scholarly journals Inverse-square law between time and amplitude for crossing tipping thresholds

2019 ◽  
Vol 475 (2222) ◽  
pp. 20180504 ◽  
Author(s):  
Paul Ritchie ◽  
Özkan Karabacak ◽  
Jan Sieber
Keyword(s):  
Dynamical System ◽  
Feedback Loop ◽  
Tipping Point ◽  
Dimensional System ◽  
Stochastic Forcing ◽  
Inverse Square Law ◽  
System A ◽  
Asymptotic Expressions ◽  
Higher Dimensional ◽  
Random Disturbances

A classical scenario for tipping is that a dynamical system experiences a slow parameter drift across a fold tipping point, caused by a run-away positive feedback loop. We study what happens if one turns around after one has crossed the threshold. We derive a simple criterion that relates how far the parameter exceeds the tipping threshold maximally and how long the parameter stays above the threshold to avoid tipping in an inverse-square law to observable properties of the dynamical system near the fold. For the case when the dynamical system is subject to stochastic forcing we give an approximation to the probability of tipping if a parameter changing in time reverses near the tipping point. The derived approximations are valid if the parameter change in time is sufficiently slow. We demonstrate for a higher-dimensional system, a model for the Indian summer monsoon, how numerically observed escape from the equilibrium converge to our asymptotic expressions. The inverse-square law between peak of the parameter forcing and the time the parameter spends above a given threshold is also visible in the level curves of equal probability when the system is subject to random disturbances.

How fast to turn around: preventing tipping after a system has crossed a climate tipping threshold

2020 ◽  
Author(s):  
Paul Ritchie ◽  
Peter Cox ◽  
Jan Sieber
Keyword(s):  
Dynamical System ◽  
Feedback Loop ◽  
Tipping Point ◽  
Climate System ◽  
Tipping Points ◽  
Simple Criterion ◽  
Inverse Square Law ◽  
Parameter Drift ◽  
Simple Models

<p>A classical scenario for tipping is that a dynamical system experiences a slow parameter drift across a fold tipping point, caused by a run-away positive<br>feedback loop. We study what happens if one turns around after one has crossed the threshold. We derive a simple criterion that relates how far the parameter exceeds the tipping threshold maximally and how long the parameter stays above the threshold to avoid tipping in an inverse-square law to observable properties of the dynamical system near the fold. We demonstrate the inverse-square law relationship using simple models of recognised potential future tipping points in the climate system. </p>

scholarly journals Revealing topology with transformation optics

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Lizhen Lu ◽  
Kun Ding ◽  
Emanuele Galiffi ◽  
Xikui Ma ◽  
Tianyu Dong ◽  
...  
Keyword(s):  
Physical Space ◽  
Real Space ◽  
Virtual Space ◽  
Transformation Optics ◽  
Coordinate Space ◽  
Dimensional System ◽  
Edge States ◽  
And Topology ◽  
Higher Dimensional ◽  
Insight Into

AbstractSymmetry deepens our insight into a physical system and its interplay with topology enables the discovery of topological phases. Symmetry analysis is conventionally performed either in the physical space of interest, or in the corresponding reciprocal space. Here we borrow the concept of virtual space from transformation optics to demonstrate how a certain class of symmetries can be visualised in a transformed, spectrally related coordinate space, illuminating the underlying topological transitions. By projecting a plasmonic system in a higher-dimensional virtual space onto a lower-dimensional system in real space, we show how transformation optics allows us to construct a topologically non-trivial system by inspecting its modes in the virtual space. Interestingly, we find that the topological invariant can be controlled via the singularities in the conformal mapping, enabling the intuitive engineering of edge states. The confluence of transformation optics and topology here can be generalized to other wave realms beyond photonics.

Non-Standard Reduction of Noisy Structural Systems: Shallow Arches

2000 ◽  
Author(s):  
Lalit Vedula ◽  
N. Sri Namachchivaya
Keyword(s):  
Markov Process ◽  
Random Perturbation ◽  
Exit Time ◽  
Stochastic Averaging ◽  
Dimensional System ◽  
Shallow Arches ◽  
Higher Dimensional ◽  
Mean Exit Time ◽  
Stationary Probability Density Function ◽  
Low Dimensional

Abstract The dynamics of a shallow arch subjected to small random external and parametric excitation is invegistated in this work. We develop rigorous methods to replace, in some limiting regime, the original higher dimensional system of equations by a simpler, constructive and rational approximation – a low-dimensional model of the dynamical system. To this end, we study the equations as a random perturbation of a two-dimensional Hamiltonian system. We achieve the model-reduction through stochastic averaging and the reduced Markov process takes its values on a graph with certain glueing conditions at the vertex of the graph. Examination of the reduced Markov process on the graph yields many important results such as mean exit time, stationary probability density function.

scholarly journals Contractibility of a persistence map preimage

2020 ◽  
Vol 4 (4) ◽  
pp. 509-523
Author(s):  
Jacek Cyranka ◽  
Konstantin Mischaikow ◽  
Charles Weibel
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Time Series ◽  
Data Driven ◽  
Dimensional System ◽  
Multiple Time ◽  
Multiple Time Series ◽  
Topological Features ◽  
Persistence Diagram ◽  
Persistence Diagrams

Abstract This work is motivated by the following question in data-driven study of dynamical systems: given a dynamical system that is observed via time series of persistence diagrams that encode topological features of snapshots of solutions, what conclusions can be drawn about solutions of the original dynamical system? We address this challenge in the context of an N dimensional system of ordinary differential equation defined in $${\mathbb {R}}^N$$ R N . To each point in $${\mathbb {R}}^N$$ R N (e.g. an initial condition) we associate a persistence diagram. The main result of this paper is that under this association the preimage of every persistence diagram is contractible. As an application we provide conditions under which multiple time series of persistence diagrams can be used to conclude the existence of a fixed point of the differential equation that generates the time series.

Sampling time scheduling for a CAN-based dynamical system: A simulation practice

2015 ◽  
Author(s):  
Amira Sarayati Ahmad Dahalan ◽  
Abdul Rashid Husaint ◽  
Mohd Badril NorShah ◽  
Muhammad Iqbal Zakaria ◽  
Muhammad Nizam Kamarudin
Keyword(s):  
Dynamical System ◽  
Sampling Time ◽  
System A ◽  
Time Scheduling ◽  
Simulation Practice

Composer's Notebook: Star Networks at the Singing Point

2004 ◽  
Vol 14 ◽  
pp. 81-82
Author(s):  
Ralph Jones
Keyword(s):  
Chaos Theory ◽  
Analog Circuits ◽  
Feedback Loop ◽  
Tipping Point ◽  
Feedback Circuit ◽  
Star Network ◽  
Star Networks ◽  
Gain Stage

Star Networks at the Singing Point, poetic though it sounds, is merely a description in engineering terminology of the sound-producing method that forms the basis for the piece. A “star network” is a circuit node having three or more connections. The “singing point” is the particular tuning at which the gain in a feedback circuit produces oscillation. In Star Networks at the Singing Point, the performer creates analog circuits composed of multiple nodes, each of which has three or more connections—in essence, “mazes” having a number of paths through which current can flow. Connecting such a circuit in a feedback loop around a gain stage produces an oscillator that is inherently unstable. Tuned to what is called in chaos theory a “tipping point,” the circuit sings unpredictably of its own accord.

scholarly journals Extending Characters of Fixed Point Algebras

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 79
Author(s):  
Stefan Wagner
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Topological Group ◽  
Continuous Action ◽  
Group Of Units ◽  
System A ◽  
Fixed Point Algebra ◽  
Continuous Inverse ◽  
Locally Convex ◽  
Locally Convex Algebra

A dynamical system is a triple ( A , G , α ) consisting of a unital locally convex algebra A, a topological group G, and a group homomorphism α : G → Aut ( A ) that induces a continuous action of G on A. Furthermore, a unital locally convex algebra A is called a continuous inverse algebra, or CIA for short, if its group of units A × is open in A and the inversion map ι : A × → A × , a ↦ a − 1 is continuous at 1 A . Given a dynamical system ( A , G , α ) with a complete commutative CIA A and a compact group G, we show that each character of the corresponding fixed point algebra can be extended to a character of A.

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