## What is/are Local Lyapunov?

Local Lyapunov - In this study, the nonlinear local Lyapunov exponent and nonlinear error growth dynamics are employed to estimate the predictability limit of oceanic mesoscale eddy (OME) tracks quantitatively using three datasets.^{[1]}For various MJO indices, nonlinear local Lyapunov exponents are computed to quantify the MJO predictability under the easterly and westerly phases of the Quasi-Biennial Oscillation (easterly: EQBO and westerly: WQBO).

^{[2]}Using the calculation of local Lyapunov exponents the influence of the stationary noise on statistical characteristics of intermittent generalized synchronization and critical coupling parameter value corresponding to the generalized synchronization regime onset has been studied.

^{[3]}Here we apply the predictability study with the Nonlinear Local Lyapunov Exponent and Attractor Radius to the products of multiple re-analyses and forecast models in several operational centers to realize general predictability of the atmosphere in the Earth system.

^{[4]}The property of local Lyapunov reducibility is introduced for the closed-loop system.

^{[5]}It is shown that this set induces a local Lyapunov function strictly decreasing at the sampling instants and that it is an estimate of the RAO of the continuous-time closed-loop system.

^{[6]}In the present paper, the nonlinear local Lyapunov exponent (NLLE) is proposed as a new early warning signal for an abrupt climate change.

^{[7]}Using the backward nonlinear local Lyapunov exponent method, the prediction lead time, also called local backward predictability limit (LBPL), of given states induced by the two types of errors can be quantitatively estimated.

^{[8]}The γ -basin of attraction of the zero solution of a nonlinear stochastic differential equation can be determined through a pair of a local and a non-local Lyapunov function.

^{[9]}Based on this model we design a controller using the total hydraulic-mechanical energy as a local Lyapunov function.

^{[10]}The basis control scheme involves usual stabilizing constraints comprising of a terminal set and a terminal cost in the form of a local Lyapunov function.

^{[11]}In this paper, on the example of the Rossler systems, the application of the Pyragas time-delay feedback control technique for verification of Eden’s conjecture on the maximum of local Lyapunov dimension, and for the estimation of the topological entropy is demonstrated.

^{[12]}Time series points extracted from the paths were used to calculate the largest local Lyapunov exponents (LLLEs).

^{[13]}In this work, two types of predictability are proposed—forward and backward predictability—and then applied in the nonlinear local Lyapunov exponent approach to the Lorenz63 and Lorenz96 models to quantitatively estimate the local forward and backward predictability limits of states in phase space.

^{[14]}Time‐varying predictability is assessed by quantifying the divergence of trajectories in the phase space with time, using Local Lyapunov Exponents.

^{[15]}In this paper, system properties of a physics-based model of a supercapacitor are computed using a local Lyapunov analysis and the solution of linear matrix inequalities.

^{[16]}Here we focus on decadal timescales and apply the nonlinear local Lyapunov exponent method to the Community Climate System Model comparing control simulations with IE simulations.

^{[17]}In this work, a global Lyapunov function is constructed by combining known local Lyapunov functions for simpler, local subproblems via an explicit formula that depends on the problem parameters.

^{[18]}Dynamical characteristics of the fractional-order system are analyzed fully through numerical simulations, mainly including finite-time local Lyapunov exponents, bifurcation diagram, and the basins of attraction.

^{[19]}

## Nonlinear Local Lyapunov

In this study, the nonlinear local Lyapunov exponent and nonlinear error growth dynamics are employed to estimate the predictability limit of oceanic mesoscale eddy (OME) tracks quantitatively using three datasets.^{[1]}For various MJO indices, nonlinear local Lyapunov exponents are computed to quantify the MJO predictability under the easterly and westerly phases of the Quasi-Biennial Oscillation (easterly: EQBO and westerly: WQBO).

^{[2]}Here we apply the predictability study with the Nonlinear Local Lyapunov Exponent and Attractor Radius to the products of multiple re-analyses and forecast models in several operational centers to realize general predictability of the atmosphere in the Earth system.

^{[3]}In the present paper, the nonlinear local Lyapunov exponent (NLLE) is proposed as a new early warning signal for an abrupt climate change.

^{[4]}Using the backward nonlinear local Lyapunov exponent method, the prediction lead time, also called local backward predictability limit (LBPL), of given states induced by the two types of errors can be quantitatively estimated.

^{[5]}In this work, two types of predictability are proposed—forward and backward predictability—and then applied in the nonlinear local Lyapunov exponent approach to the Lorenz63 and Lorenz96 models to quantitatively estimate the local forward and backward predictability limits of states in phase space.

^{[6]}Here we focus on decadal timescales and apply the nonlinear local Lyapunov exponent method to the Community Climate System Model comparing control simulations with IE simulations.

^{[7]}

## local lyapunov exponent

In this study, the nonlinear local Lyapunov exponent and nonlinear error growth dynamics are employed to estimate the predictability limit of oceanic mesoscale eddy (OME) tracks quantitatively using three datasets.^{[1]}For various MJO indices, nonlinear local Lyapunov exponents are computed to quantify the MJO predictability under the easterly and westerly phases of the Quasi-Biennial Oscillation (easterly: EQBO and westerly: WQBO).

^{[2]}Using the calculation of local Lyapunov exponents the influence of the stationary noise on statistical characteristics of intermittent generalized synchronization and critical coupling parameter value corresponding to the generalized synchronization regime onset has been studied.

^{[3]}Here we apply the predictability study with the Nonlinear Local Lyapunov Exponent and Attractor Radius to the products of multiple re-analyses and forecast models in several operational centers to realize general predictability of the atmosphere in the Earth system.

^{[4]}In the present paper, the nonlinear local Lyapunov exponent (NLLE) is proposed as a new early warning signal for an abrupt climate change.

^{[5]}Using the backward nonlinear local Lyapunov exponent method, the prediction lead time, also called local backward predictability limit (LBPL), of given states induced by the two types of errors can be quantitatively estimated.

^{[6]}Time series points extracted from the paths were used to calculate the largest local Lyapunov exponents (LLLEs).

^{[7]}In this work, two types of predictability are proposed—forward and backward predictability—and then applied in the nonlinear local Lyapunov exponent approach to the Lorenz63 and Lorenz96 models to quantitatively estimate the local forward and backward predictability limits of states in phase space.

^{[8]}Time‐varying predictability is assessed by quantifying the divergence of trajectories in the phase space with time, using Local Lyapunov Exponents.

^{[9]}Here we focus on decadal timescales and apply the nonlinear local Lyapunov exponent method to the Community Climate System Model comparing control simulations with IE simulations.

^{[10]}Dynamical characteristics of the fractional-order system are analyzed fully through numerical simulations, mainly including finite-time local Lyapunov exponents, bifurcation diagram, and the basins of attraction.

^{[11]}

## local lyapunov function

It is shown that this set induces a local Lyapunov function strictly decreasing at the sampling instants and that it is an estimate of the RAO of the continuous-time closed-loop system.^{[1]}The γ -basin of attraction of the zero solution of a nonlinear stochastic differential equation can be determined through a pair of a local and a non-local Lyapunov function.

^{[2]}Based on this model we design a controller using the total hydraulic-mechanical energy as a local Lyapunov function.

^{[3]}The basis control scheme involves usual stabilizing constraints comprising of a terminal set and a terminal cost in the form of a local Lyapunov function.

^{[4]}In this work, a global Lyapunov function is constructed by combining known local Lyapunov functions for simpler, local subproblems via an explicit formula that depends on the problem parameters.

^{[5]}