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Register a new userLinear response for macroscopic observables in high-dimensional systems
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The long-term average response of observables of chaotic systems to dynamical perturbations can often be predicted using linear response theory, but not all chaotic systems possess a linear response. Macroscopic observables of complex dissipative chaotic systems, however, are widely assumed to have a linear response even if the microscopic variables do not, but the mechanism for this is not well-understood. We present a comprehensive picture for the linear response of macroscopic observables in high-dimensional coupled deterministic dynamical systems, where the coupling is via a mean field and the microscopic subsystems may or may not obey linear response theory. We derive stochastic reductions of the dynamics of these observables from statistics of the microscopic system, and provide conditions for linear response theory to hold in finite dimensional systems and in the thermodynamic limit. In particular, we show that for large systems of finite size, linear response is induced via self-generated noise. We present examples in the thermodynamic limit where the macroscopic observable satisfies LRT, although the microscopic subsystems individually violate LRT, as well a converse example where the macroscopic observable does not satisfy LRT despite all microscopic subsystems satisfying LRT when uncoupled. This latter, maybe surprising, example is associated with emergent non-trivial dynamics of the macroscopic observable. We provide numerical evidence for our results on linear response as well as some analytical intuition.
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For general dissipative dynamical systems we study what fraction of solutions exhibit chaotic behavior depending on the dimensionality $d$ of the phase space. We find that a system of $d$ globally coupled ODEs with quadratic and cubic non-linearities with random coefficients and initial conditions, the probability of a trajectory to be chaotic increases universally from $sim 10^{-5} - 10^{-4}$ for $d=3$ to essentially one for $dsim 50$. In the limit of large $d$, the invariant measure of the dynamical systems exhibits universal scaling that depends on the degree of non-linearity but does not depend on the choice of coefficients, and the largest Lyapunov exponent converges to a universal scaling limit. Using statistical arguments, we provide analytical explanations for the observed scaling and for the probability of chaos.
The authors present two results on infinite-dimensional linear dynamical systems with chaoticity. One is about the chaoticity of the backward shift map in the space of infinite sequences on a general Fr{e}chet space. The other is about the chaoticity of a translation map in the space of real continuous functions. The chaos is shown in the senses of both Li-Yorke and Wiggins. Treating dimensions as freedoms, the two results imply that in the case of an infinite number of freedoms, a system may exhibit complexity even when the action is linear. Finally, the authors discuss physical applications of infinite-dimensional linear chaotic dynamical systems.
A central question in verification is characterizing when a system has invariants of a certain form, and then synthesizing them. We say a system has a $k$ linear invariant, $k$-LI in short, if it has a conjunction of $k$ linear (non-strict) inequalities -- equivalently, an intersection of $k$ (closed) half spaces -- as an invariant. We present a sufficient condition -- solely in terms of eigenvalues of the $A$-matrix -- for an $n$-dimensional linear dynamical system to have a $k$-LI. Our proof of sufficiency is constructive, and we get a procedure that computes a $k$-LI if the condition holds. We also present a necessary condition, together with many example linear systems where either the sufficient condition, or the necessary is tight, and which show that the gap between the conditions is not easy to overcome. In practice, the gap implies that using our procedure, we synthesize $k$-LI for a larger value of $k$ than what might be necessary. Our result enables analysis of continuous and hybrid systems with linear dynamics in their modes solely using reasoning in the theory of linear arithmetic (polygons), without needing reasoning over nonlinear arithmetic (ellipsoids).
Phase space structures such as dividing surfaces, normally hyperbolic invariant manifolds, their stable and unstable manifolds have been an integral part of computing quantitative results such as transition fraction, stability erosion in multi-stable mechanical systems, and reaction rates in chemical reaction dynamics. Thus, methods that can reveal their geometry in high dimensional phase space (4 or more dimensions) need to be benchmarked by comparing with known results. In this study, we assess the capability of one such method called Lagrangian descriptor for revealing the types of high dimensional phase space structures associated with index-1 saddle in Hamiltonian systems. The Lagrangian descriptor based approach is applied to two and three degree-of-freedom quadratic Hamiltonian systems where the high dimensional phase space structures are known, that is as closed-form analytical expressions. This leads to a direct comparison of features in the Lagrangian descriptor plots and the phase space structures intersection with an isoenergetic two-dimensional surface and hence provides a validation of the approach.
The susceptibility of an overdamped Markov system fluctuating in a bistable potential of general form is obtained by analytic solution of the Fokker-Planck equation (FPE) for low noise intensities. The results are discussed in the context of the LRT theory of stochastic resonance. They go over into recent results (Gang Hu et al {em Phys. Lett. A} {bf 172}, 21, 1992) obtained from the FPE for the case of a symmetrical potential, and they coincide with the LRT results (Dykman et al, {em Phys. Rev. Lett.} {bf 65}, 2606, 1990; {em JETP Lett} {bf 52}, 144, 1990; {em Phys. Rev. Lett.} {bf 68}, 2985, 1992) obtained for the general case of bistable systems.
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