Asymptotic Attractor Research Articles

Nonlinear bulk viscosity in FRW cosmology: a phase space analysis.

We consider a Friedmann–Robertson–Walker spacetime filled with both viscous radiation and nonviscous dust. The former has a bulk viscosity that is proportional to an arbitrary power of the energy density, i.e. and viscous pressure satisfying a nonlinear evolution equation. The analysis is carried out in the context of dynamical systems and the properties of solutions corresponding to the fixed points are discussed. For some ranges of the relevant parameter ν we find that the trajectories in the phase space evolve from a FRW singularity towards an asymptotic de Sitter attractor, confirming and extending previous analysis in the literature.

Asymptotic global fast dynamics and attractor fractal dimension of the TDGL-equations of superconductivity

In this paper, we study a model system of equations of the time dependent Ginzburg–Landau equations of superconductivity in a Lorentz gauge, in scale of Hilbert spaces \(E^{\alpha }\) with initial data in \(E^{\beta }\) satisfying \(3\alpha + \beta \ge \frac{N}{2}\), where \(N=2,3\) is such that the spatial domain of the equations Open image in new window. We show in the asymptotic dynamics of the equations, well-posedness of the dynamical system for a global exponential attractor \({\mathcal {U}}\subset E^{\alpha }\) compact in \(E^{\beta }\) if \(\alpha >\beta \), uniform differentiability of orbits on the attractor in \(E^{0}\cong L^{2}\), and the existence of an explicit finite bounding estimate on the fractal dimension of the attractor yielding that its Hausdorff dimension is as well finite. Uniform boundedness in \((0,\infty )\times \Omega \) of solutions in \(E^{\frac{1}{2}}\cong H^{1}(\Omega )\) is in addition investigated.

Exactly returning boomerangs

Differential equations of motion of a boomerang can be integrated numerically given its aerodynamic and inertial properties and initial conditions. We use the dynamic model and experimental aerodynamic data of a typical boomerang in still air studied by Hess (Boomerangs, aerodynamics and motion. PhD thesis, University of Groningen, 1975). The trajectory size and shape are well-defined functions of five initial conditions. Beginning with a nominal guessed set, an iterative search finds release conditions that result in exact return. The distance to the point on the trajectory closest to the desired return point and its gradient with respect to the release conditions are calculated. Release conditions are then modified iteratively using Newton’s method to decrease the miss distance. Exact return conditions are presented for constant values of initial angle of attack and advance ratio. A variant of the algorithm calculates release conditions that ensure return at “turnaround” where the speed is lowest and thus catching is easiest. Although in general the set of exact return release conditions is five dimensional, it is thin in the sense that certain variables must lie in a fairly narrow range. Some initial conditions are more easily modified than others, accounting for the not-inconsiderable skill required to achieve exact return in practice. A discussion is also included of the stable asymptotic helical attractor approached by the boomerang after turnaround.

Asymptotic Regularity and Exponential Attractors for Nonclassical Diusion Equations With Critical Exponent

In this paper, we consider the dynamical behavior of the nonclassical diffusion equation when nonlinearity is critical for both two cases: the forcing term belongs to H − 1 (Ω) and L 2 (Ω). For the case the forcing term only belongs to H − 1 (Ω), based on the asymptotic regularity in Dynamical Systems: An International Journal , 26 (4), (2011), 391–400, we prove the existence of exponential attractors in weak topological space H 1 (Ω). For the case the forcing term belongs to L 2 (Ω), we prove the asymptotic regularity of the solutions and exponential attractors.

Inertial focusing of small particles in wavy channels: Asymptotic analysis at weak particle inertia

The motion of tiny non-Brownian inertial particles in a two-dimensional channel flow with periodic corrugations is investigated analytically, to determine the trapping rate as well as the exact position of the attractor, and understand the conditions under which particle trapping and long-term suspension occur. This phenomenon has been observed numerically in previous works and happens under the combined effects of confinement and inertia. Starting from the particle motion equations, a Poincaré map is constructed analytically in the limit of weak inertia and weak channel corrugations. It enables to derive the equation of the attractor, if any, and the corresponding trapping rate. The attractor is close to a streamline, the so-called “attracting streamline”, and is shown to persist in the presence of transverse gravity, provided the channel Froude number is large enough. Particles which are trapped by this streamline can therefore travel over long distances, avoiding deposition. Numerical simulations confirm the theoretical results at small particle response times τ and reveal some non-linear effects at larger τ: the asymptotic attractor becomes unstable at some critical value and splits into multiple branches each with its own basin of attraction.

Asymptotic Smoothing and Global Attractors for a Class of Nonlinear Evolution Equations

We prove the asymptotic regularity of global solutions for a class of semilinear evolution equations in H01(Ω)×H01(Ω). Moreover, we study the long-time behavior of the solutions. It is proved that, under the natural assumptions, these equations possess the compact attractor 𝒜 which is bounded in H2(Ω)×H2(Ω), where the nonlinear term f satisfies a critical exponential growth condition.

Stable and periodic solutions of an exergy-based model of population dynamics

The long-term qualitative behavior of most real ecological populations can be mathematically classified as either asymptotic equilibrium or periodic. The reasons underlying these behaviors have been, and are still, debated. Aim of this paper is to show that it is possible to characterize both of these behavious through a novel (in general non-autonomous) population dynamics model, proposed and developed in previous work by the authors, in which the resource consumption is quantified solely in terms of exergy flows. A limited but significant number of reasonable assumptions on the phenomenological characteristics of the interacting species result in different evolutionary scenarios, corresponding respectively to asymptotic stability or periodic attractors. In the case of asymptotic periodic motion, the system is described by a set of non-linear, non-autonomous differential equations and an analytical investigation becomes arduous even in the simplest cases. Explicit results, both analytical and numerical, are discussed in this paper. The theoretical description is compared with the behavior of real, well-known biological systems: the snowshoe hare-lynx predator-prey system for the periodic case and the reindeer herds of the Pribilof Islands for the case of asymptotic stability. The assumptions that must be posited to obtain a periodic behavior in ecological populations appear to have been never recognized before and the possibility to adopt them as an universal mechanism of population cycles is discussed in the conclusions.

Attractor Bifurcation for Extended Fisher-Kolmogorov Equation

We consider the asymptotic stability and attractor bifurcation of the extended Fisher-Kolmogorov equation on the one-dimensional domain <svg style="vertical-align:-2.21957pt;width:37.924999px;" id="M1" height="16.200001" version="1.1" viewBox="0 0 37.924999 16.200001" width="37.924999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.625,12.687)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,6.507,12.687)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105&#xA;q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g><g transform="matrix(.017,-0,0,-.017,14.666,12.687)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,21.363,12.687)"><path id="x1D70B" d="M574 449q-23 -58 -38 -79q-16 -4 -79 -4q-27 -100 -46 -219q-14 -87 7 -87t74 36l13 -27q-74 -81 -139 -81q-48 0 -48 62q0 22 8 59l60 258l-154 4q-21 -103 -54.5 -209t-64.5 -151q-38 -23 -83 -23l-7 15q46 36 94 152.5t64 216.5q-81 0 -138 -54l-18 23q22 27 39.5 43&#xA;t61.5 33.5t100 17.5q43 0 131.5 -2.5t129.5 -2.5q23 0 33.5 5t24.5 25z" /></g><g transform="matrix(.017,-0,0,-.017,31.41,12.687)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g> </svg> with Dirichlet or periodic boundary conditions. The novelty of this paper is that, based on a new method called attractor bifurcation, we investigate the existence of an attractor bifurcated from the trivial solution and give an explicit description of the bifurcated attractor. Moreover, the stability of the bifurcated branches is discussed.

Dynamical Behaviors of Stochastic Hopfield Neural Networks with Both Time-Varying and Continuously Distributed Delays

This paper investigates dynamical behaviors of stochastic Hopfield neural networks with both time-varying and continuously distributed delays. By employing the Lyapunov functional theory and linear matrix inequality, some novel criteria on asymptotic stability, ultimate boundedness, and weak attractor are derived. Finally, an example is given to illustrate the correctness and effectiveness of our theoretical results.

Dynamical Behaviors of the Stochastic Hopfield Neural Networks with Mixed Time Delays

This paper investigates dynamical behaviors of the stochastic Hopfield neural networks with mixed time delays. The mixed time delays under consideration comprise both the discrete time-varying delays and the distributed time-delays. By employing the theory of stochastic functional differential equations and linear matrix inequality (LMI) approach, some novel criteria on asymptotic stability, ultimate boundedness, and weak attractor are derived. Finally, a numerical example is given to illustrate the correctness and effectiveness of our theoretical results.

Optimal model reduction by empirical spectral methods via sampling of chaotic orbits

Proper orthogonal decomposition (POD) coupled with the Galerkin method is applied to the one-dimensional model of a tubular reactor with external heat recycle, which exhibits periodic and chaotic solutions. The effect of the cooling medium temperature on the system dynamics is considered. The issue of the optimal construction of the POD basis is addressed by sampling of the chaotic orbits, with the aim of constructing a global basis for a reduced-order model (ROM). To demonstrate that such orbits are the most appropriate because they incorporate the maximum amount of information about the system behavior, the entropy of the orbit is calculated. Sampling of the chaotic solutions allows for the determination of the POD basis to be employed in the POD/Galerkin method. The accuracy of the ROMs is compared by means of the Hausdorff distance, computed between the asymptotic regime of the reference solution and each of the ROM-reconstructed asymptotic attractors. A norm computed on the sampled time series is employed to compare transient solutions. The POD-based ROMs work well even for values of the parameter for which the model behavior is far from chaotic, i.e. periodic orbits or fixed points. Moreover, the POD-based ROMs successfully compare with a classic orthogonal collocation method.

Asymptotic regularity and attractors of the reaction–diffusion equation with nonlinear boundary condition

We consider the dynamical behavior of the reaction–diffusion equation with nonlinear boundary condition for both autonomous and non-autonomous cases. For the autonomous case, under the assumption that the internal nonlinear term f is dissipative and the boundary nonlinear term g is non-dissipative, the asymptotic regularity of solutions is proved. For the non-autonomous case, we obtain the existence of a compact uniform attractor in H 1 ( Ω ) with dissipative internal and boundary nonlinearities.

A minimal dynamical model for tidal synchronization and orbit circularization

We study tidal synchronization and orbit circularization in a minimal model that takes into account only the essential ingredients of tidal deformation and dissipation in the secondary body. In previous work we introduced the model (Escribano et al. in Phys. Rev. E, 78:036216, 2008); here we investigate in depth the complex dynamics that can arise from this simplest model of tidal synchronization and orbit circularization. We model an extended secondary body of mass m by two point masses of mass m/2 connected with a damped spring. This composite body moves in the gravitational field of a primary of mass M ≫ m located at the origin. In this simplest case oscillation and rotation of the secondary are assumed to take place in the plane of the Keplerian orbit. The gravitational interactions of both point masses with the primary are taken into account, but that between the point masses is neglected. We perform a Taylor expansion on the exact equations of motion to isolate and identify the different effects of tidal interactions. We compare both sets of equations and study the applicability of the approximations, in the presence of chaos. We introduce the resonance function as a resource to identify resonant states. The approximate equations of motion can account for both synchronization into the 1:1 spin-orbit resonance and the circularization of the orbit as the only true asymptotic attractors, together with the existence of relatively long-lived metastable orbits with the secondary in p:q (p and q being co-prime integers) synchronous rotation.

Effect of smoothing on robust chaos

In piecewise-smooth dynamical systems, situations can arise where the asymptotic attractors of the system in an open parameter interval are all chaotic (e.g., no periodic windows). This is the phenomenon of robust chaos. Previous works have established that robust chaos can occur through the mechanism of border-collision bifurcation, where border is the phase-space region where discontinuities in the derivatives of the dynamical equations occur. We investigate the effect of smoothing on robust chaos and find that periodic windows can arise when a small amount of smoothness is present. We introduce a parameter of smoothing and find that the measure of the periodic windows in the parameter space scales linearly with the parameter, regardless of the details of the smoothing function. Numerical support and a heuristic theory are provided to establish the scaling relation. Experimental evidence of periodic windows in a supposedly piecewise linear dynamical system, which has been implemented as an electronic circuit, is also provided.

Power-law solutions and accelerated expansion in scalar-tensor theories

We find exact power-law solutions for scalar-tensor theories and clarify the conditions under which they can account for an accelerated expansion of the Universe. These solutions have the property that the signs of both the Hubble rate and the deceleration parameter in the Jordan frame may be different from the signs of their Einstein-frame counterparts. For special parameter combinations we identify these solutions with asymptotic attractors that have been obtained in the literature through dynamical-system analysis. We establish an effective general-relativistic description for which the geometrical equivalent of dark energy is associated with a time dependent equation of state. The present value of the latter is consistent with the observed cosmological ``constant". We demonstrate that this type of power-law solutions for accelerated expansion cannot be realized in f(R) theories.

A moving boundary model motivated by electric breakdown: II. Initial value problem

An interfacial approximation of the streamer stage in the evolution of sparks and lightning can be formulated as a Laplacian growth model regularized by a ‘kinetic undercooling’ boundary condition. Using this model we study both the linearized and the full nonlinear evolution of small perturbations of a uniformly translating circle. Within the linear approximation analytical and numerical results show that perturbations are advected to the back of the circle, where they decay. An initially analytic interface stays analytic for all finite times, but singularities from outside the physical region approach the interface for t → ∞ , which results in some anomalous relaxation at the back of the circle. For the nonlinear evolution numerical results indicate that the circle is the asymptotic attractor for small perturbations, but larger perturbations may lead to branching. We also present results for more general initial shapes, which demonstrate that regularization by kinetic undercooling cannot guarantee smooth interfaces globally in time.

Exact quantification of the complexity of spacewise pattern growth in cellular automata

We analyze the two possible ways of simulating complex systems with cellular automata: by using the familiar timewise updating or by using the complementary spacewise updating. Both updating algorithms operate on identical sets of initial conditions defining the state of the automaton. While timewise growth generally probes just vanishingly small sets of initial conditions producing statistical samples of the asymptotic attractors, spacewise growth operates with much restricted sets which allow one to simulate them all, exhaustively. Our main result is the derivation of an exact analytical formula to quantify precisely one of the two sources of algorithmic complexity of spacewise detection of the complete set of attractors for elementary 1D cellular automata with generic non-periodic architectures of any arbitrary size. The formula gives the total number of initial conditions that need to be investigated to locate rigorously all possible patterns for any given rule. As simple applications, we illustrate how this knowledge may be used (i) to uncover missing patterns in previous classifications in the literature and (ii) to obtain surprisingly novel patterns that are totally unreachable with the time-honored technique of artificially imposing spatially periodic boundary conditions.

Total Stability Properties Based on Fixed Point Theory for a Class of Hybrid Dynamic Systems

AbstractRobust stability results for nominally linear hybrid systems are obtained from total stability theorems for purely continuous-time and discrete-time systems by using the powerful tool of fixed point theory. The class of hybrid systems dealt consists, in general, of coupled continuous-time and digital systems subject to state perturbations whose nominal (i.e., unperturbed) parts are linear and, in general, time-varying. The obtained sufficient conditions on robust stability under a wide class of harmless perturbations are dependent on the values of the parameters defining the over-bounding functions of those perturbations. The weakness of the coupling dynamics in terms of norm among the analog and digital substates of the whole dynamic system guarantees the total stability provided that the corresponding uncoupled nominal subsystems are both exponentially stable. Fixed point stability theory is used for the proofs of stability. A generalization of that result is given for the case that sampling is not uniform. The boundedness of the state-trajectory solution at sampling instants guarantees the global boundedness of the solutions for all time. The existence of a fixed point for the sampled state-trajectory solution at sampling instants guarantees the existence of a fixed point of an extended auxiliary discrete system and the existence of a global asymptotic attractor of the solutions which is either a fixed point or a limit "Equation missing" globally stable asymptotic oscillation.

Dynamics of tidal synchronization and orbit circularization of celestial bodies

We take a dynamical-systems approach to study the qualitative dynamical aspects of the tidal locking of the rotation of secondary celestial bodies with their orbital motion around the primary. We introduce a minimal model including the essential features of gravitationally induced elastic deformation and tidal dissipation that demonstrates the details of the energy transfer between the orbital and rotovibrational degrees of freedom. Despite its simplicity, our model can account for both synchronization into the 1:1 spin-orbit resonance and the circularization of the orbit as the only true asymptotic attractors, together with the existence of relatively long-lived metastable orbits with the secondary in p:q synchronous rotation.

Asymptotic Compactness and Attractors for Phase-Field Equations in ℝ3

In this paper we study the asymptotic behaviour of solutions of the phase-field system on an unbounded domain. We do not assume conditions on the non-linear term ensuring the uniqueness of the Cauchy problem, so that we have to work with multivalued semiflows rather than with semigroups of operators. In this way we prove the existence of a global attractor by considering the convergence in an appropriate weighted space. This result is also new for more restrictive conditions, which guarantee the uniqueness of solutions.