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Journal articles on the topic 'Partially Hyperbolic System'

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Published: 11 March 2023

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1

RIOS, I., and J. SIQUEIRA. "On equilibrium states for partially hyperbolic horseshoes." Ergodic Theory and Dynamical Systems 38, no. 1 (July 4, 2016): 301–35. http://dx.doi.org/10.1017/etds.2016.21.

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We prove the existence and uniqueness of equilibrium states for a family of partially hyperbolic systems, with respect to Hölder continuous potentials with small variation. The family comes from the projection, on the center-unstable direction, of a family of partially hyperbolic horseshoes introduced by Díaz et al [Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes. Ergod. Th. & Dynam. Sys.29 (2009), 433–474]. For the original three-dimensional system we consider potentials with small variation, constant on local stable manifolds, obtaining existence and uniqueness of equilibrium states.

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2

RODRIGUEZ HERTZ, F., M. A. RODRIGUEZ HERTZ, A. TAHZIBI, and R. URES. "Maximizing measures for partially hyperbolic systems with compact center leaves." Ergodic Theory and Dynamical Systems 32, no. 2 (December 5, 2011): 825–39. http://dx.doi.org/10.1017/s0143385711000757.

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AbstractWe obtain the following dichotomy for accessible partially hyperbolic diffeomorphisms of three-dimensional manifolds having compact center leaves: either there is a unique entropy-maximizing measure, this measure has the Bernoulli property and its center Lyapunov exponent is 0, or there are a finite number of entropy-maximizing measures, all of them with non-zero center Lyapunov exponents (at least one with a negative exponent and one with a positive exponent), that are finite extensions of a Bernoulli system. In the first case of the dichotomy, we obtain that the system is topologically conjugated to a rotation extension of a hyperbolic system. This implies that the second case of the dichotomy holds for an open and dense set of diffeomorphisms in the hypothesis of our result. As a consequence, we obtain an open set of topologically mixing diffeomorphisms having more than one entropy-maximizing measure.

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3

ANDERSSON, MARTIN, and CARLOS H. VÁSQUEZ. "On mostly expanding diffeomorphisms." Ergodic Theory and Dynamical Systems 38, no. 8 (May 2, 2017): 2838–59. http://dx.doi.org/10.1017/etds.2017.17.

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In this work, we study the class of mostly expanding partially hyperbolic diffeomorphisms. We prove that such a class is$C^{r}$-open,$r>1$, among the partially hyperbolic diffeomorphisms and we prove that the mostly expanding condition guarantees the existence of physical measures and provides more information about the statistics of the system. Mañé’s classical derived-from-Anosov diffeomorphism on$\mathbb{T}^{3}$belongs to this set.

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4

Xu, Jiang, and Shuichi Kawashima. "Global Classical Solutions for Partially Dissipative Hyperbolic System of Balance Laws." Archive for Rational Mechanics and Analysis 211, no. 2 (October 8, 2013): 513–53. http://dx.doi.org/10.1007/s00205-013-0679-8.

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5

BUZZI, J., T. FISHER, M. SAMBARINO, and C. VÁSQUEZ. "Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems." Ergodic Theory and Dynamical Systems 32, no. 1 (June 10, 2011): 63–79. http://dx.doi.org/10.1017/s0143385710000854.

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AbstractWe show that a class of robustly transitive diffeomorphisms originally described by Mañé are intrinsically ergodic. More precisely, we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic and structurally stable, but nevertheless have the following stability with respect to their entropy. Their topological entropy is constant and they each have a unique measure of maximal entropy with respect to which periodic orbits are equidistributed. Moreover, equipped with their respective measure of maximal entropy, these diffeomorphisms are pairwise isomorphic. We show that the method applies to several classes of systems which are similarly derived from Anosov, i.e. produced by an isotopy from an Anosov system, namely, a mixed Mañé example and one obtained through a Hopf bifurcation.

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6

Ondich, Jeffrey. "The reducibility of partially invariant solutions of systems of partial differential equations." European Journal of Applied Mathematics 6, no. 4 (August 1995): 329–54. http://dx.doi.org/10.1017/s0956792500001881.

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Ovsiannikov's partially invariant solutions of differential equations generalize Lie's group invariant solutions. A partially invariant solution is only interesting if it cannot be discovered more readily as an invariant solution. Roughly, a partially invariant solution that can be discovered more directly by Lie's method is said to be reducible. In this paper, I develop conditions under which a partially invariant solution or a class of such solutions must be reducible, and use these conditions both to obtain non-reducible solutions to a system of hyperbolic conservation laws, and to demonstrate that some systems have no non-reducible solutions. I also demonstrate that certain elliptic systems have no non-reducible solutions.

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7

Vergara-Hermosilla, G., G. Leugering, and Y. Wang. "Boundary controllability of a system modelling a partially immersed obstacle." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 80. http://dx.doi.org/10.1051/cocv/2021076.

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In this paper, we address the problem of boundary controllability for the one-dimensional nonlinear shallow water system, describing the free surface flow of water as well as the flow under a fixed gate structure. The system of differential equations considered can be interpreted as a simplified model of a particular type of wave energy device converter called oscillating water column. The physical requirements naturally lead to the problem of exact controllability in a prescribed region. In particular, we use the concept of nodal profile controllability in which at a given point (the node) time-dependent profiles for the states are required to be reachable by boundary controls. By rewriting the system into a hyperbolic system with nonlocal boundary conditions, we at first establish the semi-global classical solutions of the system, then get the local controllability and nodal profile using a constructive method. In addition, based on this constructive process, we provide an algorithmic concept to calculate the required boundary control function for generating a solution for solving these control problem.

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8

Klar, A., and O. Tse. "An entropy functional and explicit decay rates for a nonlinear partially dissipative hyperbolic system." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 95, no. 5 (March 10, 2014): 469–75. http://dx.doi.org/10.1002/zamm.201300275.

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9

GUPTA, CHINMAYA. "Extreme-value distributions for some classes of non-uniformly partially hyperbolic dynamical systems." Ergodic Theory and Dynamical Systems 30, no. 3 (July 17, 2009): 757–71. http://dx.doi.org/10.1017/s0143385709000406.

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AbstractIn this note, we obtain verifiable sufficient conditions for the extreme-value distribution for a certain class of skew-product extensions of non-uniformly hyperbolic base maps. We show that these conditions, formulated in terms of the decay of correlations on the product system and the measure of rapidly returning points on the base, lead to a distribution for the maximum of Φ(p)=−log(d(p,p0)) that is of the first type. In particular, we establish the type I distribution for S1 extensions of piecewise C2 uniformly expanding maps of the interval, non-uniformly expanding maps of the interval modeled by a Young tower, and a skew-product extension of a uniformly expanding map with a curve of neutral points.

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10

Rempel, M., and D. Przybylski. "Efficient Numerical Treatment of Ambipolar and Hall Drift as Hyperbolic System." Astrophysical Journal 923, no. 1 (December 1, 2021): 79. http://dx.doi.org/10.3847/1538-4357/ac2c6d.

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Abstract Partially ionized plasmas, such as the solar chromosphere, require a generalized Ohm’s law including the effects of ambipolar and Hall drift. While both describe transport processes that arise from the multifluid equations and are therefore of hyperbolic nature, they are often incorporated in models as a diffusive, i.e., parabolic process. While the formulation as such is easy to include in standard MHD models, the resulting diffusive time-step constraints do require often a computationally more expensive implicit treatment or super-time-stepping approaches. In this paper we discuss an implementation that retains the hyperbolic nature and allows for an explicit integration with small computational overhead. In the case of ambipolar drift, this formulation arises naturally by simply retaining a time derivative of the drift velocity that is typically omitted. This alone leads to time-step constraints that are comparable to the native MHD time-step constraint for a solar setup including the region from photosphere to lower solar corona. We discuss an accelerated treatment that can further reduce time-step constraints if necessary. In the case of Hall drift we propose a hyperbolic formulation that is numerically similar to that for the ambipolar drift and we show that the combination of both can be applied to simulations of the solar chromosphere at minimal computational expense.

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11

AFRAIMOVICH, V. S., SHUI-NEE CHOW, and WENXIAN SHEN. "HYPERBOLIC HOMOCLINIC POINTS OF ℤd-ACTIONS IN LATTICE DYNAMICAL SYSTEMS." International Journal of Bifurcation and Chaos 06, no. 06 (June 1996): 1059–75. http://dx.doi.org/10.1142/s0218127496000576.

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We study ℤd action on a set of equilibrium solutions of a lattice dynamical system, i.e., a system with discrete spatial variables, and the stability and hyperbolicity of the equilibrium solutions. Complicated behavior of ℤd-action corresponds to the existence of an infinite number of equilibrium solutions which are randomly situated along spatial coordinates. We prove that the existence of a homoclinic point of a ℤd-action implies complicated behavior, provided the hyperbolicity of the homoclinic solution with respect to the lattice dynamical system (this is a generalization of the previous work of the first two authors). Similar result holds for hyperbolic partially homoclinic and heteroclinic points. We show the equivalence of stability for any equilibrium solutions and the equivalence of hyperbolicity for homoclinic points under various norms.

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12

CHOQUET, CATHERINE. "PARABOLIC AND DEGENERATE PARABOLIC MODELS FOR PRESSURE-DRIVEN TRANSPORT PROBLEMS." Mathematical Models and Methods in Applied Sciences 20, no. 04 (April 2010): 543–66. http://dx.doi.org/10.1142/s0218202510004337.

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We consider two models of flow and transport in porous media, the first one for consolidational flow in compressible sedimentary basins, the second one for flow in partially saturated media. Despite the differences in these physical settings, they lead to quite similar mathematical models with a strong pressure coupling. The first model is a coupled system of pde's of parabolic type. The second one involves a coupled system of pdes of degenerate parabolic–hyperbolic type. We state an existence result of weak solutions for both models.

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13

Nazir, Umar, Muhammad Sohail, Muhammad Bilal Hafeez, and Marek Krawczuk. "Significant Production of Thermal Energy in Partially Ionized Hyperbolic Tangent Material Based on Ternary Hybrid Nanomaterials." Energies 14, no. 21 (October 21, 2021): 6911. http://dx.doi.org/10.3390/en14216911.

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Nanoparticles are frequently used to enhance the thermal performance of numerous materials. This study has many practical applications for activities that have to minimize losses of energy due to several impacts. This study investigates the inclusion of ternary hybrid nanoparticles in a partially ionized hyperbolic tangent liquid passed over a stretched melting surface. The fluid motion equation is presented by considering the rotation effect. The thermal energy expression is derived by the contribution of Joule heat and viscous dissipation. Flow equations were modeled by using the concept of boundary layer theory, which occurs in the form of a coupled system of partial differential equations (PDEs). To reduce the complexity, the derived PDEs (partial differential equations) were transformed into a set of ordinary differential equations (ODEs) by engaging in similarity transformations. Afterwards, the converted ODEs were handled via a finite element procedure. The utilization and effectiveness of the methodology are demonstrated by listing the mesh-free survey and comparative analysis. Several important graphs were prepared to show the contribution of emerging parameters on fluid velocity and temperature profile. The findings show that the finite element method is a powerful tool for handling the complex coupled ordinary differential equation system, arising in fluid mechanics and other related dissipation applications in applied science. Furthermore, enhancements in the Forchheimer parameter and the Weissenberg number are necessary to control the fluid velocity.

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14

ALVES, JOSÉ F., and HELDER VILARINHO. "Strong stochastic stability for non-uniformly expanding maps." Ergodic Theory and Dynamical Systems 33, no. 3 (August 6, 2012): 647–92. http://dx.doi.org/10.1017/s0143385712000077.

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AbstractWe consider random perturbations of discrete-time dynamical systems. We give sufficient conditions for the stochastic stability of certain classes of maps, in a strong sense. This improves the main result in Alves and Araújo [Random perturbations of non-uniformly expanding maps. Astérisque 286 (2003), 25–62], where the stochastic stability in the $\mathrm {weak}^*$ topology was proved. Here, under slightly weaker assumptions on the random perturbations, we obtain a stronger version of stochastic stability: convergence of the density of the stationary measure to the density of the Sinai–Ruelle–Bowen (SRB) measure of the unperturbed system in the $L^1$-norm. As an application of our results, we obtain strong stochastic stability for two classes of non-uniformly expanding maps. The first one is an open class of local diffeomorphisms introduced in Alves, Bonatti and Viana [SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140 (2000), 351–398] and the second one is the class of Viana maps.

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15

Medina, Virtudes, Antonio Lorenzo, and Mario Dı́az. "Electrogenic Na+-dependentl-alanine transport in the lizard duodenum. Involvement of systems A and ASC." American Journal of Physiology-Regulatory, Integrative and Comparative Physiology 280, no. 3 (March 1, 2001): R612—R622. http://dx.doi.org/10.1152/ajpregu.2001.280.3.r612.

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l-Alanine transport across the isolated duodenal mucosa of the lizard Gallotia galloti has been studied in Ussing chambers under short-circuit conditions. Net l-alanine fluxes, transepithelial potential difference (PD), and short-circuit current ( Isc) showed concentration-dependent relationships. Na+-dependent l-alanine transport was substantially inhibited by the analog α-methyl aminoisobutyric acid (MeAIB). Likewise, MeAIB fluxes were completely inhibited byl-alanine, indicating the presence of system A for neutral amino acid transport. System A transport activity was electrogenic and exhibited hyperbolic relationships for net MeAIB fluxes, PD, and Isc, which displayed similar apparent K m values. Na+-dependentl-alanine transport, but not MeAIB transport, was partially inhibited by l-serine and l-cysteine, indicating the participation of system ASC. This transport activity represents the major pathway for l-alanine absorption and seemed to operate in an electroneutral mode with a negligible contribution to the l-alanine-induced electrogenicity. It is concluded from the present study that the active Na+-dependent l-alanine transport across the isolated duodenal mucosa of Gallotia galloti results from the independent activity of systems A and ASC for neutral amino acid transport.

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16

Harrison, David K., Mario Fasching, Mona Fontana-Ayoub, and Erich Gnaiger. "Cytochrome redox states and respiratory control in mouse and beef heart mitochondria at steady-state levels of hypoxia." Journal of Applied Physiology 119, no. 10 (November 15, 2015): 1210–18. http://dx.doi.org/10.1152/japplphysiol.00146.2015.

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Mitochondrial control of cellular redox states is a fundamental component of cell signaling in the coordination of core energy metabolism and homeostasis during normoxia and hypoxia. We investigated the relationship between cytochrome redox states and mitochondrial oxygen consumption at steady-state levels of hypoxia in mitochondria isolated from beef and mouse heart (BHImt, MHImt), comparing two species with different cardiac dynamics and local oxygen demands. A low-noise, rapid spectrophotometric system using visible light for the measurement of cytochrome redox states was combined with high-resolution respirometry. Monophasic hyperbolic relationships were observed between oxygen consumption, JO2, and oxygen partial pressure, Po2, within the range <1.1 kPa (8.3 mmHg; 13 μM). P50 j (Po2 at 0.5· Jmax) was 0.015 ± 0.0004 and 0.021 ± 0.003 kPa (0.11 and 0.16 mmHg) for BHImt and MHImt, respectively. Maximum oxygen consumption, Jmax, was measured at saturating ADP levels (OXPHOS capacity) with Complex I-linked substrate supply. Redox states of cytochromes aa3 and c were biphasic hyperbolic functions of Po2. The relationship between cytochrome oxidation state and oxygen consumption revealed a separation of distinct phases from mild to severe and deep hypoxia. When cytochrome c oxidation increased from fully reduced to 45% oxidized at 0.1 Jmax, Po2 was as low as 0.002 kPa (0.02 μM), and trace amounts of oxygen are sufficient to partially oxidize the cytochromes. At higher Po2 under severe hypoxia, respiration increases steeply, whereas redox changes are small. Under mild hypoxia, the steep slope of oxidation of cytochrome c when flux remains more stable represents a cushioning mechanism that helps to maintain respiration high at the onset of hypoxia.

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17

Clark Butler and Disheng Xu. "Uniformly quasiconformal partially hyperbolic systems." Annales scientifiques de l'École normale supérieure 51, no. 5 (2018): 1085–127. http://dx.doi.org/10.24033/asens.2372.

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18

Zhang, Pengfei. "Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems." Discrete & Continuous Dynamical Systems - A 32, no. 4 (2012): 1435–47. http://dx.doi.org/10.3934/dcds.2012.32.1435.

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19

Hammerlindl, Andy. "Ergodic components of partially hyperbolic systems." Commentarii Mathematici Helvetici 92, no. 1 (2017): 131–84. http://dx.doi.org/10.4171/cmh/409.

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20

Dolgopyat, Dmitry. "Limit theorems for partially hyperbolic systems." Transactions of the American Mathematical Society 356, no. 4 (September 22, 2003): 1637–89. http://dx.doi.org/10.1090/s0002-9947-03-03335-x.

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21

Katok, A., and A. Kononenko. "Cocycles' stability for partially hyperbolic systems." Mathematical Research Letters 3, no. 2 (1996): 191–210. http://dx.doi.org/10.4310/mrl.1996.v3.n2.a6.

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22

Gogolev, Andrey, Pedro Ontaneda, and Federico Rodriguez Hertz. "New partially hyperbolic dynamical systems I." Acta Mathematica 215, no. 2 (2015): 363–93. http://dx.doi.org/10.1007/s11511-016-0135-3.

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23

XU, LAN, and BEIMEI CHEN. "TWO NOTES ABOUT THE ERGODICITY OF PARTIALLY HYPERBOLIC SYSTEMS." International Journal of Bifurcation and Chaos 23, no. 07 (July 2013): 1350123. http://dx.doi.org/10.1142/s021812741350123x.

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In this paper, two notes about the ergodicity of partially hyperbolic systems are given. First one is the ergodicity for a C2 volume preserving partially hyperbolic diffeomorphism of a smooth compact Riemannian manifold which is essentially accessible and weak central exponentially bunched. Second one is that for a C2 partially hyperbolic diffeomorphism, if both forward and backward center bunched are a full probability set, then it is center bunched in the sense of [Burns & Wilkinson, 2010].

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24

HU, HUYI, YUNHUA ZHOU, and YUJUN ZHU. "Quasi-shadowing for partially hyperbolic diffeomorphisms." Ergodic Theory and Dynamical Systems 35, no. 2 (December 15, 2014): 412–30. http://dx.doi.org/10.1017/etds.2014.126.

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AbstractA partially hyperbolic diffeomorphism $f$ has the quasi-shadowing property if for any pseudo orbit $\{x_{k}\}_{k\in \mathbb{Z}}$, there is a sequence of points $\{y_{k}\}_{k\in \mathbb{Z}}$ tracing it in which $y_{k+1}$ is obtained from $f(y_{k})$ by a motion ${\it\tau}$ along the center direction. We show that any partially hyperbolic diffeomorphism has the quasi-shadowing property, and if $f$ has a $C^{1}$ center foliation then we can require ${\it\tau}$ to move the points along the center foliation. As applications, we show that any partially hyperbolic diffeomorphism is topologically quasi-stable under $C^{0}$-perturbation. When $f$ has a uniformly compact $C^{1}$ center foliation, we also give partially hyperbolic diffeomorphism versions of some theorems which hold for uniformly hyperbolic systems, such as the Anosov closing lemma, the cloud lemma and the spectral decomposition theorem.

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25

Climenhaga, Vaughn, Yakov Pesin, and Agnieszka Zelerowicz. "Equilibrium measures for some partially hyperbolic systems." Journal of Modern Dynamics 16 (2020): 155–205. http://dx.doi.org/10.3934/jmd.2020006.

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26

Burns, Keith, and Amie Wilkinson. "On the ergodicity of partially hyperbolic systems." Annals of Mathematics 171, no. 1 (March 17, 2010): 451–89. http://dx.doi.org/10.4007/annals.2010.171.451.

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27

Wirth, Jens. "Diffusion phenomena for partially dissipative hyperbolic systems." Journal of Mathematical Analysis and Applications 414, no. 2 (June 2014): 666–77. http://dx.doi.org/10.1016/j.jmaa.2014.01.034.

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28

Tian, Xueting, and Weisheng Wu. "Unstable entropies and dimension theory of partially hyperbolic systems." Nonlinearity 35, no. 1 (December 14, 2021): 658–80. http://dx.doi.org/10.1088/1361-6544/ac3dcb.

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Abstract In this paper we define unstable topological entropy for any subsets (not necessarily compact or invariant) in partially hyperbolic systems as a Carathéodory–Pesin dimension characteristic, motivated by the work of Bowen and Pesin etc. We then establish some basic results in dimension theory for Bowen unstable topological entropy, including an entropy distribution principle and a variational principle in general setting. As applications of this new concept, we study unstable topological entropy of saturated sets and extend some results in Bowen (1973 Trans. Am. Math. Soc. 184 125–36); Pfister and Sullivan (2007 Ergod. Theor. Dynam. Syst. 27 929–56). Our results give new insights to the multifractal analysis for partially hyperbolic systems.

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29

De Simoi, Jacopo, and Carlangelo Liverani. "Limit theorems for fast–slow partially hyperbolic systems." Inventiones mathematicae 213, no. 3 (May 12, 2018): 811–1016. http://dx.doi.org/10.1007/s00222-018-0798-9.

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30

Kalinin, Boris, and Victoria Sadovskaya. "Cocycles with one exponent over partially hyperbolic systems." Geometriae Dedicata 167, no. 1 (December 2, 2012): 167–88. http://dx.doi.org/10.1007/s10711-012-9808-z.

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31

Beauchard, Karine, and Enrique Zuazua. "Large Time Asymptotics for Partially Dissipative Hyperbolic Systems." Archive for Rational Mechanics and Analysis 199, no. 1 (April 17, 2010): 177–227. http://dx.doi.org/10.1007/s00205-010-0321-y.

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Suprun, E. N. "Prediction of uncontrolled course of bronchial asthma in children based on polymorphisms of genes of signaling molecules of the immune system and detoxification genes." Bulletin Physiology and Pathology of Respiration 1, no. 86 (December 22, 2022): 56–61. http://dx.doi.org/10.36604/1998-5029-2022-86-56-61.

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Aim. To study the possibility of predicting the asthma control at various stages of the development of the disease, possibly on the basis of taking into account the genetic polymorphisms of Toll-like receptors, cytokines and detoxification system genes using the statistical method of learning neural networks.Materials and methods. We examined 167 children with bronchial asthma. The degree of asthma control was determined, the following mutations were detected: TLR2-Arg753Glu, TLR4-Asp299Gly, TLR4-Ghr399Ile, TLR9-T1237C, TLR9-A2848G; IL4-C589T, IL6- C174G, IL10-G1082A, IL10-C592A, IL10-C819T, IL12B-A1188C, TNFa-G308A; GSTM, GSTT, GSTM/GSTT, GSTP1 Ile105Val, GSTP1 Ala114Val, by PCR. The STATISTICA Automated Neural Networks package was used to model neural networks.Results. The model is based on the MLP (15-9-3) multilayer perceptron architecture with a layer of 15 input neurons (by the number of analyzed variables), a hidden intermediate layer of 9 neurons and an output layer of 3 neurons by the number of values of the classified variable (control). The training algorithm was chosen by BFGS as the most adequate to the classification task. The error function is traditionally chosen as the sum of squared deviations. The activation function of output neurons is Softmax. The activation function of the intermediate layer is hyperbolic. The volume of the training sample was 88 sets. The volume of samples for testing and quality control of the model was 36 sets. The resulting model was able to predict 79.01% of the correct values of the target variable (the degree of asthma control).Conclusion. The application of the developed program makes it possible to predict the possibility of uncontrolled or partially controlled asthma at any stage of the disease, including preclinical and pre-nosological for groups with a high risk of asthma. This allows you to individually adjust the measures of secondary and even primary prevention of asthma within the personalization of therapeutic approaches.

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33

HUA, YONGXIA, RADU SAGHIN, and ZHIHONG XIA. "Topological entropy and partially hyperbolic diffeomorphisms." Ergodic Theory and Dynamical Systems 28, no. 3 (June 2008): 843–62. http://dx.doi.org/10.1017/s0143385707000405.

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AbstractWe consider partially hyperbolic diffeomorphisms on compact manifolds. We define the notion of the unstable and stable foliations stably carrying some unique non-trivial homologies. Under this topological assumption, we prove the following two results: if the center foliation is one-dimensional, then the topological entropy is locally a constant; and if the center foliation is two-dimensional, then the topological entropy is continuous on the set of all $C^{\infty }$ diffeomorphisms. The proof uses a topological invariant we introduced, Yomdin’s theorem on upper semi-continuity, Katok’s theorem on lower semi-continuity for two-dimensional systems, and a refined Pesin–Ruelle inequality we proved for partially hyperbolic diffeomorphisms.

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34

de Simoi, Jacopo, Carlangelo Liverani, Christophe Poquet, and Denis Volk. "Fast–Slow Partially Hyperbolic Systems Versus Freidlin–Wentzell Random Systems." Journal of Statistical Physics 166, no. 3-4 (September 28, 2016): 650–79. http://dx.doi.org/10.1007/s10955-016-1628-3.

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35

Turo, Jan. "On some class of quasilinear hyperbolic systems of partial differential-functional equations of the first order." Czechoslovak Mathematical Journal 36, no. 2 (1986): 185–97. http://dx.doi.org/10.21136/cmj.1986.102083.

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36

Castorrini, Roberto, and Carlangelo Liverani. "Quantitative statistical properties of two-dimensional partially hyperbolic systems." Advances in Mathematics 409 (November 2022): 108625. http://dx.doi.org/10.1016/j.aim.2022.108625.

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37

Liu, Cunming, and Peng Qu. "Global classical solution to partially dissipative quasilinear hyperbolic systems." Journal de Mathématiques Pures et Appliquées 97, no. 3 (March 2012): 262–81. http://dx.doi.org/10.1016/j.matpur.2011.06.001.

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38

Dolgopyat, Dmitry. "On differentiability of SRB states for partially hyperbolic systems." Inventiones Mathematicae 155, no. 2 (February 1, 2004): 389–449. http://dx.doi.org/10.1007/s00222-003-0324-5.

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39

Zhou, Yi. "Global classical solutions to partially dissipative quasilinear hyperbolic systems." Chinese Annals of Mathematics, Series B 32, no. 5 (August 26, 2011): 771–80. http://dx.doi.org/10.1007/s11401-011-0666-z.

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40

WU, WEISHENG. "Local unstable entropies of partially hyperbolic diffeomorphisms." Ergodic Theory and Dynamical Systems 40, no. 8 (February 26, 2019): 2274–304. http://dx.doi.org/10.1017/etds.2019.3.

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Consider a $C^{1}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$. Following the ideas in establishing the local variational principle for topological dynamical systems, we introduce the notions of local unstable metric entropies (and local unstable topological entropy) relative to a Borel cover ${\mathcal{U}}$ of $M$. It is shown that they coincide with the unstable metric entropy (and unstable topological entropy, respectively), when ${\mathcal{U}}$ is an open cover with small diameter. We also define the unstable tail entropy in the sense of Bowen and the unstable topological conditional entropy in the sense of Misiurewicz, and demonstrate that both of them vanish. Some generalizations of these results to the case of unstable pressure are also investigated.

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41

CASTRO, ARMANDO, and TEÓFILO NASCIMENTO. "Statistical properties of the maximal entropy measure for partially hyperbolic attractors." Ergodic Theory and Dynamical Systems 37, no. 4 (January 28, 2016): 1060–101. http://dx.doi.org/10.1017/etds.2015.86.

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We show the existence and uniqueness of the maximal entropy probability measure for partially hyperbolic diffeomorphisms which are semiconjugate to non-uniformly expanding maps. Using the theory of projective metrics on cones, we then prove exponential decay of correlations for Hölder continuous observables and the central limit theorem for the maximal entropy probability measure. Moreover, for systems derived from a solenoid, we also prove the statistical stability for the maximal entropy probability measure. Finally, we use such techniques to obtain similar results in a context containing partially hyperbolic systems derived from Anosov.

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De Simoi, Jacopo, and Carlangelo Liverani. "Statistical properties of mostly contracting fast-slow partially hyperbolic systems." Inventiones mathematicae 206, no. 1 (March 4, 2016): 147–227. http://dx.doi.org/10.1007/s00222-016-0651-y.

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Zhou, Yunhua. "Non-zero Lyapunov exponents for some conservative partially hyperbolic systems." Proceedings of the American Mathematical Society 143, no. 7 (February 17, 2015): 3147–53. http://dx.doi.org/10.1090/s0002-9939-2015-12498-7.

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Mi, Zeya, Yongluo Cao, and Dawei Yang. "A note on partially hyperbolic systems with mostly expanding centers." Proceedings of the American Mathematical Society 145, no. 12 (September 7, 2017): 5299–313. http://dx.doi.org/10.1090/proc/13701.

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Kowalski, Julia, and Jim N. McElwaine. "Shallow two-component gravity-driven flows with vertical variation." Journal of Fluid Mechanics 714 (January 2, 2013): 434–62. http://dx.doi.org/10.1017/jfm.2012.489.

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AbstractGravity-driven geophysical mass flows often consist of a heterogeneous fluid–solid mixture. The complex interplay between the components leads to phenomena such as lateral levee formation in avalanches, or a granular front and an excess fluid pore pressure in debris flows. These effects are very important for predicting runout and the forces on structures, yet they are only partially represented in simplified shallow flow theories, since rearrangement of the mixture composition perpendicular to the main flow direction is neglected. In realistic flows, however, rheological properties and effective basal drag may depend strongly on the relative concentration of the components. We address this problem and present a depth-averaged model for shallow mixtures that explicitly allows for rearrangement in this direction. In particular we consider a fluid–solid mixture that experiences bulk horizontal motion, as well as internal sedimentation and resuspension of the particles, and therefore resembles the case of a debris flow. Starting from general mixture theory we derive bulk balance laws and an evolution equation for the particle concentration. Depth-integration yields a shallow mixture flow model in terms of bulk mass, depth-averaged particle concentration, the particle vertical centre of mass and the depth-averaged velocity. This new equation in this model for the particle vertical centre of mass is derived by taking the first moment, with respect to the vertical coordinate, of the particle mass conservation equation. Our approach does not make the Boussinesq approximation and results in additional terms coupling the momentum flux to the vertical centre of mass. The system is hyperbolic and reduces to the shallow-water equations in the homogeneous limit of a pure fluid or perfect mixing. We highlight the effects of sedimentation on resuspension and finally present a simple friction feedback which qualitatively resembles a large-scale experimental debris flow data set acquired at the Illgraben, Switzerland.

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Koellermeier, Julian, and Manuel Torrilhon. "Numerical Study of Partially Conservative Moment Equations in Kinetic Theory." Communications in Computational Physics 21, no. 4 (March 8, 2017): 981–1011. http://dx.doi.org/10.4208/cicp.oa-2016-0053.

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AbstractMoment models are often used for the solution of kinetic equations such as the Boltzmann equation. Unfortunately, standard models like Grad's equations are not hyperbolic and can lead to nonphysical solutions. Newly derived moment models like the Hyperbolic Moment Equations and the Quadrature-Based Moment Equations yield globally hyperbolic equations but are given in partially conservative form that cannot be written as a conservative system.In this paper we investigate the applicability of different dedicated numerical schemes to solve the partially conservative model equations. Caused by the non-conservative type of equation we obtain differences in the numerical solutions, but due to the structure of the moment systems we show that these effects are very small for standard simulation cases. After successful identification of useful numerical settings we show a convergence study for a shock tube problem and compare the results to a discrete velocity solution. The results are in good agreement with the reference solution and we see convergence considering an increasing number of moments.

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Zhao, Liang, and Shuai Xi. "Convergence rate from systems of balance laws to isotropic parabolic systems, a periodic case." Asymptotic Analysis 124, no. 1-2 (July 8, 2021): 163–98. http://dx.doi.org/10.3233/asy-211687.

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It is proved that partially dissipative hyperbolic systems converge globally-in-time to parabolic systems in a slow time scaling, when initial data are smooth and sufficiently close to constant equilibrium states. Based on this result, we establish the global-in-time error estimates between the smooth solutions to the partially dissipative hyperbolic systems and those to the isotropic parabolic limiting systems in a three dimensional torus, rather than in the one dimensional whole space (Appl. Anal. 100(5) (2021) 1079–1095). This avoids the condition raised for the strong connection between the flux and the source term and make the result obtained more generalized. In the proof, we provide a similar stream function technique which is valid for the three dimensional periodic case. Similar method is provided for the one-dimensional periodic case. As applications of the results, we give several examples arising from physical models at the end of the paper.

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Huang, Ping, Chenwei Wang, and Ercai Chen. "Unstable topological entropy in mean u-metrics for partially hyperbolic systems." Dynamical Systems 36, no. 3 (June 3, 2021): 387–403. http://dx.doi.org/10.1080/14689367.2021.1923659.

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Burguet, David, and Todd Fisher. "Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle." Discrete & Continuous Dynamical Systems - A 33, no. 6 (2013): 2253–70. http://dx.doi.org/10.3934/dcds.2013.33.2253.

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Li, Xiaolong, and Katsutoshi Shinohara. "On super-exponential divergence of periodic points for partially hyperbolic systems." Discrete & Continuous Dynamical Systems 42, no. 4 (2022): 1707. http://dx.doi.org/10.3934/dcds.2021169.

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<p style='text-indent:20px;'>We say that a diffeomorphism <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula> is super-exponentially divergent if for every <inline-formula><tex-math id="M2">\begin{document}$ b>1 $\end{document}</tex-math></inline-formula> the lower limit of <inline-formula><tex-math id="M3">\begin{document}$ \#\mbox{Per}_n(f)/b^n $\end{document}</tex-math></inline-formula> diverges to infinity, where <inline-formula><tex-math id="M4">\begin{document}$ \mbox{Per}_n(f) $\end{document}</tex-math></inline-formula> is the set of all periodic points of <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula> with period <inline-formula><tex-math id="M6">\begin{document}$ n $\end{document}</tex-math></inline-formula>. This property is stronger than the usual super-exponential growth of the number of periodic points. We show that for any <inline-formula><tex-math id="M7">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional smooth closed manifold <inline-formula><tex-math id="M8">\begin{document}$ M $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M9">\begin{document}$ n\ge 3 $\end{document}</tex-math></inline-formula>, there exists a non-empty open subset <inline-formula><tex-math id="M10">\begin{document}$ \mathcal{O} $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M11">\begin{document}$ \mbox{Diff}^1(M) $\end{document}</tex-math></inline-formula> such that diffeomorphisms with super-exponentially divergent property form a dense subset of <inline-formula><tex-math id="M12">\begin{document}$ \mathcal{O} $\end{document}</tex-math></inline-formula> in the <inline-formula><tex-math id="M13">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula>-topology. A relevant result about the growth rate of the lower limit of the number of periodic points for diffeomorphisms in a <inline-formula><tex-math id="M14">\begin{document}$ C^r $\end{document}</tex-math></inline-formula>-residual subset of <inline-formula><tex-math id="M15">\begin{document}$ \mbox{Diff}^r(M)\ (1\le r\le \infty) $\end{document}</tex-math></inline-formula> is also shown.</p>

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