Volume-preserving Systems Research Articles

The Reduction, First Integral and Kam Tori for n-Dimensional Volume-Preserving Systems

The Reduction, First Integral and Kam Tori for n-Dimensional Volume-Preserving Systems

Invariant Forms in Hybrid and Impact Systems and a Taming of Zeno

Hybrid (and impact) systems are dynamical systems experiencing both continuous and discrete transitions. In this work, we derive necessary and sufficient conditions for when a given differential form is invariant, with special attention paid to the case of the existence of invariant volumes. Particular attention is given to impact systems where the continuous dynamics are Lagrangian and subject to nonholonomic constraints. A celebrated result for volume-preserving dynamical systems is Poincaré recurrence. In order to be recurrent, trajectories need to exist for long periods of time, which can be controlled in continuous-time systems through e.g. compactness. For hybrid systems, an additional mechanism can occur which breaks long-time existence: Zeno (infinitely many discrete transitions in a finite amount of time). We demonstrate that the existence of a smooth invariant volume severely inhibits Zeno behavior; hybrid systems with the “boundary identity property” along with an invariant volume-form have almost no Zeno trajectories (although Zeno trajectories can still exist). This leads to the result that many billiards (e.g. the classical point, the rolling disk, and the rolling ball) are recurrent independent on the shape of the compact table-top.

Lyapunov Exponents for Random Perturbations of Coupled Standard Maps

In this paper, we give a quantitative estimate for the sum of the first $N$ Lyapunov exponents for random perturbations of a natural class $2N$-dimensional volume-preserving systems exhibiting strong hyperbolicity on a large but noninvariant subset of phase space. Concrete models covered by our setting include systems of coupled standard maps, in both `weak' and `strong' coupling regimes.

Power-law trapping in the volume-preserving Arnold-Beltrami-Childress map.

Understanding stickiness and power-law behavior of Poincaré recurrence statistics is an open problem for higher-dimensional systems, in contrast to the well-understood case of systems with two degrees of freedom. We study such intermittent behavior of chaotic orbits in three-dimensional volume-preserving maps using the example of the Arnold-Beltrami-Childress map. The map has a mixed phase space with a cylindrical regular region surrounded by a chaotic sea for the considered parameters. We observe a characteristic overall power-law decay of the cumulative Poincaré recurrence statistics with significant oscillations superimposed. This slow decay is caused by orbits which spend long times close to the surface of the regular region. Representing such long-trapped orbits in frequency space shows clear signatures of partial barriers and reveals that coupled resonances play an essential role. Using a small number of the most relevant resonances allows for classifying long-trapped orbits. From this the Poincaré recurrence statistics can be divided into different exponentially decaying contributions, which very accurately explains the overall power-law behavior including the oscillations.

Bifurcations in Volume-Preserving Systems

We give a survey on local and semi-local bifurcations of divergence-free vector fields. These differ for low dimensions from ‘generic’ bifurcations of structure-less ‘dissipative’ vector fields, up to a dimension-threshold that grows with the co-dimension of the bifurcation.

Lyapunov exponents of partially hyperbolic volume-preserving maps with 2-dimensional center bundle

We consider the set of partially hyperbolic symplectic diffeomorphisms which are accessible, have 2-dimensional center bundle and satisfy some pinching and bunching conditions. In this set, we prove that the non-uniformly hyperbolic maps are Cr open and there exists a Cr open and dense subset of continuity points for the center Lyapunov exponents. We also generalize these results to volume-preserving systems.

Livšic measurable rigidity for 𝒞¹ generic volume-preserving Anosov systems

In this paper, we prove that for C 1 \mathcal {C}^1 generic volume-preserving Anosov diffeomorphisms of a compact connected Riemannian manifold, the Livšic measurable rigidity theorem holds. We also give a parallel result for C 1 \mathcal {C}^1 generic volume-preserving Anosov flows.

Recurrence-time statistics in non-Hamiltonian volume-preserving maps and flows.

We analyze the recurrence-time statistics (RTS) in three-dimensional non-Hamiltonian volume-preserving systems (VPS): an extended standard map and a fluid model. The extended map is a standard map weakly coupled to an extra dimension which contains a deterministic regular, mixed (regular and chaotic), or chaotic motion. The extra dimension strongly enhances the trapping times inducing plateaus and distinct algebraic and exponential decays in the RTS plots. The combined analysis of the RTS with the classification of ordered and chaotic regimes and scaling properties allows us to describe the intricate way trajectories penetrate the previously impenetrable regular islands from the uncoupled case. Essentially the plateaus found in the RTS are related to trajectories that stay for long times inside trapping tubes, not allowing recurrences, and then penetrate diffusively the islands (from the uncoupled case) by a diffusive motion along such tubes in the extra dimension. All asymptotic exponential decays for the RTS are related to an ordered regime (quasiregular motion), and a mixing dynamics is conjectured for the model. These results are compared to the RTS of the standard map with dissipation or noise, showing the peculiarities obtained by using three-dimensional VPS. We also analyze the RTS for a fluid model and show remarkable similarities to the RTS in the extended standard map problem.

Barriers to transport and mixing in volume-preserving maps with nonzero flux

We identify some geometric structures (secondary tori) that restrict transport and prevent mixing in perturbations of integrable volume-preserving systems with nonzero net flux. Unlike the customary KAM tori, secondary tori cannot be continued to the tori present in the integrable system but are generated by resonances and have a contractible direction. We also note that secondary tori persist under the addition of a net flux, which destroys all customary KAM tori.We introduce a remarkably simple algorithm to analyze the behavior of volume preserving maps and to obtain quantitative properties of the secondary tori. We then implement the algorithm and, after running it, present assertions regarding the distribution of the escape times of the unbounded orbits, the abundance of secondary tori, the size of the resonant regions, and the robustness of the tori under the addition of a mean flux.

Normal forms of Hopf-zero singularity

The Lie algebra generated by Hopf-zero classical normal forms is decomposed into two versal Lie subalgebras. Some dynamical properties for each subalgebra are described; one is the set of all volume-preserving conservative systems while the other is the maximal Lie algebra of nonconservative systems. This introduces a unique conservative–nonconservative decomposition for the normal form systems. There exists a Lie-subalgebra that is Lie-isomorphic to a large family of vector fields with Bogdanov–Takens singularity. This gives rise to a conclusion that the local dynamics of formal Hopf-zero singularities is well-understood by the study of Bogdanov–Takens singularities. Despite this, the normal form computations of Bogdanov–Takens and Hopf-zero singularities are independent. Thus, by assuming a quadratic nonzero condition, complete results on the simplest Hopf-zero normal forms are obtained in terms of the conservative–nonconservative decomposition. Some practical formulas are derived and the results implemented using Maple. The method has been applied on the Rössler and Kuramoto–Sivashinsky equations to demonstrate the applicability of our results.

Stable weakly shadowable volume-preserving systems are volume-hyperbolic

We prove that any C1-stable weakly shadowable volume-preserving diffeomorphism de-fined on a compact manifold displays a dominated splitting E ⊕ F. Moreover, both E and F are volume-hyperbolic. Finally, we prove the version of this result for divergence-free vector fields. As a consequence, in low dimensions, we obtain global hyperbolicity.

Symmetry-Based Balance Equation for Local Entropy Density in a Dissipative Multibaker Chain System

In this study, the balance equation for local entropy density defined on each partition is obtained by the decomposition of the time-evolution operator for local entropy density, on the level of the master equation, by using symmetric and antisymmetric properties for the inversion of partition, density pairs and a given drift velocity. The resultant equation includes the following terms: convection, diffusion, entropy flow due to a thermostat and entropy production. The averaging of the four terms recover the corresponding terms in a balance equation for the macroscopic entropy density of irreversible thermodynamics for a thermostated system. Moreover, an empirical law of order estimation is introduced to explain the limiting behavior of the averaged quantities in the macroscopic limit for the bulk system. The law makes it possible to separate some minor contributions from the major four terms and, for example, to explain the positive entropy production rate in a nonequilibrium state for volume-preserving systems, even if the state is far from steady state. They are numerically confirmed on an invertible, dissipative multibaker chain system, named a circuit model. These properties are independent of partitioning.

Resonance phenomena and long-term chaotic advection in volume-preserving systems

Creating chaotic advection is the most efficient strategy to achieve mixing on microscale or in very viscous fluids. In this paper, we present a quantitative theory of the long-time resonant mixing in 3D near-integrable flows. We use the flow between two coaxial elliptic counter-rotating cylinders as a demonstrative model, where multiple scatterings on resonance result in mixing by causing the jumps of adiabatic invariants. We improve the existing estimates of the width of the mixing domain. We show that the resulting mixing both on short and long time scales can be described in terms of a single diffusion-type equation with a diffusion coefficient depending on the averaged effect of multiple passages through resonances. We discuss the exact location of the boundaries of the chaotic domain and show how it affects the properties of mixing.

Fractality of the nonequilibrium stationary states of open volume-preserving systems. II. Galton boards

Galton boards are models of deterministic diffusion in a uniform external field, akin to driven periodic Lorentz gases, here considered in the absence of dissipation mechanism. Assuming a cylindrical geometry with axis along the direction of the external field, the two-dimensional board becomes a model for one-dimensional mass transport along the direction of the external field. This is a purely diffusive process which admits fractal nonequilibrium stationary states under flux boundary conditions. Analytical results are obtained for the statistics of multibaker maps modeling such a nonuniform diffusion process. A correspondence is established between the local phase-space statistics and their macroscopic counterparts. The fractality of the invariant state is shown to be responsible for the positiveness of the entropy production rate.

Fractality of the nonequilibrium stationary states of open volume-preserving systems. I. Tagged particle diffusion

Deterministic diffusive systems, such as the periodic Lorentz gas, multibaker map, as well as spatially periodic systems of interacting particles, have nonequilibrium stationary states with fractal properties when put in contact with particle reservoirs at their boundaries. We study the macroscopic limits of these systems and establish a correspondence between the thermodynamics of the macroscopic diffusion process and the fractality of the stationary states that characterize the phase-space statistics. In particular the entropy production rate is recovered from first principles using a formalism due to Gaspard [J. Stat. Phys. 88, 1215 (1997)]. This paper is the first of two; the second article considers the influence of a uniform external field on such systems.

A decomposed equation for local entropy and entropy production in volume-preserving coarse-grained systems

In this study an equation for the local entropy is derived based on the formulation of a master equation and is applied to volume-preserving maps. The equation consists of the following terms: unsteady, convection, diffusion, probability-weighted phase space volume expansion rate, nonnegative entropy production, and residuals. The decomposition makes it possible to evaluate entropy production in terms of microscopic dynamics and is expected to be applicable to many coarse-grained systems on the phase space. When it is applied to two volume-preserving multibaker chain systems it is confirmed that the summation of the nonnegative entropy production on each site numerically coincides with the entropy production introduced by Gilbert et al. [T. Gilbert, J.R. Dorfman, P. Gaspard, Entropy production, fractals, and relaxation to equilibrium, Phys. Rev. Lett. 85 (2000) 1606–1609] and the phenomenological expression both in nonequilibrium steady and unsteady states. The coincidence is brought about by the fact that the residual terms vanish in the thermodynamic limit when they are integrated on each site. It follows that the entropy production is dominated by the nonnegative entropy production term and becomes positive in nonequilibrium states.

On passage through resonances in volume-preserving systems

Resonance processes are common phenomena in multiscale (slow-fast) systems. In the present paper we consider capture into resonance and scattering on resonance in 3D volume-preserving multiscale systems. We propose a general theory of those processes and apply it to a class of kinematic models inspired by viscous Taylor-Couette flows between two counter-rotating cylinders. We describe the phenomena during a single passage through resonance and show that multiple passages lead to the chaotic advection and mixing. We calculate the width of the mixing domain and estimate a characteristic time of mixing. We show that the resultant mixing can be described using a diffusion equation with a diffusion coefficient depending on the averaged effect of the passages through resonances.

On General Volume-Preserving Mechanical Systems via Cohomology**The project supported in part by National Natural Science Foundation of China (Grant Nos. 90103004 and 10175070) and the State Key Basic Subjects Research Program of China (G1998030601)

We present the general form of equations that generate a volume-preserving flow on a symplectic manifold via the highest Euler–Lagrange cohomology. It is shown that for every volume-preserving flow there are some 2-forms that play a similar role to the Hamiltonian in the Hamilton mechanics and the ordinary canonical equations with Hamiltonian H are included as a special case with a

The Euler–Lagrange Cohomology and General Volume-Preserving Systems

We briefly introduce the concept of Euler–Lagrange cohomology groups on a symplectic manifold (ℳ2n, ω) and systematically present the general form of volume-preserving equations on the manifold from the cohomological point of view. It is shown that for every volume-preserving flow generated by these equations there is an important two-form that plays the analog role with the Hamiltonian in the Hamilton mechanics. In addition, the ordinary canonical equations with Hamiltonian H are included as a special case with the two-form [Formula: see text]. The other volume preserving systems on (ℳ2n, ω) are studied. The relations between our approach and Feng–Shang's volume-preserving systems as well as the Nambu mechanics are also explored.

Geometric and statistical properties induced by separatrix crossings in volume-preserving systems

We consider a three-dimensional volume-preserving system inside the unit sphere. The system is a small perturbation of an integrable one. Almost all phase trajectories of the integrable system are closed curves. Under an arbitrarily small non-zero perturbation all the interior of the sphere, up to a residue of small measure, is apparently a domain of chaotic motion. The phenomenon is described as a result of jumps in an adiabatic invariant of the system occurring when a phase trajectory crosses the two-dimensional separatrix of the unperturbed (integrable) system. The dynamical properties can be understood as a consequence of splitting of invariant manifolds under the perturbation. A two-dimensional return map generated by the system possesses strong stretching properties. This makes the dynamics of the system close to hyperbolic. Numerical investigation of the statistical properties of the system demonstrates a good agreement with theoretical predictions made under the assumption that the system has a large ergodic component whose measure tends to the measure of the sphere's interior as the perturbation tends to zero. However, the system is not ergodic, and stable periodic solutions are found, surrounded by stability islands of measure exponentially small with the perturbation.