scholarly journals Physical invariant measures and tipping probabilities for chaotic attractors of asymptotically autonomous systems

2021 ◽  
Author(s):  
Peter Ashwin ◽  
Julian Newman
Keyword(s):  
Dynamical Systems ◽  
Invariant Measures ◽  
Autonomous Systems ◽  
Chaotic Attractors ◽  
Physical Measure ◽  
Multiple Attractors ◽  
The Past ◽  
Rate Dependent ◽  
Typical Behaviour ◽  
Autonomous Dynamical Systems

AbstractPhysical measures are invariant measures that characterise “typical” behaviour of trajectories started in the basin of chaotic attractors for autonomous dynamical systems. In this paper, we make some steps towards extending this notion to more general nonautonomous (time-dependent) dynamical systems. There are barriers to doing this in general in a physically meaningful way, but for systems that have autonomous limits, one can define a physical measure in relation to the physical measure in the past limit. We use this to understand cases where rate-dependent tipping between chaotic attractors can be quantified in terms of “tipping probabilities”. We demonstrate this for two examples of perturbed systems with multiple attractors undergoing a parameter shift. The first is a double-scroll system of Chua et al., and the second is a Stommel model forced by Lorenz chaos.

scholarly journals The Planar Inverse Problem for Autonomous Systems

1983 ◽  
Vol 74 ◽  
pp. 353-367 ◽  
Author(s):  
Basilis C. Xanthopoulos ◽  
George Bozis
Keyword(s):  
Inverse Problem ◽  
Dynamical Systems ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
Parametric Family ◽  
General Version ◽  
Conic Sections ◽  
Family Of Curves ◽  
Necessary And Sufficient ◽  
Autonomous Dynamical Systems

AbstractWe study the general version of the inverse problem for planar trajectories and for autonomous dynamical systems possessing three integrals, i.e., for a given three-parametric family of curves f(x,y,a,b)=c we find the potential V(x,y) for which these curves are orbits of a unit mass. All possible cases, depending on the preassigned function f, are classified and in each case the necessary and sufficient conditions for the.existence of a solution are established. Among the examples is the case of the Keplerian conic sections which is studied in detail.

The Motion Constants of Linear Autonomous Dynamical Systems

2013 ◽  
Vol 65 (4) ◽  
Author(s):  
Andrew J. Sinclair ◽  
John E. Hurtado
Keyword(s):  
Dynamical Systems ◽  
Canonical Form ◽  
Frequency Ratio ◽  
Autonomous Systems ◽  
State Transformation ◽  
Close Inspection ◽  
Ergodic Behavior ◽  
Single Exception ◽  
Comprehensive Development ◽  
Autonomous Dynamical Systems

The time-independent integrals, here referred to as motion constants, for general nth-order linear autonomous systems are developed. Although it is commonly believed that this topic has been fully addressed, close inspection of the literature reveals that a comprehensive development is missing. This paper provides a complete tutorial treatment of the calculation of these motion constants. The process involves a state transformation to a canonical form of uncoupled real subsystems. Following this, motion constants that are internal to each subsystem are found, after which motion constants that connect the subsystems to each other are computed. Complete sets of n − 1 real single-valued motion constants can be formed for all linear autonomous systems with a single exception. The exception is systems composed of undamped oscillators whose frequency ratio is irrational. Such systems are known to exhibit ergodic behavior and lack a number of analytic motion constants.

Using Generalized Cell Mapping to Approximate Invariant Measures on Compact Manifolds

1997 ◽  
Vol 07 (11) ◽  
pp. 2487-2499 ◽  
Author(s):  
Rabbijah Guder ◽  
Edwin Kreuzer
Keyword(s):  
Dynamical Systems ◽  
Invariant Measures ◽  
Nonlinear Dynamical Systems ◽  
Powerful Method ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
Nonlinear Dynamical ◽  
Long Term Behavior ◽  
Mapping Theory

In order to predict the long term behavior of nonlinear dynamical systems the generalized cell mapping is an efficient and powerful method for numerical analysis. For this reason it is of interest to know under what circumstances dynamical quantities of the generalized cell mapping (like persistent groups, stationary densities, …) reflect the dynamics of the system (attractors, invariant measures, …). In this article we develop such connections between the generalized cell mapping theory and the theory of nonlinear dynamical systems. We prove that the generalized cell mapping is a discretization of the Frobenius–Perron operator. By applying the results obtained for the Frobenius–Perron operator to the generalized cell mapping we outline for some classes of transformations that the stationary densities of the generalized cell mapping converges to an invariant measure of the system. Furthermore, we discuss what kind of measures and attractors can be approximated by this method.

Identification of Synchronicity in Deterministic Chaotic Attractors for the Downscaling Process in the Bogotá River Basin

2021 ◽  
Author(s):  
Santiago Duarte ◽  
Gerald Corzo ◽  
Germán Santos
Keyword(s):  
Time Series ◽  
Time Delay ◽  
River Basin ◽  
Spatial Scales ◽  
Climatic Variables ◽  
Chaotic Attractors ◽  
Multiple Attractors ◽  
Nearest Neighbours ◽  
Basin Scale ◽  
Non Linear

<p>Bogotá’s River Basin, it’s an important basin in Cundinamarca, Colombia’s central region. Due to the complexity of the dynamical climatic system in tropical regions, can be difficult to predict and use the information of GCMs at the basin scale. This region is especially influenced by ENSO and non-linear climatic oscillation phenomena. Furthermore, considering that climatic processes are essentially non-linear and possibly chaotic, it may reduce the effectiveness of downscaling techniques in this region. </p><p>In this study, we try to apply chaotic downscaling to see if we could identify synchronicity that will allow us to better predict. It was possible to identify clearly the best time aggregation that can capture at the best the maximum relations between the variables at different spatial scales. Aside this research proposes a new combination of multiple attractors. Few analyses have been made to evaluate the existence of synchronicity between two or more attractors. And less analysis has considered the chaotic behaviour in attractors derived from climatic time series at different spatial scales. </p><p>Thus, we evaluate general synchronization between multiple attractors of various climate time series. The Mutual False Nearest Neighbours parameter (MFNN) is used to test the “Synchronicity Level” (existence of any type of synchronization) between two different attractors. Two climatic variables were selected for the analysis: Precipitation and Temperature. Likewise, two information sources are used: At the basin scale, local climatic-gauge stations with daily data and at global scale, the output of the MPI-ESM-MR model with a spatial resolution of 1.875°x1.875° for both climatic variables (1850-2005). In the downscaling process, two RCP (Representative Concentration Pathways)  scenarios are used, RCP 4.5 and RCP 8.5.</p><p>For the attractor’s reconstruction, the time-delay is obtained through the  Autocorrelation and the Mutual Information functions. The False Nearest Neighbors method (FNN) allowed finding the embedding dimension to unfold the attractor. This information was used to identify deterministic chaos at different times (e.g. 1, 2, 3 and 5 days) and spatial scales using the Lyapunov exponents. These results were used to test the synchronicity between the various chaotic attractor’s sets using the MFNN method and time-delay relations. An optimization function was used to find the attractor’s distance relation that increases the synchronicity between the attractors.  These results provided the potential of synchronicity in chaotic attractors to improve rainfall and temperature downscaling results at aggregated daily-time steps. Knowledge of loss information related to multiple reconstructed attractors can provide a better construction of downscaling models. This is new information for the downscaling process. Furthermore, synchronicity can improve the selection of neighbours for nearest-neighbours methods looking at the behaviour of synchronized attractors. This analysis can also allow the classification of unique patterns and relationships between climatic variables at different temporal and spatial scales.</p>

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