Unstable Directions Research Articles

Bifurcations and degenerate periodic points in a three dimensional chaotic fluid flow.

Analysis of the periodic points of a conservative periodic dynamical system uncovers the basic kinematic structure of the transport dynamics and identifies regions of local stability or chaos. While elliptic and hyperbolic points typically govern such behaviour in 3D systems, degenerate (parabolic) points also play an important role. These points represent a bifurcation in local stability and Lagrangian topology. In this study, we consider the ramifications of the two types of degenerate periodic points that occur in a model 3D fluid flow. (1) Period-tripling bifurcations occur when the local rotation angle associated with elliptic points is reversed, creating a reversal in the orientation of associated Lagrangian structures. Even though a single unstable point is created, the bifurcation in local stability has a large influence on local transport and the global arrangement of manifolds as the unstable degenerate point has three stable and three unstable directions, similar to hyperbolic points, and occurs at the intersection of three hyperbolic periodic lines. The presence of period-tripling bifurcation points indicates regions of both chaos and confinement, with the extent of each depending on the nature of the associated manifold intersections. (2) The second type of bifurcation occurs when periodic lines become tangent to local or global invariant surfaces. This bifurcation creates both saddle-centre bifurcations which can create both chaotic and stable regions, and period-doubling bifurcations which are a common route to chaos in 2D systems. We provide conditions for the occurrence of these tangent bifurcations in 3D conservative systems, as well as constraints on the possible types of tangent bifurcation that can occur based on topological considerations.

SRB measures for a class of partially hyperbolic attractors in Hilbert spaces

In this paper, we study the existence of SRB measures and their properties for infinite dimensional dynamical systems in a Hilbert space. We show several results including (i) if the system has a partially hyperbolic attractor with nontrivial finite dimensional unstable directions, then it has at least one SRB measure; (ii) if the attractor is uniformly hyperbolic and the system is topological mixing and the splitting is Hölder continuous, then there exists a unique SRB measure which is mixing; (iii) if the attractor is uniformly hyperbolic and the system is non-wondering and the splitting is Hölder continuous, then there exist at most finitely many SRB measures; (iv) for a given hyperbolic measure, there exist at most countably many ergodic components whose basin contains an observable set.

Lyapunov theorems for exponential dichotomies in Hilbert spaces

For a nonautonomous dynamics defined by a sequence of linear operators, we obtain a complete characterization of the notion of a uniform exponential dichotomy in terms of the existence of appropriate Lyapunov sequences. In sharp contrast to previous results, we consider the case of noninvertible dynamics, thus requiring only the invertibility of operators along the unstable direction. Furthermore, we deal with operators acting on an arbitrary Hilbert space. As a nontrivial application of our work, we study the persistence of uniform exponential behavior under small linear and nonlinear perturbations.

A minimization principle for the description of modes associated with finite-time instabilities.

We introduce a minimization formulation for the determination of a finite-dimensional, time-dependent, orthonormal basis that captures directions of the phase space associated with transient instabilities. While these instabilities have finite lifetime, they can play a crucial role either by altering the system dynamics through the activation of other instabilities or by creating sudden nonlinear energy transfers that lead to extreme responses. However, their essentially transient character makes their description a particularly challenging task. We develop a minimization framework that focuses on the optimal approximation of the system dynamics in the neighbourhood of the system state. This minimization formulation results in differential equations that evolve a time-dependent basis so that it optimally approximates the most unstable directions. We demonstrate the capability of the method for two families of problems: (i) linear systems, including the advection-diffusion operator in a strongly non-normal regime as well as the Orr-Sommerfeld/Squire operator, and (ii) nonlinear problems, including a low-dimensional system with transient instabilities and the vertical jet in cross-flow. We demonstrate that the time-dependent subspace captures the strongly transient non-normal energy growth (in the short-time regime), while for longer times the modes capture the expected asymptotic behaviour.

Head Direction Cell Activity Is Absent in Mice without the Horizontal Semicircular Canals.

Head direction (HD) cells fire when an animal faces a particular direction in its environment, and they are thought to represent the neural correlate of the animal's perceived spatial orientation. Previous studies have shown that vestibular information is critical for generating the HD signal but have not delineated whether information from all three semicircular canals or just the horizontal canals, which are primarily sensitive to angular head rotation in the horizontal (yaw) plane, are critical for the signal. Here, we monitored cell activity in the anterodorsal thalamus (ADN), an area known to contain HD cells, in epstatic circler (Ecl) mice, which have a bilateral malformation of the horizontal (lateral) semicircular canals. Ecl mice and their littermates that did not express the mutation (controls) were implanted with recording electrodes in the ADN. Results confirm the important role the horizontal canals play in forming the HD signal. Although normal HD cell activity (Raleigh's r > 0.4) was recorded in control mice, no such activity was found in Ecl mice, although some cells had activity that was mildly modulated by HD (0.4 > r > 0.2). Importantly, we also observed activity in Ecl mice that was best characterized as bursty--a pattern of activity similar to an HD signal but without any preferred firing direction. These results suggest that the neural structure for the HD network remains intact in Ecl mice, but the absence of normal horizontal canals results in an inability to control the network properly and brings about an unstable HD signal. Significance statement: Cells in the anterior dorsal thalamic nucleus normally fire in relation to the animal's directional heading with respect to the environment--so-called head direction cells. To understand how these head direction cells generate their activity, we recorded single-unit activity from the anterior dorsal thalamus in transgenic mice that lack functional horizontal semicircular canals. We show that the neural network for the head direction signal remains intact in these mice, but that the absence of normal horizontal canals results in an inability to control the network properly and brings about an unstable head direction signal.

On the unstable directions and Lyapunov exponents of Anosov endomorphisms

Despite the invertible setting, Anosov endomorphisms may have infinitely many unstable directions. Here we prove, under transitivity assumption, that an Anosov endomorphism on a closed manifold $M,$ is either special (that is, every $x \in M$ has only one unstable direction) or for a typical point in $M$ there are infinitely many unstable directions. Other result of this work is the semi rigidity of the unstable Lyapunov exponent of a $C^{1+\alpha}$ codimension one Anosov endomorphism and $C^1$ close to a linear endomorphism of $\mathbb{T}^n$ for $(n \geq 2).$ In the appendix we give a proof for ergodicity of $C^{1+\alpha}, \alpha > 0,$ conservative Anosov endomorphism.

160 W laser-diode end-pumped Nd:YVO4 slab laser with positive branch hybrid resonator

In this paper, we present an experiment on a continuous-wave Nd:YVO4 Innoslab laser diode-pumped at 808 nm. The LD stack is composed of six bars, with the central wavelength fixed at 808 nm by adjusting the cooling water temperature. The emission from each diode laser bar is individually collimated by micro lens, which is coupled into a coupling system. The coupling system includes a planar waveguide, four cylindrical lenses and a spherical lens. The planar waveguide is used to shape the emitting beams of LD to obtain uniform distribution. The coupling system leads to a pump power loss of ~12%. By the coupling system, we obtain a homogeneous pumping line of ~0.4 mm22 mm coupled into the 0.3 at. % Nd:YVO4 (22 mm10 mm1 mm) crystal. The Nd:YVO4 crystal is a-cut with c axis along 22 mm direction. Indium foil is used for uniform thermal contact and cooling. The laser crystal is mounted between two water-cooled copper heat sinks with two large faces 22 mm10 mm. The heat conduction inside the laser crystal is quasionedimensional. The two 22 mm1 mm surfaces are polished and antireflectioncoated for the pump light and the laser light. Temperature of LD stack and laser crystal are controlled by cooling circulating water. The resonator consists of the input mirror (M1) and the output mirror (M2). M1 is a concave mirror with a radius of R1=500 mm, which is coated for high refection (HR) at 1064 nm and high transmission (HT) at 808 nm. The output mirror (M2) is a cylindrical mirror with a radius of R2=-350 mm, which is coated for HR at 1064 nm. M2 is cut and polished at one edge where the large beam exits. M1 and M2 constitute a stable resonator in vertical direction and off-axis unstable positive confocal resonator in the horizontal direction. In theory, the length of the resonator is L=(R1+R2)/2=75 mm. In fact, the length of the resonator is the same as the theoretical value. The equivalent transmission of the resonator is T=1-|R2/R1|=30%. At a pumping power of 462 W, a maximum power of 160 W continuous wave laser output is obtained, with the stability being 2.6%. Considering 88% of the coupling efficiency and 95% of absorbed efficiency, the optical-to-optical efficiency and slope efficiency are 41.5% and 47.7%, respectively. When the output power is 145 W, the beam quality M2 factors in the stable direction and unstable direction are 2.21 and 1.37, respectively. With the help of the ANSYS software, the temperature distribution in the crystal at the pumped power of 462 W is demonstrated. The temperature distributions are analogous to exponential decays in the Z-direction and parabola decay in the Y-direction, respectively. The maximum temperature difference is 71.6 K in our experiment. The thermal lens is negligible in the unstable direction because the temperature distribution is uniform. That is why the Innoslab laser is beneficial to the power scaling, as it keeps the power density constant, and enlarges the size of gain medium in the unstable direction to inject bigger power to obtain a higher power output, and maintain the constant beam quality.

Equilibrium Measures at Temperature Zero for Hénon-Like Maps at the First Bifurcation

We develop a thermodynamic formalism for a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. For any $t\in\mathbb R$ we prove the existence of an invariant Borel probability measure which minimizes the free energy associated with a noncontinuous geometric potential $-t\log J^u$, where $J^u$ denotes the Jacobian in the unstable direction. We characterize accumulation points of these measures as $t\to\pm\infty$ in terms of the unstable Lyapunov exponent.

Admissibility and nonuniformly hyperbolic sets

We obtain a characterization of two classes of dynamics with nonuniformly hyperbolic behavior in terms of an admissibility property. Namely, we consider exponential dichotomies with respect to a sequence of norms and nonuniformly hyperbolic sets. We note that the approach to establishing exponential bounds along the stable and the unstable directions differs from the standard technique of substituting test sequences. Moreover, we obtain the bounds in a single step.

Coexistence and Dynamical Connections between Hyperchaos and Chaos in the 4D Rössler System: A Computer-Assisted Proof

It has recently been reported [P. C. Reich, Neurocomputing, 74 (2011), pp. 3361--3364] that it is quite difficult to distinguish between chaos and hyperchaos in numerical simulations which are frequently “noisy.” For the classical four-dimensional (4D) Rössler model [O. E. Rössler, Phys. Lett. A, 71 (1979), pp. 155--157] we show that the coexistence of two invariant sets with different nature (a global hyperchaotic invariant set and a chaotic attractor) and heteroclinic connections between them give rise to long hyperchaotic transient behavior, and therefore it provides a mechanism for noisy simulations. The same phenomena is expected in other 4D and higher-dimensional systems. The proof combines topological and smooth methods with rigorous numerical computations. The existence of (hyper)chaotic sets is proved by the method of covering relations [P. Zgliczyński and M. Gidea, J. Differential Equations, 202 (2004), pp. 32--58]. We extend this method to the case of a nonincreasing number of unstable directions which is necessary to study hyperchaos to chaos transport. The cone condition [H. Kokubu, D. Wilczak, and P. Zgliczyński, Nonlinearity, 20 (2007), pp. 2147--2174] is used to prove the existence of homoclinic and heteroclinic orbits between some periodic orbits which belong to both hyperchaotic and chaotic invariant sets. In particular, the existence of a countable infinity of heteroclinic orbits linking hyperchaos with chaos justifies the presence of long transient behavior.

Nonautonomous Dynamics with Discrete Time and Topological Equivalence

For evolution families with discrete time, we show that any exponential dichotomy is topologically equivalent to a certain normal form, in which the exponential behavior along the stable and unstable directions are multiples of the identity. We consider the general case of a generalized exponential dichotomy in which the usual exponential behavior is replaced by an arbitrary growth rate. In addition, we show that the topological equivalence between two evolution families with generalized exponential dichotomies can be completely characterized in terms of a notion of equivalence between the growth rates.

Exact coherent states and connections to turbulent dynamics in minimal channel flow

Several new families of nonlinear three-dimensional travelling wave solutions to the Navier–Stokes equation, also known as exact coherent states, are computed for Newtonian plane Poiseuille flow. The symmetries and streak/vortex structures are reported and their possible connections to critical layer dynamics are examined. While some of the solutions clearly display fluctuations that are localized around the critical layer (the surface on which the streamwise velocity matches the wave speed of the solution), for others this connection is not as clear. Dynamical trajectories along unstable directions of the solutions are computed. Over certain ranges of Reynolds number, two solution families are shown to lie on the basin boundary between laminar and turbulent flow. Direct comparison of nonlinear travelling wave solutions to turbulent flow in the same channel is presented. The state-space dynamics of the turbulent flow is organized around one of the newly identified travelling wave families, and in particular the lower-branch solutions of this family are closely approached during transient excursions away from the dominant behaviour. These observations provide a firm dynamical-systems foundation for prior observations that minimal channel turbulence displays time intervals of ‘active’ turbulence punctuated by brief periods of ‘hibernation’ (see, e.g., Xi & Graham, Phys. Rev. Lett., vol. 104, 2010, 218301). The hibernating intervals are approaches to lower-branch nonlinear travelling waves. Representing these solutions on a Prandtl–von Kármán plot illustrates how their bulk flow properties are related to those of Newtonian turbulence as well as the universal asymptotic state called maximum drag reduction (MDR) found in viscoelastic turbulent flow. In particular, the lower- and upper-branch solutions of the family around which the minimal channel dynamics is organized appear to approach the MDR asymptote and the classical Newtonian result respectively, in terms of both bulk velocity and mean velocity profile.

High-Power Continuous-Wave Nd:GdVO4 Solid-State Laser Dual-End-Pumped at 880 nm

A high-power cw all-solid-state Nd:GdVO4 laser operating at 880 nm is reported. The laser consists of a low doped level Nd:GdVO4 crystal dual-end-pumped by two high-power diode lasers and a compact negative confocal unstable–stable hybrid resonator. At an incident pump power of 820 W, a maximum cw output of 240 W at 1064 nm is obtained. The optical-to-optical efficiency and slope efficiency are 40.7% and 53.2%, respectively. The M2 factors in the unstable direction and in the stable direction are 4.38 and 5.44, respectively.

Statistical energy conservation principle for inhomogeneous turbulent dynamical systems

Understanding the complexity of anisotropic turbulent processes over a wide range of spatiotemporal scales in engineering shear turbulence as well as climate atmosphere ocean science is a grand challenge of contemporary science with important societal impact. In such inhomogeneous turbulent dynamical systems there is a large dimensional phase space with a large dimension of unstable directions where a large-scale ensemble mean and the turbulent fluctuations exchange energy and strongly influence each other. These complex features strongly impact practical prediction and uncertainty quantification. A systematic energy conservation principle is developed here in a Theorem that precisely accounts for the statistical energy exchange between the mean flow and the related turbulent fluctuations. This statistical energy is a sum of the energy in the mean and the trace of the covariance of the fluctuating turbulence. This result applies to general inhomogeneous turbulent dynamical systems including the above applications. The Theorem involves an assessment of statistical symmetries for the nonlinear interactions and a self-contained treatment is presented below. Corollary 1 and Corollary 2 illustrate the power of the method with general closed differential equalities for the statistical energy in time either exactly or with upper and lower bounds, provided that the negative symmetric dissipation matrix is diagonal in a suitable basis. Implications of the energy principle for low-order closure modeling and automatic estimates for the single point variance are discussed below.

Experimental measurement of the Melnikov function

We study the transport properties of a genuine two-dimensional flow with a large mean velocity perturbed periodically in time by means of an original experimental technique. The flow generated by the co-rotation of two cylinders is both stratified with a linear density gradient using salted water and viscous in order to prevent Ekman pumping and centrifugal instabilities. Thus, the mean flow contains a hyperbolic point with a homoclinic streamline, which we perturb periodically by an extra oscillation. A blob of scalar injected close to the stagnation point contracts on the stable manifold and stretches in the unstable direction. The distance between the stable and the unstable manifolds is measured as the distance between the maximum and the minimum of the dye undulating pattern and is recorded as a function of the perturbation frequency. This distance, also called the Melnikov function, presents a maximum when the residence time of a fluid particle in the mean flow is about half a perturbation period. This resonance criterion is recovered with good quantitative agreement by the theoretical prediction of the Melnikov function computed for this flow.

Diode-end-pumped Nd:YVO4 slab laser with a compact negative confocal hybrid resonator

Laser diode-end-pumped Nd:YVO4 slab laser with a hybrid resonator operating at 1064 nm was carried out. The size of Nd:YVO4 crystal with 0.3 at.% doped concentration was 14 mm × 10 mm × 1 mm, which was cut along the a axis and the c axis was parallel with the 14 mm direction. By using a compact negative confocal unstable-stable hybrid resonator, 134 W 1064 nm laser output was realized with the M2 factors in the unstable direction and in the stable direction were 2.1 and 1.7 respectively. The slope efficiency and optical-to-optical efficiency were 49.3% and 40.2%, respectively.

Uniform positivity and continuity of Lyapunov exponents for a class of [formula omitted] quasiperiodic Schrödinger cocycles

We show that for a class of C2 quasiperiodic potentials and for any fixed Diophantine frequency, the Lyapunov exponent of the corresponding Schrödinger cocycles, as a function of energies, are uniformly positive and weakly Hölder continuous. As a corollary, we obtain that the corresponding integrated density of states is weakly Hölder continuous as well. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to more general SL(2,R) cocycles, which in turn can be applied to get uniform positivity and continuity of Lyapunov exponents around unique nondegenerate extremal points of any smooth potential, and to a certain class of C2 Szegő cocycles.

Laser diode double-end-direct-pumped slab laser with hybrid resonator

To reduce the influence of thermal lensing and attain the high power laser with high beam quality, a laser diode stack dual-end-direct-pumped Nd:YVO4 slab laser with negative branch confocal folded hybrid cavity has been developed. Its maximum output power is 416 W, optical conversion efficiency is 54.3%, and slope efficiency is 61.6%. When the output laser power is 370 W, M2 values in the unstable resonator direction and stable resonator direction are 3.9 and 4.7, respectively.

Forecasting transitions in systems with high-dimensional stochastic complex dynamics: a linear stability analysis of the tangled nature model.

We propose a new procedure to monitor and forecast the onset of transitions in high-dimensional complex systems. We describe our procedure by an application to the tangled nature model of evolutionary ecology. The quasistable configurations of the full stochastic dynamics are taken as input for a stability analysis by means of the deterministic mean-field equations. Numerical analysis of the high-dimensional stability matrix allows us to identify unstable directions associated with eigenvalues with a positive real part. The overlap of the instantaneous configuration vector of the full stochastic system with the eigenvectors of the unstable directions of the deterministic mean-field approximation is found to be a good early warning of the transitions occurring intermittently.

Robustness of dichotomies and trichotomies for difference equations

We give a short proof of the robustness of the notion of a nonuniform exponential dichotomy for a sequence of linear operators acting on a Banach space. This means that any sufficiently small linear perturbation of a nonuniform exponential dichotomy exhibits the same type of exponential behavior. The method of proof is based on the notion of admissibility introduced by Perron in the special case of a uniform exponential behavior. In strong contrast to former proofs, we do not need to construct explicitly projections on the stable and unstable directions. As an application, we also give a short proof of the robustness of the notion of a nonuniform exponential trichotomy.