220,389 research outputs found
Nonintegrability, Chaos, and Complexity
Two-dimensional driven dissipative flows are generally integrable via a
conservation law that is singular at equilibria. Nonintegrable dynamical
systems are confined to n*3 dimensions. Even driven-dissipative deterministic
dynamical systems that are critical, chaotic or complex have n-1 local
time-independent conservation laws that can be used to simplify the geometric
picture of the flow over as many consecutive time intervals as one likes. Those
conserevation laws generally have either branch cuts, phase singularities, or
both. The consequence of the existence of singular conservation laws for
experimental data analysis, and also for the search for scale-invariant
critical states via uncontrolled approximations in deterministic dynamical
systems, is discussed. Finally, the expectation of ubiquity of scaling laws and
universality classes in dynamics is contrasted with the possibility that the
most interesting dynamics in nature may be nonscaling, nonuniversal, and to
some degree computationally complex
Chaos, Complexity, and Random Matrices
Chaos and complexity entail an entropic and computational obstruction to
describing a system, and thus are intrinsically difficult to characterize. In
this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE)
Hamiltonians and analytically compute out-of-time-ordered correlation functions
(OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and
circuit complexity. While our random matrix analysis gives a qualitatively
correct prediction of the late-time behavior of chaotic systems, we find
unphysical behavior at early times including an scrambling
time and the apparent breakdown of spatial and temporal locality. The salient
feature of GUE Hamiltonians which gives us computational traction is the
Haar-invariance of the ensemble, meaning that the ensemble-averaged dynamics
look the same in any basis. Motivated by this property of the GUE, we introduce
-invariance as a precise definition of what it means for the dynamics of a
quantum system to be described by random matrix theory. We envision that the
dynamical onset of approximate -invariance will be a useful tool for
capturing the transition from early-time chaos, as seen by OTOCs, to late-time
chaos, as seen by random matrix theory.Comment: 61 pages, 14 figures; v2: references added, typos fixe
Nonlinear and Complex Dynamics in Economics
This paper is an up-to-date survey of the state-of-the-art in dynamical systems theory relevant to high levels of dynamical complexity, characterizing chaos and near chaos, as commonly found in the physical sciences. The paper also surveys applications in economics and �finance. This survey does not include bifurcation analyses at lower levels of dynamical complexity, such as Hopf and transcritical
bifurcations, which arise closer to the stable region of the parameter space. We discuss the geometric approach (based on the theory of differential/difference
equations) to dynamical systems and make the basic notions of complexity, chaos, and other related concepts precise, having in mind their (actual or potential) applications to economically motivated questions. We also introduce specifi�c
applications in microeconomics, macroeconomics, and �finance, and discuss the policy relevancy of chaos
Chaos and Complexity of quantum motion
The problem of characterizing complexity of quantum dynamics - in particular
of locally interacting chains of quantum particles - will be reviewed and
discussed from several different perspectives: (i) stability of motion against
external perturbations and decoherence, (ii) efficiency of quantum simulation
in terms of classical computation and entanglement production in operator
spaces, (iii) quantum transport, relaxation to equilibrium and quantum mixing,
and (iv) computation of quantum dynamical entropies. Discussions of all these
criteria will be confronted with the established criteria of integrability or
quantum chaos, and sometimes quite surprising conclusions are found. Some
conjectures and interesting open problems in ergodic theory of the quantum many
problem are suggested.Comment: 45 pages, 22 figures, final version, at press in J. Phys. A, special
issue on Quantum Informatio
Black holes, complexity and quantum chaos
We study aspects of black holes and quantum chaos through the behavior of
computational costs, which are distance notions in the manifold of unitaries of
the theory. To this end, we enlarge Nielsen geometric approach to quantum
computation and provide metrics for finite temperature/energy scenarios and
CFT's. From the framework, it is clear that costs can grow in two different
ways: operator vs `simple' growths. The first type mixes operators associated
to different penalties, while the second does not. Important examples of simple
growths are those related to symmetry transformations, and we describe the
costs of rotations, translations, and boosts. For black holes, this analysis
shows how infalling particle costs are controlled by the maximal Lyapunov
exponent, and motivates a further bound on the growth of chaos. The analysis
also suggests a correspondence between proper energies in the bulk and average
`local' scaling dimensions in the boundary. Finally, we describe these
complexity features from a dual perspective. Using recent results on SYK we
compute a lower bound to the computational cost growth in SYK at infinite
temperature. At intermediate times it is controlled by the Lyapunov exponent,
while at long times it saturates to a linear growth, as expected from the
gravity description.Comment: 30 page
Intrinsic chaos and external noise in population dynamics
We address the problem of the relative importance of the intrinsic chaos and
the external noise in determining the complexity of population dynamics. We use
a recently proposed method for studying the complexity of nonlinear random
dynamical systems. The new measure of complexity is defined in terms of the
average number of bits per time-unit necessary to specify the sequence
generated by the system. This measure coincides with the rate of divergence of
nearby trajectories under two different realizations of the noise. In
particular, we show that the complexity of a nonlinear time-series model
constructed from sheep populations comes completely from the environmental
variations. However, in other situations, intrinsic chaos can be the crucial
factor. This method can be applied to many other systems in biology and
physics.Comment: 13 pages, Elsevier styl
Topological and Dynamical Complexity of Random Neural Networks
Random neural networks are dynamical descriptions of randomly interconnected
neural units. These show a phase transition to chaos as a disorder parameter is
increased. The microscopic mechanisms underlying this phase transition are
unknown, and similarly to spin-glasses, shall be fundamentally related to the
behavior of the system. In this Letter we investigate the explosion of
complexity arising near that phase transition. We show that the mean number of
equilibria undergoes a sharp transition from one equilibrium to a very large
number scaling exponentially with the dimension on the system. Near
criticality, we compute the exponential rate of divergence, called topological
complexity. Strikingly, we show that it behaves exactly as the maximal Lyapunov
exponent, a classical measure of dynamical complexity. This relationship
unravels a microscopic mechanism leading to chaos which we further demonstrate
on a simpler class of disordered systems, suggesting a deep and underexplored
link between topological and dynamical complexity
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