certain special cases involving flow in rigid tubes, unless we introduce in the momentum equation a suitable additional term depending on the outflow. The influence of such a term on the pressure- and velocity waves calculated from an existing model of the aorta is investigated.

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Zusammenfassung Eindimensionale mathematische Modelle des Blutstromes in Arterien verwenden meist das Konzept von distribuiertem Abfluß um den Blutverlust über Verzweigungen zu simulieren. Es wird gezeigt, daß die übliche Behandlungsweise in speziellen Fällen von Strömungen in starren Röhren zu physikalisch unrealistischen Resultaten führt, es sei denn, daß in der Impulsgleichung ein passendes zusätzliches Glied eingeführt wird, welches vom Abfluß abhängt. Der Einfluß dieses Gliedes auf die Druck- und Geschwindigkeitswellen wird mit Hilfe eines bestehenden Modelles der Aorta untersucht.

     

Time lag in disease models with recruitment

In modelling sexually transmitted diseases, it is standard to divide the population into a very active core group and a larger but much less active noncore group. Transitions between these two groups may be affected by the prevalence of disease in the core group. There may be a time-lag in this effect, and we explore the relation between such a time-lag and the stability of an endemic equilibrium.     

The importance of constitutive equations in mechanochemical models of pattern formation

We analyze a one-dimensional mechanical model which has been proposed to account for the pattern formation which arises in various processes in developmental biology. We compare the effect that different constitutive equations have on the resulting spatial pattern. In this way, we conclude that a more detailed knowledge of the mechanical properties of tissues is need if accurate predictions of developing spatial patterns are to be made.     

Blow-up and pattern formation in hyperbolic models for chemotaxis in 1-D

In this paper we study finite time blow-up of solutions of a hyperbolic model for chemotaxis. Using appropriate scaling this hyperbolic model leads to a parabolic model as studied by Othmer and Stevens (1997) and Levine and Sleeman (1997). In the latter paper, explicit solutions which blow-up in finite time were constructed. Here, we adapt their method to construct a corresponding blow-up solution of the hyperbolic model. This construction enables us to compare the blow-up times of the corresponding models. We find that the hyperbolic blow-up is always later than the parabolic blow-up. Moreover, we show that solutions of the hyperbolic problem become negative near blow-up. We calculate the zero-turning-rate time explicitly and we show that this time can be either larger or smaller than the parabolic blow-up time. The blow-up models as discussed here and elsewhere are limiting cases of more realistic models for chemotaxis. At the end of the paper we discuss the relevance to biology and exhibit numerical solutions of more realistic models.     

Blow-up and pattern formation in hyperbolic models for chemotaxis in 1-D

In this paper we study finite time blow-up of solutions of a hyperbolic model for chemotaxis. Using appropriate scaling this hyperbolic model leads to a parabolic model as studied by Othmer and Stevens (1997) and Levine and Sleeman (1997). In the latter paper, explicit solutions which blow-up in finite time were constructed. Here, we adapt their method to construct a corresponding blow-up solution of the hyperbolic model. This construction enables us to compare the blow-up times of the corresponding models. We find that the hyperbolic blow-up is always later than the parabolic blow-up. Moreover, we show that solutions of the hyperbolic problem become negative near blow-up. We calculate the zero-turning-rate time explicitly and we show that this time can be either larger or smaller than the parabolic blow-up time. The blow-up models as discussed here and elsewhere are limiting cases of more realistic models for chemotaxis. At the end of the paper we discuss the relevance to biology and exhibit numerical solutions of more realistic models.     

From one-dimensional to infinite-dimensional dynamical systems: Ideal turbulence

There is a very short chain that joins dynamical systems with the simplest phase space (real line) and dynamical systems with the “most complicated” phase space containing random functions, as well. This statement is justified in this paper. By using “simple” examples of dynamical systems (one-dimensional and two-dimensional boundary-value problems), we consider notions that generally characterize the phenomenon of turbulence—first of all, the emergence of structures (including the cascade process of emergence of coherent structures of decreasing scales) and self-stochasticity.     

Thresholds for shock formation in traffic flow models with nonlocal-concave-convex flux

We identify sub-thresholds for finite time shock formation in a class of non-local conservation law with concavity changing flux. From a class of non-local conservation laws, the Riccati-type ODE system that governs a solution's gradient is obtained. The changes in concavity of the flux function correspond to the sign changes in the leading coefficient functions of the ODE system. We identify the blow up condition of this structurally generalized Riccati-type ODE. The method is illustrated via the traffic flow models with nonlocal-concave-convex flux. The techniques and ideas developed in this paper is applicable to a large class of non-local conservation laws.     

Analysis of pattern formation in reaction diffusion models with spatially inhomogenous diffusion coefficients

We consider a reaction diffusion system in one spatial dimension in which the diffusion coefficients are spatially varying. We present a non-standard linear analysis for a certain class of spatially varying diffusion coefficients and show that it accurately predicts the behaviour of the full nonlinear system near bifurcation. We show that the steady state solutions exhibit qualitatively different behaviour to that observed in the usual case with constant diffusion coefficients. Specifically, the modified system can generate patterns with spatially varying amplitude and wavelength. Application to chondrogenesis in the limb is discussed.     

Parameter domains for spots and stripes in mechanical models for biological pattern formation

Summary Mechanochemical models for biological pattern formation have been applied to the development of a variety of patterning problems, such as feather germ primordia and cartilage formation in the vertebrate limb. Linear analysis has been the main technique for assessing the pattern formation potential of these models to date. In this paper we carry out a nonlinear analysis and numerical simulations of a generic model in two spatial dimensions. With these methods, we obtain conditions for generating specific spatial patterns such as stripes and spots, and divide the parameter space into domains giving rise to distinct types of pattern. We accomplish our goal through a study of model parameter domains by showing how different mechanical forces affect spatial patterning.     

Wasserstein statistics in one-dimensional location scale models

Annals of the Institute of Statistical Mathematics - Wasserstein geometry and information geometry are two important structures to be introduced in a manifold of probability distributions....     

Shock waves in one-dimensional models with cubic nonlinearity

Shock waves are described qualitatively for a class of one-dimensional models with cubic nonlinearity (of the type of the modified Korteweg-de Vries equation):u _t–6u ²u_x+u _xxx=vu _xx. Both the integrable and the nonintegrable case are considered. The behavior of a shock wave in the limitt is considered.St. Petersburg Branch of the V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 2, pp. 191–212, November, 1993.     

Time lag size in multiple operations flow shop scheduling heuristics

This paper considers a multistage flow shop where jobs require multiple operations at each stage and a finish-to-start time lag between any two consecutive operations of a job: the next operation of a job cannot start until the time lag after the former operation of that job has elapsed. The effect of the size of this time lag is considered when studying the effectiveness of solution approaches for this problem. Since the problem of minimizing the makespan is shown to be NP-hard even for the two-stage case, we present a lower bound based heuristic approach that is used to construct several heuristic procedures. These heuristics use lower bounds on the minimum makespan to solve the problem. The effectiveness of these heuristics is empirically evaluated for various time lag sizes by solving a large number of randomly generated problems. We show that the relative performance of the heuristics depends on the size of the time lag. If the ratio of mean time lag and mean processing time is 20% or more, heuristics that construct an active schedule perform less well than heuristics that construct a non-delay schedule. The opposite holds true if this ratio is smaller. The performance of the widely used Shortest Processing Time heuristic (SPT) deteriorates quickly if the size of the time lags increases. We propose instead to use the Earliest Finish Time heuristic (EFT) in case time lags are present. EFT performs much better in this case and is identical to SPT if all time lags are zero. The use of the lower bound based heuristics results in an improvement of the makespan performance of up to 50% as compared with the performance of some simple dispatching heuristics that take the presence of multiple operations and time lags into account. This effect increases with the size of the time lags.     

Robust synchronization of chaotic behavior in unidirectional coupled RCLSJ models subject to uncertainties

This paper deals with the problem of robust synchronization for a class of unidirectional coupled RCL-shunted Josephson junction (RCLSJ) models. A nonlinear controller is proposed based on variable structure control technique to ensure that these coupled RCLSJ models with different parameters can be asymptotically synchronized even when uncertainties are present in the coupled system. Finally, a comparative example is given to emphasize the simplicity and robustness of the proposed method.     

From discrete flow of Beckner to continuous flow of Janson in complex hypercontractivity

We show how Beckner's montonicity result on Hamming cube easily implies the monotonicity of a flow introduced by Janson in Hausdorff–Young inequality     

Some predictability in one-dimensional chaotic dynamical systems arising in population models

Mathematical framework is given to “resolved chaos” studied numerically by Vandermeer in population biology, which means some kind of predictability in the chaotic dynamical systems. A general theory about one-dimensional unimodal maps is constructed. A quantity called “sojourning time,” which is the duration of staying in an interval by iteration of a map, is considered. Predictability is formulated as the size of error by fluctuation from the deterministic system. Topological entropy is used as the degree of chaos and a relation between topological entropy and sojourning time is obtained. Also, some conditions for the coexistence of chaotic behavior and predictability of sojourning time are given generally. In conclusion, many of the unimodal maps with high degree of chaos are predictable on the sojourning time.