On the chaotic behaviour of size structured cell populations

We show that the solution of a model describing size structured cell populations exhibits chaotic behaviour, for a certain range of parameters. The analysis depends on a uniqueness property in ?², treated in Appendix A.     

Generalized synchronization of identical chaotic systems on the route from an independent dynamics to the complete synchrony

The transition from asynchronous hyperchaos to complete synchrony in coupled identical chaotic systems may either occur directly or be mediated by a preliminary stage of generalized synchronization. In the present paper we investigate the underlying mechanisms of realization of the both scenarios. It is shown that a generalized synchronization arises when the manifold of identically synchronous states M is transversally unstable, while the local transversal contraction of phase volume first appears in the areas of phase space separated from M and being visited by the chaotic trajectories. On the other hand, a direct transition from an asynchronous hyperchaos to the complete synchronization occurs, under variation of the controlling parameter, if the transversal stability appears first on the manifold M, and only then it extends upon the neighboring phase volume. The realization of one or another scenario depends upon the choice of the coupling function. This result is valid for both unidirectionally and mutually coupled systems, that is confirmed by theoretical analysis of the discrete models and numerical simulations of the physically realistic flow systems.     

Cyclic and unstable chaotic dynamics in models of two populations of sturgeon fish

This paper considers nonlinear effects in the dynamics of biological models. We describe two dynamic systems elaborated for simulating populations of Russian sturgeon and stellate sturgeon and based on formalization of the relationship between the spawning stock and recruitment according to the analysis of observational data. For the numerical study of differential equations with a structurally changing right-hand side, we use the method of representing models based on maps of states with conditional transitions. For dynamic systems, the presence of qualitatively different modes of the behavior of trajectories is revealed: stable periodic oscillations (sturgeon model) and unstable chaotic oscillations (stellate model) realized in a limited time interval due to a chaotic subset not being an attractor, which is present in the phase space.     

Control of chaotic vibration in automotive wiper systems

Chaotic vibration has been identified in the automotive wiper system at certain wiping speeds. This irregular vibration not only decreases the wiping efficiency, but also degrades the driving comfort. The purpose of this paper is to propose a new approach to stabilize the chaotic vibration in the wiper system. The key is to employ the extended time-delay feedback control in such a way that the applied voltage of the wiper motor is online adjusted according to its armature current feedback. Based on a practical wiper system, it is verified that the proposed approach can successfully stabilize the chaotic vibration, and provide a wide range of wiping speeds.     

Control and identification of chaotic systems by altering their energy

We developed the control technique for (non)linear oscillators when repellors are stabilized by adjusting the system to energy levels corresponding to their stable counterparts. The technique does not require knowledge of the system equations. Two control strategies are possible. Following the first one, we simply test the systems by changing the feedback strength. This strategy does not require any computation of the control signal, and, hence, can be useful for control as well as identification of unknown systems. If the desired target can be identified (say, from the system time series), one can use another strategy based on goal-oriented control of the desired target. We analyze how the perturbation shape can be tuned so as to preserve the system natural response and discuss how to calculate the minimal strength of the perturbation required for stabilization of a priori chosen orbit. Generally, the control represents addition of an extra nonlinear damping to the system. In two limits of the perturbation slope, it manifests itself in (i) changing the oscillator natural damping; (ii) suppressing (enhancing) the external driving force. In the case of large deviations between phases of the chaotic oscillator and the driving force, only first scenario holds. Generalization of the technique to the case of oscillator networks and 3D autonomous dynamical systems is considered.     

Control of discrete-time chaotic systems via feedback linearization

An approach for controlling discrete-time chaotic systems by feedback linearization is proposed. This method can not only stabilize unstable periodic orbits embedded in a strange attractor, but also can be applied even if the real trajectory is far from the target one. A Hénon map with different operation conditions is implemented to demonstrate the feasibility of the proposed method.     

On generalizing transitivity, persistence, and sensitivity

A subgroup property $\alpha $ is transitive in a group $G$ if $U \alpha V$ and $V \alpha G$ imply that $U \alpha G$ whenever $U \le V \le G$ , and $\alpha $ is persistent in $G$ if $U \alpha G$ implies that $U \alpha V$ whenever $U \le V \le G$ . Even though a subgroup property $\alpha $ may be neither transitive nor persistent, a given subgroup $U$ may have the property that each $\alpha $ -subgroup of $U$ is an $\alpha $ -subgroup of $G$ , or that each $\alpha $ -subgroup of $G$ in $U$ is an $\alpha $ -subgroup of $U$ . We call these subgroup properties $\alpha $ -transitivity and $\alpha $ -persistence, respectively. We introduce and develop the notions of $\alpha $ -transitivity and $\alpha $ -persistence, and we establish how the former property is related to $\alpha $ -sensitivity. In order to demonstrate how these concepts can be used, we apply the results to the cases in which $\alpha $ is replaced with “normal” and the “cover-avoidance property.” We also suggest ways in which the theory can be developed further.     

Control of chaotic behaviour and prevention of extinction using constant proportional feedback

We study a strategy to control the dynamics of one dimensional discrete maps known as the proportional feedback control method. We completely characterize the maps for which it is possible to stabilize the unstable or even chaotic dynamics towards an asymptotically stable equilibrium employing this method.Additionally, under conditions commonly assumed in modelling population dynamics, we show that the strategy drives the system to the optimal situation from a practical point of view, that is, to a global stable equilibrium since in that case the basin of attraction covers all the possible initial conditions. We also show that in some situations the strategy can be used to prevent the extinction of the population when controlling some models with the Allee effect.     

Control and adaptive modified function projective synchronization of Liu chaotic dynamical system

In this work, the feedback control method is proposed to control the behaviour of Liu chaotic dynamical system. The controlled system is stable under some conditions on the parameters of the system determined by Routh-Hurwitz criterion. This paper also presents the adaptive modified function projective synchronization (AMFPS) between two identical Liu chaotic dynamical systems. Based on the Lyapunov stability theorem, adaptive control laws are designed to achieving the AMFPS. Finally, some numerical simulations are obtained to validate the proposed methods.     

Lyapunov exponent,Liao perturbation and persistence

Science China Mathematics - Consider a C1 vector field together with an ergodic invariant probability that has ? nonzero Lyapunov exponents. Using orthonormal moving frames along a generic...     

On the homology group of chaotic manifolds and chaotic quantum field theories

In this paper the homology group of the chaotic manifolds are discussed. Relations between the homology of the geometric manifold and the homology of its chaotic manifolds are deduced. Theorems governing these relations are achieved. The work could be relevant to vacuum fluctuation and chaotic quantum field theories.     

Adiabatic divergence of the chaotic layer width and acceleration of chaotic and noise-induced transport

We show that, in spatially periodic Hamiltonian systems driven by a time-periodic coordinate-independent (AC) force, the upper energy of the chaotic layer grows unlimitedly as the frequency of the force goes to zero. This remarkable effect is absent in any other physically significant systems. It gives rise to the divergence of the rate of the spatial chaotic transport. We also generalize this phenomenon for the presence of a weak noise and weak dissipation. We demonstrate for the latter case that the adiabatic AC force may greatly accelerate the spatial diffusion and the reset rate at a given threshold.     

The strong persistence property and symbolic strong persistence property

Czechoslovak Mathematical Journal - Let I be an ideal in a commutative Noetherian ring R. Then the ideal I has the strong persistence property if and only if (Ik+1: RI) = Ik for all k, and I has...     

Lyapunov function,parameter estimation,synchronization and chaotic cryptography

In this paper we have investigated a new method of synchronization between two non linear systems based upon lyapunov function and parameter estimation through modulational equations. The driving and response systems, both are different and their parameters are unknown.We have constructed the parameter modulation equations and control laws to achieve the synchronization. This method is well applied to two different three dimensional systems and the transverse lyapunov exponents show the effectiveness of the method. Further more we have investigated the cryptographical applications with the help of the above two systems.     

On the chaotic character of the stochastic heat equation, II

Consider the stochastic heat equation ${\partial_t u = (\varkappa/2)\Delta u+\sigma(u)\dot{F}}$ , where the solution u := u _t (x) is indexed by ${(t, x) \in (0, \infty) \times {\bf R}^d}$ , and ${\dot{F}}$ is a centered Gaussian noise that is white in time and has spatially-correlated coordinates. We analyze the large- ${\|x\|}$ fixed-t behavior of the solution u in different regimes, thereby study the effect of noise on the solution in various cases. Among other things, we show that if the spatial correlation function f of the noise is of Riesz type, that is ${f(x)\propto \|x\|^{-\alpha}}$ , then the “fluctuation exponents” of the solution are ${\psi}$ for the spatial variable and ${2\psi-1}$ for the time variable, where ${\psi:=2/(4-\alpha)}$ . Moreover, these exponent relations hold as long as ${\alpha \in (0, d \wedge 2)}$ ; that is precisely when Dalang’s theory [Dalang, Electron J Probab 4:(Paper no. 6):29, 1999] implies the existence of a solution to our stochastic PDE. These findings bolster earlier physical predictions [Kardar et al., Phys Rev Lett 58(20):889–892, 1985; Kardar and Zhang, Phys Rev Lett 58(20):2087–2090, 1987].     

Optimistic problem-solving activity: enacting confidence,persistence, and perseverance

Optimism supports creative mathematical problem-solving. To elaborate its nature, empirical data were analyzed to identify relationships between optimism and more commonly researched constructs, confidence, and persistence. To do so, theoretical links between these constructs were first explored. Theoretically, confidence and persistence were found to be mutually exclusive personal characteristics possessed by optimistic students. Then, five elementary school students were purposefully selected from a broader longitudinal video-stimulated interview study of the role of optimism in collaborative problem-solving to find whether all combinations of confidence and persistence existed. Activity of students possessing different combinations of confidence and persistence was analyzed to determine whether there were differences in their problem-solving activity. Perseverance emerged as a third mutually exclusive characteristic within optimism. By distinguishing between persistence and perseverance, the crucial nature of perseverance in creative mathematical thinking was illuminated. These findings should inform teachers, teacher educators, and researchers interested in building optimism to increase problem-solving capacity.     

Parameters estimation,mixed synchronization,and antisynchronization in chaotic systems

Mixed synchronization between two Hindmarsh–Rose neuron models is realized by optimizing the scheme of Lyapunov function with two selectable gain coefficients. Based on the Lyapunov stability theory, the distribution of synchronization region and the nonsynchronization region in the two‐parameter phase space is calculated, respectively. And then the optimized parameter observers and controllers are approached analytically. All unknown parameters with different orders of magnitude are identified accurately, and the error function for corresponding variables decreases to stable value when the two gain coefficients are given values in the synchronization region. Otherwise, only the four larger unknown parameters are estimated exactly and the error function of corresponding variables decreases stably to certain minimal value with an order about 1 × 10^?6, whereas the smallest unknown parameter is approached greatly although the error of corresponding variables are stabilized within certain transient period. © 2014 Wiley Periodicals, Inc. Complexity 20: 64–73, 2014     

Control of chaotic instabilities in a spinning spacecraft with dissipation using Lyapunov''s method

Control of chaotic instability in a simplified model of a spinning spacecraft with dissipation is achieved using an algorithm derived using Lyapunov's second method. The control method is implemented on a realistic spacecraft parameter configuration which has been found to exhibit chaotic instability for a range of forcing amplitudes and frequencies when a sinusoidally varying torque is applied to the spacecraft. Such a torque, may arise in practice from an unbalanced rotor or from vibrations in appendages. Numerical simulations are performed and the results are studied by means of time history, phase space, Poincaré map, Lyapunov characteristic exponents and bifurcation diagrams.