Ruelle Takens Newhouse Scenario Research Articles

Phase-locking flows between orthogonally stretching parallel plates

In this paper, we explore the stability and dynamical relevance of a wide variety of steady, time-periodic, quasiperiodic, and chaotic flows arising between orthogonally stretching parallel plates. We first explore the stability of all the steady flow solution families formerly identified by Ayats et al. [“Flows between orthogonally stretching parallel plates,” Phys. Fluids 33, 024103 (2021)], concluding that only the one that originates from the Stokesian approximation is actually stable. When both plates are shrinking at identical or nearly the same deceleration rates, this Stokesian flow exhibits a Hopf bifurcation that leads to stable time-periodic regimes. The resulting time-periodic orbits or flows are tracked for different Reynolds numbers and stretching rates while monitoring their Floquet exponents to identify secondary instabilities. It is found that these time-periodic flows also exhibit Neimark–Sacker bifurcations, generating stable quasiperiodic flows (tori) that may sometimes give rise to chaotic dynamics through a Ruelle–Takens–Newhouse scenario. However, chaotic dynamics is unusually observed, as the quasiperiodic flows generally become phase-locked through a resonance mechanism before a strange attractor may arise, thus restoring the time-periodicity of the flow. In this work, we have identified and tracked four different resonance regions, also known as Arnold tongues or horns. In particular, the 1 : 4 strong resonance region is explored in great detail, where the identified scenarios are in very good agreement with normal form theory.

ДИНАМИКА ГЕОМЕТРИЧЕСКИ И ФИЗИЧЕСКИ НЕЛИНЕЙНОГО ЧУВСТВИТЕЛЬНОГО ЭЛЕМЕНТА НАНОЭЛЕКТРОМЕХАНИЧЕСКОГО ДАТЧИКА В ВИДЕ НЕОДНОРОДНОЙ НАНОБАЛКИ, НАХОДЯЩЕЙСЯ В ТЕМПЕРАТУРНОМ И ШУМОВОМ ПОЛЯХ

Актуальность исследования. Наноэлектромеханические системы,будучи высокочувствительными датчиками, имеющими малыеразмеры, и надежными в эксплуатации, находят все более широкое применение в нефтегазовой промышленности для мониторинга различных процессов в нефтедобыче – от разведки до повышения нефтеотдачи, а также при бурении скважин, очистке, фракционировании и переработке до вывода их из эксплуатации. Одним из примеров применения наноэлектромеханических систем является сейсмические исследование месторождений. Применениенаноэлектромеханических систем позволяет улучшить производительность в дополнение к существенной экономии средств и времени для широкого спектра технологий нефтегазовой промышленности. Благодаря возможности непрерывного контроля эти технологии могут стать основой «умных» месторождений. Цель: построение математической модели, наиболее полно описывающей нелинейную динамику чувствительного элемента наноэлектромеханического датчика под действием знакопеременной нагрузки. Для этогонеобходимо учесть наиболее распространённые в настоящее время кинематические гипотезы, масштабные эффекты с помощью моментной теории упругости,нелинейную зависимость между напряжениями и деформациями, неоднородность материала, шумовые и тепловые поля.А также исследовать характер сложных нелинейных колебаний и выявитьзакономерности перехода их от гармонических к хаотическим. Объекты: геометрически и физически нелинейная нанобалка, описываемая кинематической моделью первого приближения, на которую воздействует равномерно распределенная знакопеременная поперечная нагрузка с гармонической составляющей, температурное поле и аддитивный внешний шум. Методы: вариационные методы,метод конечных разностей второго порядка точности для сведения системы нелинейных дифференциальных уравнений в частных производных к задаче Коши, метод Ньюмарка для решения задачи Коши, метод переменных параметров упругости Биргерадля решения физически нелинейной задачи, метод вариационных итераций для получения аналитического решения двумерного уравнения теплопроводности. Результаты.Для получения аналитического решения теплопроводности применяется метод вариационных итераций. Построенаматематическая модельколебаний чувствительного элемента наноэлектромеханического датчика в виде размерно-зависимой балки, на которую действует равномерно распределенная поперечная нагрузка с гармонической составляющей. Помимо переменной нагрузки учитывалось влияние температурного поля и аддитивного внешнего шумового воздействия. Геометрическая нелинейность принята по теории Теодора фонКармана (связь между деформациями и перемещениями). Для учета физической нелинейностиматериалабалки применяются деформационная теория пластичности и метод переменных параметров упругости. Уравнения движения элемента механической системы, а также соответствующие граничные и начальные условия выведены исходя из принципа Остроградского–Гамильтона на базе модифицированной моментной теории с учетом гипотезы Эйлера–Бернулли.Выявлено,что температурное и шумовое поляуменьшают нагрузку, при которой происходит переход в хаотическое состояние системы. Переход от гармонических колебаний к хаотическим происходит по сценарию Рюэля–Такенса–Ньюхауза.

Transition to chaos in the flow-induced vibration of a pitching–plunging airfoil at low Reynolds numbers: Ruelle–Takens–Newhouse scenario

This study focuses on numerically analyzing the transition from periodic to chaotic dynamics in the fluid-elastic response of a 2-dof flexibly-mounted airfoil with chord-wise rigidity. The computational framework is composed of a high fidelity Navier–Stokes solver, weakly coupled with a structural model having geometric nonlinearity represented by cubic order stiffness terms. A low Reynolds number flow regime and a very low structure-to-fluid added mass ratio have been considered to simulate the flying conditions of very light-weight unmanned devices. A bifurcation analysis of the system, in the absence of actuation or control forces, is undertaken with the wind velocity as the control parameter. The route to chaos – identified to be the Ruelle–Takens–Newhouse quasi-periodic route – is established for the first time for a flexible pitch–plunge flapping system. Robust nonlinear time series analysis techniques have been implemented to characterize different complex dynamical states present in the system.

Nonresonant Hopf–Hopf bifurcation and a chaotic attractor in neutral functional differential equations

An algorithm for calculating the third-order normal form of a nonresonant Hopf–Hopf singularity in a neutral functional differential equation (NFDE) is established. The van der Pol equation with extended delay feedback is investigated as an NFDE of second order. The existence of Hopf–Hopf bifurcation is studied and the unfolding near these critical points is given by applying this algorithm. Periodic solutions and quasi-periodic solutions are found with the aid of the bifurcation diagram, and corresponding numerical illustrations are presented. With the breaking down of the 3-torus, a chaotic attractor appears in this NFDE of second order, following the Ruelle–Takens–Newhouse scenario which usually arises for an ordinary differential equation of order at least 4. This transition is shown via both theoretical and numerical approaches.

Direct numerical simulation of fluid flow and heat transfer in periodic wavy channels with rectangular cross-sections

Fully-developed flow and heat transfer in periodic wavy channels with rectangular cross sections are studied using direct numerical simulation, for increasing Reynolds numbers spanning from the steady laminar to transitional flow regimes. The results show that steady flow is characterized by the formation of symmetric secondary flow or Dean vortices when liquid flows past the bends. It is found that the patterns of Dean vortices may evolve along the flow direction, thus leading to chaotic advection, which can greatly enhance the convective fluid mixing and heat transfer. With increasing Reynolds numbers, the flow undergoes transition from a steady state to a periodic one with a single frequency, and subsequently to a quasiperiodic flow with two incommensurate fundamental frequencies. Within these unsteady regimes, the flow is characterized by very complex Dean vortices patterns which evolve temporally and spatially along the flow direction, and the flow symmetry may even be lost. Further increase in Reynolds number leads to chaotic flow, where the Fourier spectrum of the velocity evolution becomes broadband. The bifurcation scenario in wavy channels may thus share some common features with the well-known Ruelle–Takens–Newhouse scenario. Heat transfer simulation in all flow regimes is carried out with constant wall temperature condition and liquid water as the coolant. It is found that due to the efficient mixing in wavy channels, the heat transfer performance is always significantly more superior to that of straight channels with the same cross sections; at the same time the pressure drop penalty of wavy channels can be much smaller than the heat transfer enhancement. The present study shows that these wavy channels may have advantages over straight channels and thus serve as promising candidates for incorporation into efficient heat transfer devices.

Double-diffusive Marangoni convection in a rectangular cavity: Transition to chaos

Transition to chaos in double-diffusive Marangoni convection in a rectangular cavity with horizontal temperature and concentration gradients is considered. Attention is restricted to the special case when the resultant thermal and solutal Marangoni effects are equal and opposing. Direct numerical simulation is used and some techniques from nonlinear dynamics are adopted to identify the different dynamic regimes. It is found that the supercritical solution branch takes a quasi-periodicity and phase locking route to chaos while the subcritical branch follows the Ruelle–Takens–Newhouse scenario. Transient intermittency in the supercritical branch is observed and physical instability mechanisms of the subcritical branch are identified.

Chaotic dynamics in an unstirred ferroin catalyzed Belousov–Zhabotinsky reaction

The Belousov–Zhabotinsky (BZ) reaction is the best known example of far from equilibrium self-organizing chemical reaction. Among the many dynamical behaviors that this reaction can exhibit, spatio-temporal chaos attracted particular interest, both for the ferroin and cerium catalyzed systems. In recent years transient chaos was found in the cerium catalyzed BZ reaction, when conducted in batch and unstirred reactors. It was established that the chaotic oscillations, originated by the coupling among chemical kinetics and transport phenomena, appeared and disappeared following a Ruelle–Takens–Newhouse scenario. In this Letter, we show results about the ferroin catalyzed system conducted in batch and unstirred reactors. An aperiodic transient regime was detected and we show that this is a genuine manifestation of spatio-temporal chaos.

Ruelle–Takens–Newhouse scenario in reaction-diffusion-convection system

Direct numerical simulations of the transition process from periodic to chaotic dynamics are presented for two variable Oregonator-diffusion model coupled with convection. Numerical solutions to the corresponding reaction-diffusion-convection system of equations show that natural convection can change in a qualitative way, the evolution of concentration distribution, as compared with convectionless conditions. The numerical experiments reveal distinct bifurcations as the Grashof number is increased. A transition to chaos similar to Ruelle-Takens-Newhouse scenario is observed. Numerical results are in agreement with the experiments.

Atmospheric behavior of oil-shale combustion fly ash in a chamber study : Teinemaa, E. et al. Atmospheric Environment, 2002, 36, (5), 813–824

Our objective was to study an unstirred Belousov–Zhabotinsky (BZ) closed system that shows transient spatio-temporal chaos before reaching a periodic attractor. The following sequence of well-defined dynamic regimes was observed during the reaction: chaotic, quasiperiodic, periodic and steady. This is the first time that an inverse transient Ruelle–Takens–Newhouse scenario has been observed during the temporal evolution of an unstirred closed BZ-system.

A route to chaos in a conservative system of three interacting Langmuir waves

This paper presents the results of numerical calculations of a route to chaos in a conservative Hamiltonian system of three Langmuir waves interacting with each other through three-wave couplings. The route is investigated by studying time series, power spectra, phase space portraits and Lyapnov exponents of wave variables for several combinations of wave vectors. The results show that the system follows a route which is very similar to the Ruelle–Takens–Newhouse scenario observed in dissipative systems, and widths and shifts of peaks in power spectra appeared due to the three moderate strength wave interactions. The breaks of tori in the system are also numerically investigated by studying the dependency of Maximum Lyapnov exponents for wave-variables on a parameter which represents the nonlinearity of the system.

Inverse Ruelle–Takens–Newhouse scenario in a closed unstirred cerium-catalysed Belousov–Zhabotinsky system

Our objective was to study an unstirred Belousov–Zhabotinsky (BZ) closed system that shows transient spatio-temporal chaos before reaching a periodic attractor. The following sequence of well-defined dynamic regimes was observed during the reaction: chaotic, quasiperiodic, periodic and steady. This is the first time that an inverse transient Ruelle–Takens–Newhouse scenario has been observed during the temporal evolution of an unstirred closed BZ-system.

Emission dynamics of coupled Nd3+ : YAG lasers with a shared population inversion source

A calculation is made of the frequency spectrum of relaxation oscillations of a cw Nd3+ : YAG laser with several lasing channels intersecting in the same active element. It is shown that the highest frequency of relaxation oscillations of isolated channels is virtually retained when these channels are coupled by population inversion, but the other frequencies are reduced. The results of these calculations are in agreement with measurements carried out on a double-beam laser. Experiments are reported on the transition of such a system to dynamic chaos. As in the case of a solid-state laser which emits multimode (in respect of the longitudinal index) radiation, near the frequencies of relaxation oscillations the transition of the investigated system to dynamic chaos follows the Ruelle—Takens—Newhouse scenario.

Transition to chaos in converging–diverging channel flows: Ruelle–Takens–Newhouse scenario

Direct numerical simulations of the transition process from laminar to chaotic flow in converging–diverging channels are presented. The chaotic flow regime is reached after a sequence of successive supercritical Hopf bifurcations to periodic, quasiperiodic, and chaotic self-sustained flow regimes. The numerical experiments reveal three distinct bifurcations as the Reynolds number is increased, each adding a new fundamental frequency to the velocity spectrum. In addition, frequency-locked periodic solutions with independent but synchronized periodic functions are obtained. A scenario similar to the Ruelle–Takens–Newhouse scenario of the onset of chaos is verified in this forced convective open system flow. The results are illustrated for different Reynolds numbers using time-velocity histories, Fourier power spectra, and phase space trajectories. The global structure of the self-sustained oscillatory flow for a periodic regime is also discussed.

Quasiperiodic transition to chaos in a multimode YAG:Nd3+laser with modulated parameters

Numerical and experimental investigations were made of the nature of the transition to chaos in the case of YAG:Nd3+ laser with modulated parameters. It was found that the interaction of longitudinal modes via a spatially distributed population inversion was responsible for the transition to chaos in accordance with the Ruelle–Takens–Newhouse scenario. The chaos appearance threshold was minimal near the frequency of purely relaxational oscillations.