Non Asymptotic Properties of Transport and Mixing

We study relative dispersion of passive scalar in non-ideal cases, i.e. in situations in which asymptotic techniques cannot be applied; typically when the characteristic length scale of the Eulerian velocity field is not much smaller than the domain size. Of course, in such a situation usual asymptotic quantities (the diffusion coefficients) do not give any relevant information about the transport mechanisms. On the other hand, we shall show that the Finite Size Lyapunov Exponent, originally introduced for the predictability problem, appears to be rather powerful in approaching the non-asymptotic transport properties. This technique is applied in a series of numerical experiments in simple flows with chaotic behaviors, in experimental data analysis of drifter and to study relative dispersion in fully developed turbulence.Comment: 19 RevTeX pages + 8 figures included, submitted on Chaos special issue on Transport and Mixin

Predictability: a way to characterize Complexity

Different aspects of the predictability problem in dynamical systems are reviewed. The deep relation among Lyapunov exponents, Kolmogorov-Sinai entropy, Shannon entropy and algorithmic complexity is discussed. In particular, we emphasize how a characterization of the unpredictability of a system gives a measure of its complexity. Adopting this point of view, we review some developments in the characterization of the predictability of systems showing different kind of complexity: from low-dimensional systems to high-dimensional ones with spatio-temporal chaos and to fully developed turbulence. A special attention is devoted to finite-time and finite-resolution effects on predictability, which can be accounted with suitable generalization of the standard indicators. The problems involved in systems with intrinsic randomness is discussed, with emphasis on the important problems of distinguishing chaos from noise and of modeling the system. The characterization of irregular behavior in systems with discrete phase space is also considered.Comment: 142 Latex pgs. 41 included eps figures, submitted to Physics Reports. Related information at this http://axtnt2.phys.uniroma1.i

Quantum dynamical phase transition in a system with many-body interactions

We introduce a microscopic Hamiltonian model of a two level system with many-body interactions with an environment whose excitation dynamics is fully solved within the Keldysh formalism. If a particle starts in one of the states of the isolated system, the return probability oscillates with the Rabi frequency $\omega_{0}$. For weak interactions with the environment $1/\tau_{\mathrm{SE}}<2\omega_{0},$ we find a slower oscillation whose amplitude decays with a decoherence rate $1/\tau_{\phi}=1/(2\tau_{\mathrm{SE}})$. However, beyond a finite critical interaction with the environment, $1/\tau_{\mathrm{SE}}>2\omega_{0}$, the decoherence rate becomes $1/\tau_{\phi}\propto(\omega_{0}^{2})\tau_{\mathrm{SE}}$. The oscillation period diverges showing a \emph{quantum dynamical phase transition}to a Quantum Zeno phase.Comment: 5 pages, 3 figures, minor changes, fig.2 modified, added reference

Chaos and chaotic phase mixing in cuspy triaxial potentials

This paper investigates chaos and chaotic phase mixing in triaxial Dehnen potentials which have been proposed to describe realistic ellipticals. Earlier work is extended by exploring the effects of (1) variable axis ratios, (2) `graininess' associated with stars and bound substructures, idealised as friction and white noise, and (3) large-scale organised motions presumed to induce near-random forces idealised as coloured noise with finite autocorrelation time. Three important conclusions are: (1) not all the chaos can be attributed to the cusp; (2) significant chaos can persist even for axisymmetric systems; and (3) introducing a supermassive black hole can increase both the relative number of chaotic orbits and the size of the largest Lyapunov exponent. Sans perturbations, distribution functions associated with initially localised chaotic ensembles evolve exponentially towards a nearly time-independent form at a rate L that correlates with the finite time Lyapunov exponents associated with the evolving orbits. Perturbations accelerate phase space transport by increasing the rate of phase mixing in a given phase space region and by facilitating diffusion along the Arnold web that connects different phase space regions, thus facilitating an approach towards a true equilibrium. The details of the perturbation appear unimportant. All that matters are the amplitude and the autocorrelation time, upon which there is a weak logarithmic dependence. Even comparatively weak perturbations can increase L by a factor of three or more, a fact that has potentially significant implications for violent relaxation.Comment: 17 pages, 17 figures -- revised and extended manuscript to appear in Monthly Notices of the Royal Astronomical Societ

Modularity and the spread of perturbations in complex dynamical systems

We propose a method to decompose dynamical systems based on the idea that modules constrain the spread of perturbations. We find partitions of system variables that maximize 'perturbation modularity', defined as the autocovariance of coarse-grained perturbed trajectories. The measure effectively separates the fast intramodular from the slow intermodular dynamics of perturbation spreading (in this respect, it is a generalization of the 'Markov stability' method of network community detection). Our approach captures variation of modular organization across different system states, time scales, and in response to different kinds of perturbations: aspects of modularity which are all relevant to real-world dynamical systems. It offers a principled alternative to detecting communities in networks of statistical dependencies between system variables (e.g., 'relevance networks' or 'functional networks'). Using coupled logistic maps, we demonstrate that the method uncovers hierarchical modular organization planted in a system's coupling matrix. Additionally, in homogeneously-coupled map lattices, it identifies the presence of self-organized modularity that depends on the initial state, dynamical parameters, and type of perturbations. Our approach offers a powerful tool for exploring the modular organization of complex dynamical systems

Chaotic Disintegration of the Inner Solar System

On timescales that greatly exceed an orbital period, typical planetary orbits evolve in a stochastic yet stable fashion. On even longer timescales, however, planetary orbits can spontaneously transition from bounded to unbound chaotic states. Large-scale instabilities associated with such behavior appear to play a dominant role in shaping the architectures of planetary systems, including our own. Here we show how such transitions are possible, focusing on the specific case of the long-term evolution of Mercury. We develop a simple analytical model for Mercury's dynamics and elucidate the origins of its short term stochastic behavior as well as of its sudden progression to unbounded chaos. Our model allows us to estimate the timescale on which this transition is likely to be triggered, i.e. the dynamical lifetime of the Solar System as we know it. The formulated theory is consistent with the results of numerical simulations and is broadly applicable to extrasolar planetary systems dominated by secular interactions. These results constitute a significant advancement in our understanding of the processes responsible for sculpting of the dynamical structures of generic planetary systems.Comment: 18 pages, 7 figures, accepted to Ap

고차원 로렌츠 시스템, 대기예측성 및 자료동화

학위논문(박사) -- 서울대학교대학원 : 자연과학대학 지구환경과학부, 2021.8. 문승주.로렌츠 시스템은 레일리 베나르 대류 현상의 단순한 모형으로 처음 고안되었으나, 이후 야릇한 끌개의 발견 및 혼돈 이론의 급속한 발전에 대한 기여 등을 통해 그 중요성이 꾸준히 부각되어 왔다. 본 연구에서는 두 가지 접근 방식을 통해 로렌츠 시스템을 고차원으로 확장하고자 하였다. 첫번째 접근 방식은 유도 과정에서 비롯되는 푸리에 급수의 절단에 있어 추가 모드를 통해 차수를 확장하는 방법으로, 본 연구에서는 이를 일반화 하여 임의의 자연수 $N$에 대한 $(3N)$ 및 $(3N+2)$차 로렌츠 시스템을 유도하였다. 두번째는 물리적 확장이라 불리는 방식으로, 레일리 베나르 대류 현상을 관장하는 지배방정식에 나타내고자 하는 물리 성분을 추가하여 더 높은 차수의 방정식계를 얻는 과정이다. 이에 추가 물리 성분으로 모형 프레임의 회전과 내부에 부유하는 오염 물질 따위의 스칼라를 고려하여 새로운 6차 로렌츠 시스템을 유도 할 수 있었다. 이렇게 얻어진 고차원 로렌츠 시스템은 비선형성, 대칭성, 소산성 등의 공통된 특징을 지닌다. 새롭게 확장된 로렌츠 시스템의 해의 특성 및 그들이 나타내는 다양한 비선형 현상의 규명은 수치 적분을 통해 얻은 해의 분석을 바탕으로 이루어졌다. 이를 위해 카오스 이론에 입각한 여러가지 분석 방법이 활용되었는데, 이러한 분석방법에는 파라미터 공간 상의 주기성도표, 분기도표 및 리아푸노프 지수 그리고 위상 공간 내 해의 궤도 및 프렉탈 흡인경계 등이 있다. 밝혀진 비선형 동역학적 현상 중 특히 주목할 만한 현상에는 파라미터 값에 따른 분기 구조의 변동, 하나의 위상 공간 내 존재하는 여러 타입의 해의 공존, 카오스의 동기화 등이 있다. 본 연구에서는 이러한 현상의 수학적~$\cdot$~수치적 분석뿐만 아니라 이것이 대기과학 특히 자료동화와 대기예측성 분야에 함의하는 바가 무엇인지도 탐구하였다. 본 연구에서 제안된 일반화 방식에 따라 로렌츠 시스템의 차수를 올리면 분기 구조에 변동이 일어나 임계 레일리 파라미터의 증가가 비롯된다. 여기서 임계 레일리 파라미터는 카오스가 처음 발생하는 가장 낮은 레일리 파라미터 값이므로 이것이 차수에 따라 증가한다는 것은 즉 고차원 로렌츠 시스템에서는 카오스의 발생이 저차원에서보다 더 어렵다는 것을 의미한다. 차수 및 파라미터 공간에 그려진 주기성 도표를 보면 카오스가 존재하는 영역이 차수에 따라 점점 줄어들고, 어느 차수 이상부터는 사라지는 것을 확인 할 수 있다. 마찬가지로 물리적으로 확장된 로렌츠 시스템에서는 임계 레일리 파라미터가 추가된 물리현상을 나타내는 새로운 파라미터의 값을 증가시킴에 따라 증가하는 것을 알 수 있다. 한편 유체 내 스칼라 효과와 연관된 파라미터만 점진적으로 올릴 경우에는 시스템의 불안정을 야기하는 레일리 파라미터와 안정을 야기하는 스칼라 관련 파라미터 간의 경쟁으로 인해 시스템이 완전히 안정화 되기 전 카오스 해가 한번 더 발생하는 현상이 일어난다. 이 두번째 카오스에 대응되는 끌개는 기존에 알려진 로렌츠 끌개와는 사뭇 다른 모양새를 보인다. 해의 공존 현상은 로렌츠에 의해 밝혀진 해의 초기조건에 대한 민감도와는 구분되는 개념으로, 초기조건으로 인한 카오스 해 간의 차이가 증폭되는 이른바 나비효과와는 달리 초기조건에 따라 완전히 다른 타입의 해를 나타내는 끌개가 같은 위상공간에 공존함을 의미한다. 따라서 만약 실제 날씨를 나타내는 시스템이 상존하는 위상공간에서 이러한 해의 공존이 실제한다면 이것은 카오스의 초기조건에 대한 민감성과 더불어 대기예측성 특히 앙상블 예보에 이론적으로 시사하는 바가 클 것으로 생각된다. 물리적으로 확장된 로렌츠 시스템에서는 기존 로렌츠 시스템과 같이 레일리 파라미터의 변화에 따른 분기 구조의 부정합으로 인해 비롯되는 해의 공존이 나타난다. 해의 공존 가능성이 높은 파라미터 조합을 찾아내기 위해 물리적으로 확장된 6차 로렌츠 시스템의 분기구조를 수치적~$\cdot$~해석적 방법으로 도출하였고 초기조건에 따라 서로 다른 두가지 종류의 분기 즉 호프 및 헤테로클리닉 분기가 엇갈리는 구간을 집중적으로 분석하였다. 기존 3차 로렌츠 시스템에서 하나의 변수에 대한 정보 전달 만으로도 자기동기화 현상이 일어남은 이미 잘 알려진 사실이다. 본 연구에서는 물리적으로 확장된 로렌츠 시스템에서도 기존 로렌츠 시스템과 같은 조건 하에서 카오스의 자기동기화가 일어나는 점을 적절한 리아푸노프 함수의 제시를 통해 증명하였다. 일반화된 로렌츠 시스템의 자기동기화에 대해서는 비록 수학적 증명이 동반되지는 않았지만 수치적 방법을 통해 역시 같은 조건 하에서 자기동기화가 일어남을 뒷받침 할 근거를 제시하였다. 또한 수치 실험을 통해 서로 다른 차수를 가진 일반화된 로렌츠 시스템 간 동기화가 일어나는 정도가 상호 차수를 기반으로 한 두 시스템 사이의 거리와 음의 상관관계를 가진다는 점도 확인하였다. 추가 푸리에 모드를 포함하여 더 작은 스케일의 운동을 분해할 수 있는 고차원 로렌츠 시스템과 그렇게 하지 못하는 저차원 로렌츠 시스템 간 동기화의 가능성은 대기과학에서 특히 대기 모형 및 자료동화에 있어 중요한 개념적인 함의를 가진다. 본 연구에서는 특별히 앙상블 칼만 필터 자료동화 기법을 일례로 일반화된 로렌츠 시스템이 자료동화 기법의 비교적 단순한 테스트베드로써의 역할을 할 수 있는지 탐구하였다. 카오스 동기화 현상에 기반을 둔 개념적 도식으로 발신자를 실제 대기 현상, 수신자를 대기 모형, 그리고 발신자에서 수신자로 전달되는 정보를 관측에 대응시킴으로써 수신자와 발신자 간의 오차, 발신자에서 수신자로 전달할 정보 추출 과정에서 비롯되는 오차 등을 통해 실제 대기 모형과 관측의 불완전함을 개념적으로 모의할 수 있었다. 일반화된 로렌츠 시스템에서 초기조건에 아주 작은 섭동을 준 해와 그렇지 않은 해 간의 비교를 통해 이것이 대기예측성에 함의하는 바가 무엇인지 탐구하였다. 이때 이렇게 두 해가 벌어지는 정도가 기준값을 넘게 되는 시간을 편차시간이라 칭한다. 본 연구에서는 편차시간이 적어도 주어진 파라미터 값 하에서는 로렌츠 시스템의 차수에 대한 강한 비단조적 의존성을 보임을 발견하였다. 또한 이렇게 정의된 편차시간을 활용하여 실제 날씨 사례의 수치 예보 모의에서 나타나는 대기예측성을 측정하였을때, 마찬가지로 대기예측성이 연직해상도에 대한 비단조적 의존성을 보이는 것을 확인할 수 있었다. 이에 이러한 비단조적 의존성의 근본적인 원인은 모형의 대기 나아가 실제 날씨에 내재된 카오스에 있을 수 있음을 제안하였다.The Lorenz system is a simplified model of Rayleigh--B\'{e}nard convection whose importance lies not only in understanding the fluid convection problem but also in its formative role in the discovery of strange attractors and the subsequent development of the modern theory of chaos. In this dissertation, two different approaches to extending the Lorenz system to higher dimensions are considered. First, by including additional wavenumber modes at the series truncation stage of the derivation, the so-called high-order Lorenz systems are obtained up to dimension 11, which are then generalized into $(3N)$ and $(3N+2)$ dimensions for any positive integer $N$. Second, by incorporating additional physical ingredients, namely, rotation and density-affecting scalar in the governing equations, a new 6-dimensional physically extended Lorenz system is derived. All of these high-dimensional extensions of the Lorenz system are shown to share some basic properties such as nonlinearity, symmetry, and volume contraction. The numerically obtained solutions of the extended Lorenz systems are studied through periodicity diagrams, bifurcation diagrams, and Lyapunov exponent spectra in parameter spaces and also through solution trajectories and basin boundaries in the phase space, illuminating various nonlinear dynamical phenomena such as shifts in the bifurcation structures, attractor coexistence, and chaos synchronization. Accompanying these results are discussions about their applicability and theoretical implications, particularly in the context of data assimilation and atmospheric predictability. The shifts in bifurcation structures induced by raising the dimension lead to higher critical Rayleigh parameter values, implying that it gets more difficult for chaos to emerge at higher dimensions. Periodicity diagrams reveal that the parameter ranges in which chaos resides tend to diminish with rising dimensions, eventually vanishing altogether. Likewise, simultaneously increasing the newly added parameters in the physically extended Lorenz system leads to higher critical Rayleigh parameter values; however, raising only the scalar-related parameter leads to an eventual return of chaos albeit with an attractor with qualitatively distinct features from the Lorenz attractor. The peculiar bifurcation structure shaped by the competition between the opposing effects of raising the Rayleigh and the scalar-related parameters helps explain this second onset of chaos. Attractor coexistence refers to the partition of the phase space by basin boundaries so that different types of attractors emerge depending on the initial condition. Similar to the original Lorenz system, the physically extended Lorenz system is found to exhibit attractor coexistence stemming from mismatches between the Hopf and heteroclinic bifurcations. If the atmosphere is found to exhibit such behavior, it can have grave implications for atmospheric predictability and ensemble forecasting beyond mere sensitive dependence on initial conditions, which only applies to chaotic solutions. Chaos synchronization is another curious phenomenon known to occur in the Lorenz system. By finding an appropriate Lyapunov function, the physically extended Lorenz system is shown to self-synchronize under the same condition that guarantees self-synchronization in the original Lorenz system. Regarding the generalized Lorenz systems, numerical evidence in support of self- as well as some degree of generalized synchronization, that is, synchronization between two Lorenz systems differing in their dimensions, is provided. Numerical results suggest that the smaller the dimensional difference between the two, the stronger they tend to synchronize. Some conceptual implications of such results are discussed in relation to atmospheric modeling and data assimilation. Especially, the feasibility of using the $(3N)$-dimensional Lorenz systems as a testbed for data assimilation methods is explored. For demonstration, the ensemble Kalman filter method is implemented to assimilate observations with ensembles of model outputs generated using the generalized Lorenz systems, whose imperfections are simulated through varying the severity of ensemble over- or underdispersion, dimensional differences, random forcing, and model or observation biases. Further investigation of the generalized Lorenz systems is carried out from the perspective of predictability, showing that predictability measured by deviation time, which is the time when the threshold-exceeding deviations among ensemble members occur, can respond non-monotonically to increases in the system's dimension. Accordingly, deviation time is put forward as a direct measure of predictability due to weather's sensitive dependence on initial conditions. Raising the dimension under the proposed generalizations is thought to be analogous to resolving smaller-scale motions in the vertical direction. The estimated deviation times in an ensemble of real-case simulations using a realistic numerical weather forecasting model reveal that the predictability of real-case simulations also depend non-monotonically on model vertical resolution. It is suggested that beneath this non-monotonicity fundamentally lies chaos inherent to the model atmospheres and, by extension, weather at large.1 Overview 1 1.1 Chaos and the Lorenz system 1 1.2 Extending the Lorenz system 6 1.3 Bifurcations and related phenomena 8 1.4 Chaos in the atmosphere 14 1.5 Organization of the dissertation 16 2 Chaos and Periodicity of the High-Order Lorenz Systems 18 2.1 Introduction 18 2.2 The high-order Lorenz systems 20 2.2.1 Derivation 22 2.2.2 Some properties of the Lorenz systems 24 2.3 Numerical methods 26 2.4 Results 32 2.4.1 Periodicity diagrams 32 2.4.2 Bifurcation diagrams and phase portraits 34 2.5 Discussion 40 3 A Physically Extended Lorenz System with Rotation and Density-Affecting Scalar 42 3.1 Introduction 42 3.2 Derivation 45 3.3 Effects of rotation and scalar 49 3.3.1 Fixed points and stability 49 3.3.2 Bifurcation structure in the rT-σ space 52 3.3.3 Bifurcations along rC and s 55 3.4 The case when β &lt; 0 65 3.4.1 Bifurcation and the onset of chaos 67 3.4.2 Chaotic attractors and associated flow patterns 73 3.5 Self-synchronization 81 3.6 Discussion 85 4 Coexisting Attractors in the Physically Extended Lorenz System 87 4.1 Introduction 87 4.2 Methodology 89 4.3 Results 92 4.3.1 Coexisting attractors in the LorenzStenflo system 92 4.3.2 Coexisting attractors under rotation and scalar 100 4.4 Discussion 110 5 The (3N)- and (3N + 2)-Dimensional Generalizations of the Lorenz System 113 5.1 Introduction 113 5.2 The generalized Lorenz systems 115 5.2.1 The Pk- and Qk-sets for nonlinear terms 115 5.2.2 The (3N)- and (3N + 2)-dimensional systems 116 5.2.3 Choosing the nonlinear pairs 117 5.3 Derivation 119 5.3.1 The (3N)-dimensional generalization 121 5.3.2 The (3N + 2)-dimensional generalization 126 5.4 Effects of dimension in parameter spaces 126 5.4.1 Linear stability analysis 126 5.4.2 Chaos in dimension-parameter spaces 130 5.5 Perspectives on predictability 136 5.5.1 Notions of predictability 136 5.5.2 Twin experiments and deviation time 138 5.6 Discussion 144 6 Chaos Synchronization in the Generalized Lorenz Systems 147 6.1 Introduction 147 6.2 Self-synchronization 149 6.2.1 Numerical evidence 149 6.2.2 Error subsystems 155 6.3 Application in image encryption 157 6.3.1 Demonstration: A simple approach 157 6.3.2 Demonstration: An alternative approach 168 6.4 Beyond self-synchronization 172 6.5 Discussion 180 7 The Generalized Lorenz Systems as a Testbed for Data Assimilation: The Ensemble Kalman Filter 182 7.1 Introduction 182 7.2 Methodology 187 7.2.1 Implementation of the ensemble Kalman filter 188 7.3 Results 191 7.3.1 Effects of ensemble size and model accuracy 191 7.3.2 Effects of observation frequency and accuracy 205 7.3.3 Effects of observation and model biases 214 7.4 Discussion 218 8 Can Chaos Theory Explain Non-Monotonic Dependence of Atmospheric Predictability on Model Vertical Resolution 220 8.1 Introduction 220 8.2 Background 222 8.2.1 Lorenz's ideas about atmospheric predictability 222 8.2.2 Model vertical resolution and predictability in numerical weather prediction 224 8.3 Results 229 8.3.1 Deviation time in the Lorenz systems revisited 229 8.3.2 WRF model control simulations 232 8.3.3 WRF model ensemble experiments and deviation time 241 8.3.4 Spatial distribution of deviation time 254 8.4 Discussion 261 9 Summary and Final Remarks 264 Bibliography 271 Abstract in Korean 295 Acknowledgments 299 Index 303박