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Peter Giesl - One of the best experts on this subject based on the ideXlab platform.

  • computation of the stochastic Basin of Attraction by rigorous construction of a lyapunov function
    Discrete and Continuous Dynamical Systems-series B, 2019
    Co-Authors: Hjortur Bjornsson, Peter Giesl, Sigurdur F Hafstein, Enrico Scalas, Skuli Gudmundsson
    Abstract:

    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. In this paper, we construct a non-local Lyapunov function by solving a second-order PDE using meshless collocation. We provide a-posteriori error estimates which guarantee that the constructed function is indeed a non-local Lyapunov function. Combining this method with the computation of a local Lyapunov function for the linearisation around an equilibrium of the stochastic differential equation in question, a problem which is much more manageable than computing a Lyapunov function in a large area containing the equilibrium, we provide a rigorous estimate of the stochastic γ-Basin of Attraction of the equilibrium.

  • determination of the Basin of Attraction of a periodic orbit in two dimensions using meshless collocation
    ACM Journal of Computer Documentation, 2017
    Co-Authors: Peter Giesl, James Mcmichen
    Abstract:

    A contraction metric for an autonomous ordinary differential equation is a Riemannian metric such that the distance between adjacent solutions contracts over time. A contraction metric can be used to determine the Basin of Attraction of a periodic orbit without requiring information about its position or stability. Moreover, it is robust to small perturbations of the system. In two-dimensional systems, a contraction metric can be characterised by a scalar-valued function. In [9], the function was constructed as solution of a first-order linear Partial Differential Equation (PDE), and numerically constructed using meshless collocation. However, information about the periodic orbit was required, which needed to be approximated. In this paper, we overcome this requirement by studying a second-order PDE, which does not require any information about the periodic orbit. We show that the second-order PDE has a solution, which defines a contraction metric. We use meshless collocation to approximate the solution and prove error estimates. In particular, we show that the approximation itself is a contraction metric, if the collocation points are dense enough. The method is applied to two examples.

  • towards calculating the Basin of Attraction of non smooth dynamical systems using radial basis functions
    2011
    Co-Authors: Peter Giesl
    Abstract:

    We consider a special type of non-smooth dynamical systems, namely x? = f(t, x), where x ? R, f is t-periodic with period T and non-smooth at x = 0. In [6] a sufficient Borg-like condition to determine a subset of its Basin of Attraction was given. The condition involves a function W and its partial derivatives; the function W is t-periodic and non-smooth at x = 0.

  • Numerical determination of the Basin of Attraction for exponentially asymptotically autonomous dynamical systems
    Nonlinear Analysis: Theory Methods & Applications, 2011
    Co-Authors: Peter Giesl, Holger Wendland
    Abstract:

    Numerical methods to determine the Basin of Attraction for autonomous equations focus on a bounded subset of the phase space. For non-autonomous systems, any relevant subset of the phase space, which now includes the time as one coordinate, is unbounded in the t-direction. Hence, a numerical method would have to use infinitely many points. To overcome this problem, we introduce a transformation of the phase space. Restricting ourselves to exponentially asymptotically autonomous systems, we can map the infinite time interval to a finite, compact one. The Basin of Attraction of a solution becomes the Basin of Attraction of an exponentially stable equilibrium for an autonomous system. Now we are able to generalise numerical methods from the autonomous case. More precisely, we characterise a Lyapunov function as a solution of a suitable linear first-order partial differential equation and approximate it using radial basis functions.

  • approximating the Basin of Attraction of time periodic odes by meshless collocation
    Discrete and Continuous Dynamical Systems, 2009
    Co-Authors: Peter Giesl, Holger Wendland
    Abstract:

    In this paper we study a periodic solution of a general time-periodic ordinary differential equation (ODE) and determine its Basin of Attraction using a time-periodic Lyapunov function. We show the existence of a Lyapunov function satisfying a certain linear partial differential equation and approximate it using meshless collocation. Therefore, we establish error estimates for the approximate reconstruction and collocation of functions [V(t,x)] which are periodic with respect to [t] .

Hiroshi Kokubu - One of the best experts on this subject based on the ideXlab platform.

  • disappearance of chaotic attractor of passive dynamic walking by stretch bending deformation in Basin of Attraction
    Intelligent Robots and Systems, 2020
    Co-Authors: Kota Okamoto, Shinya Aoi, Ippei Obayashi, Hiroshi Kokubu, Kei Senda, Kazuo Tsuchiya
    Abstract:

    Passive dynamic walking is a model that walks down a shallow slope without any control or input. This model has been widely used to investigate how stable walking is generated from a dynamic viewpoint, which is useful to provide design principles for developing energy-efficient biped robots. However, the Basin of Attraction is very small and thin, and it has a fractal-like complicated shape. This makes it difficult to produce stable walking. Furthermore, the passive dynamic walking shows chaotic attractor through a period-doubling cascade by increasing the slope angle, and the chaotic attractor suddenly disappears at a critical slope angle. These make it further difficult to produce stable walking. In our previous work, we used the simplest walking model and investigated the fractal-like Basin of Attraction based on dynamical systems theory by focusing on the hybrid dynamics of the model composed of the continuous dynamics with saddle hyperbolicity and the discontinuous dynamics by the impact at foot contact. We elucidated that the fractal-like Basin of Attraction is generated through iterative stretch and bending deformations of the domain of the Poincare map by sequential inverse images of the Poincare map. In this study, we investigated the mechanism for the disappearance of the chaotic attractor by improving our previous analysis. In particular, we focused on the range of the Poincare map to specify the regions to be stretched and bent by the inverse image of the Poincare map. We clarified the condition for the chaotic attractor to disappear and the mechanism why the chaotic attractor disappears based on the stretch-bending deformation in the Basin of Attraction.

  • Fractal mechanism of Basin of Attraction in passive dynamic walking
    Bioinspiration & biomimetics, 2020
    Co-Authors: Kota Okamoto, Shinya Aoi, Ippei Obayashi, Hiroshi Kokubu, Kei Senda, Kazuo Tsuchiya
    Abstract:

    Passive dynamic walking is a model that walks down a shallow slope without any control or input. This model has been widely used to investigate how humans walk with low energy consumption and provides design principles for energy-efficient biped robots. However, the Basin of Attraction is very small and thin and has a fractal-like complicated shape, which makes producing stable walking difficult. In our previous study, we used the simplest walking model and investigated the fractal-like Basin of Attraction based on dynamical systems theory by focusing on the hybrid dynamics of the model composed of the continuous dynamics with saddle hyperbolicity and the discontinuous dynamics caused by the impact upon foot contact. We clarified that the fractal-like Basin of Attraction is generated through iterative stretching and bending deformations of the domain of the Poincar\'e map by sequential inverse images. However, whether the fractal-like Basin of Attraction is actually fractal, i.e., whether infinitely many self-similar patterns are embedded in the Basin of Attraction, is dependent on the slope angle, and the mechanism remains unclear. In the present study, we improved our previous analysis in order to clarify this mechanism. In particular, we newly focused on the range of the Poincar\'e map and specified the regions that are stretched and bent by the sequential inverse images of the Poincar\'e map. Through the analysis of the specified regions, we clarified the conditions and mechanism required for the Basin of Attraction to be fractal.

  • Formation mechanism of a Basin of Attraction for passive dynamic walking induced by intrinsic hyperbolicity.
    Proceedings. Mathematical physical and engineering sciences, 2016
    Co-Authors: Ippei Obayashi, Shinya Aoi, Kazuo Tsuchiya, Hiroshi Kokubu
    Abstract:

    Passive dynamic walking is a useful model for investigating the mechanical functions of the body that produce energy-efficient walking. The Basin of Attraction is very small and thin, and it has a fractal-like shape; this explains the difficulty in producing stable passive dynamic walking. The underlying mechanism that produces these geometric characteristics was not known. In this paper, we consider this from the viewpoint of dynamical systems theory, and we use the simplest walking model to clarify the mechanism that forms the Basin of Attraction for passive dynamic walking. We show that the intrinsic saddle-type hyperbolicity of the upright equilibrium point in the governing dynamics plays an important role in the geometrical characteristics of the Basin of Attraction; this contributes to our understanding of the stability mechanism of bipedal walking.

  • Common formation mechanism of Basin of Attraction for bipedal walking models by saddle hyperbolicity and hybrid dynamics
    Japan Journal of Industrial and Applied Mathematics, 2015
    Co-Authors: Ippei Obayashi, Kazuo Tsuchiya, Hiroshi Kokubu
    Abstract:

    In this paper, we investigate the mathematical structures and mechanisms of bipedal walking from a dynamical viewpoint. Especially, we focus on the Basin of Attraction since it determines the stability of bipedal walking. We treat two similar but different bipedal walking models (passive and active dynamic walking models) and examine common mathematical structure between these models. We find that the saddle hyperbolicity and hybrid system play important roles for the shape of the Basin of Attraction in both models, which are quite common for more general bipedal models and important for understanding the stability mechanism of bipedal walking.

Kazuo Tsuchiya - One of the best experts on this subject based on the ideXlab platform.

  • disappearance of chaotic attractor of passive dynamic walking by stretch bending deformation in Basin of Attraction
    Intelligent Robots and Systems, 2020
    Co-Authors: Kota Okamoto, Shinya Aoi, Ippei Obayashi, Hiroshi Kokubu, Kei Senda, Kazuo Tsuchiya
    Abstract:

    Passive dynamic walking is a model that walks down a shallow slope without any control or input. This model has been widely used to investigate how stable walking is generated from a dynamic viewpoint, which is useful to provide design principles for developing energy-efficient biped robots. However, the Basin of Attraction is very small and thin, and it has a fractal-like complicated shape. This makes it difficult to produce stable walking. Furthermore, the passive dynamic walking shows chaotic attractor through a period-doubling cascade by increasing the slope angle, and the chaotic attractor suddenly disappears at a critical slope angle. These make it further difficult to produce stable walking. In our previous work, we used the simplest walking model and investigated the fractal-like Basin of Attraction based on dynamical systems theory by focusing on the hybrid dynamics of the model composed of the continuous dynamics with saddle hyperbolicity and the discontinuous dynamics by the impact at foot contact. We elucidated that the fractal-like Basin of Attraction is generated through iterative stretch and bending deformations of the domain of the Poincare map by sequential inverse images of the Poincare map. In this study, we investigated the mechanism for the disappearance of the chaotic attractor by improving our previous analysis. In particular, we focused on the range of the Poincare map to specify the regions to be stretched and bent by the inverse image of the Poincare map. We clarified the condition for the chaotic attractor to disappear and the mechanism why the chaotic attractor disappears based on the stretch-bending deformation in the Basin of Attraction.

  • Fractal mechanism of Basin of Attraction in passive dynamic walking
    Bioinspiration & biomimetics, 2020
    Co-Authors: Kota Okamoto, Shinya Aoi, Ippei Obayashi, Hiroshi Kokubu, Kei Senda, Kazuo Tsuchiya
    Abstract:

    Passive dynamic walking is a model that walks down a shallow slope without any control or input. This model has been widely used to investigate how humans walk with low energy consumption and provides design principles for energy-efficient biped robots. However, the Basin of Attraction is very small and thin and has a fractal-like complicated shape, which makes producing stable walking difficult. In our previous study, we used the simplest walking model and investigated the fractal-like Basin of Attraction based on dynamical systems theory by focusing on the hybrid dynamics of the model composed of the continuous dynamics with saddle hyperbolicity and the discontinuous dynamics caused by the impact upon foot contact. We clarified that the fractal-like Basin of Attraction is generated through iterative stretching and bending deformations of the domain of the Poincar\'e map by sequential inverse images. However, whether the fractal-like Basin of Attraction is actually fractal, i.e., whether infinitely many self-similar patterns are embedded in the Basin of Attraction, is dependent on the slope angle, and the mechanism remains unclear. In the present study, we improved our previous analysis in order to clarify this mechanism. In particular, we newly focused on the range of the Poincar\'e map and specified the regions that are stretched and bent by the sequential inverse images of the Poincar\'e map. Through the analysis of the specified regions, we clarified the conditions and mechanism required for the Basin of Attraction to be fractal.

  • Formation mechanism of a Basin of Attraction for passive dynamic walking induced by intrinsic hyperbolicity.
    Proceedings. Mathematical physical and engineering sciences, 2016
    Co-Authors: Ippei Obayashi, Shinya Aoi, Kazuo Tsuchiya, Hiroshi Kokubu
    Abstract:

    Passive dynamic walking is a useful model for investigating the mechanical functions of the body that produce energy-efficient walking. The Basin of Attraction is very small and thin, and it has a fractal-like shape; this explains the difficulty in producing stable passive dynamic walking. The underlying mechanism that produces these geometric characteristics was not known. In this paper, we consider this from the viewpoint of dynamical systems theory, and we use the simplest walking model to clarify the mechanism that forms the Basin of Attraction for passive dynamic walking. We show that the intrinsic saddle-type hyperbolicity of the upright equilibrium point in the governing dynamics plays an important role in the geometrical characteristics of the Basin of Attraction; this contributes to our understanding of the stability mechanism of bipedal walking.

  • Common formation mechanism of Basin of Attraction for bipedal walking models by saddle hyperbolicity and hybrid dynamics
    Japan Journal of Industrial and Applied Mathematics, 2015
    Co-Authors: Ippei Obayashi, Kazuo Tsuchiya, Hiroshi Kokubu
    Abstract:

    In this paper, we investigate the mathematical structures and mechanisms of bipedal walking from a dynamical viewpoint. Especially, we focus on the Basin of Attraction since it determines the stability of bipedal walking. We treat two similar but different bipedal walking models (passive and active dynamic walking models) and examine common mathematical structure between these models. We find that the saddle hyperbolicity and hybrid system play important roles for the shape of the Basin of Attraction in both models, which are quite common for more general bipedal models and important for understanding the stability mechanism of bipedal walking.

Ippei Obayashi - One of the best experts on this subject based on the ideXlab platform.

  • disappearance of chaotic attractor of passive dynamic walking by stretch bending deformation in Basin of Attraction
    Intelligent Robots and Systems, 2020
    Co-Authors: Kota Okamoto, Shinya Aoi, Ippei Obayashi, Hiroshi Kokubu, Kei Senda, Kazuo Tsuchiya
    Abstract:

    Passive dynamic walking is a model that walks down a shallow slope without any control or input. This model has been widely used to investigate how stable walking is generated from a dynamic viewpoint, which is useful to provide design principles for developing energy-efficient biped robots. However, the Basin of Attraction is very small and thin, and it has a fractal-like complicated shape. This makes it difficult to produce stable walking. Furthermore, the passive dynamic walking shows chaotic attractor through a period-doubling cascade by increasing the slope angle, and the chaotic attractor suddenly disappears at a critical slope angle. These make it further difficult to produce stable walking. In our previous work, we used the simplest walking model and investigated the fractal-like Basin of Attraction based on dynamical systems theory by focusing on the hybrid dynamics of the model composed of the continuous dynamics with saddle hyperbolicity and the discontinuous dynamics by the impact at foot contact. We elucidated that the fractal-like Basin of Attraction is generated through iterative stretch and bending deformations of the domain of the Poincare map by sequential inverse images of the Poincare map. In this study, we investigated the mechanism for the disappearance of the chaotic attractor by improving our previous analysis. In particular, we focused on the range of the Poincare map to specify the regions to be stretched and bent by the inverse image of the Poincare map. We clarified the condition for the chaotic attractor to disappear and the mechanism why the chaotic attractor disappears based on the stretch-bending deformation in the Basin of Attraction.

  • Fractal mechanism of Basin of Attraction in passive dynamic walking
    Bioinspiration & biomimetics, 2020
    Co-Authors: Kota Okamoto, Shinya Aoi, Ippei Obayashi, Hiroshi Kokubu, Kei Senda, Kazuo Tsuchiya
    Abstract:

    Passive dynamic walking is a model that walks down a shallow slope without any control or input. This model has been widely used to investigate how humans walk with low energy consumption and provides design principles for energy-efficient biped robots. However, the Basin of Attraction is very small and thin and has a fractal-like complicated shape, which makes producing stable walking difficult. In our previous study, we used the simplest walking model and investigated the fractal-like Basin of Attraction based on dynamical systems theory by focusing on the hybrid dynamics of the model composed of the continuous dynamics with saddle hyperbolicity and the discontinuous dynamics caused by the impact upon foot contact. We clarified that the fractal-like Basin of Attraction is generated through iterative stretching and bending deformations of the domain of the Poincar\'e map by sequential inverse images. However, whether the fractal-like Basin of Attraction is actually fractal, i.e., whether infinitely many self-similar patterns are embedded in the Basin of Attraction, is dependent on the slope angle, and the mechanism remains unclear. In the present study, we improved our previous analysis in order to clarify this mechanism. In particular, we newly focused on the range of the Poincar\'e map and specified the regions that are stretched and bent by the sequential inverse images of the Poincar\'e map. Through the analysis of the specified regions, we clarified the conditions and mechanism required for the Basin of Attraction to be fractal.

  • Formation mechanism of a Basin of Attraction for passive dynamic walking induced by intrinsic hyperbolicity.
    Proceedings. Mathematical physical and engineering sciences, 2016
    Co-Authors: Ippei Obayashi, Shinya Aoi, Kazuo Tsuchiya, Hiroshi Kokubu
    Abstract:

    Passive dynamic walking is a useful model for investigating the mechanical functions of the body that produce energy-efficient walking. The Basin of Attraction is very small and thin, and it has a fractal-like shape; this explains the difficulty in producing stable passive dynamic walking. The underlying mechanism that produces these geometric characteristics was not known. In this paper, we consider this from the viewpoint of dynamical systems theory, and we use the simplest walking model to clarify the mechanism that forms the Basin of Attraction for passive dynamic walking. We show that the intrinsic saddle-type hyperbolicity of the upright equilibrium point in the governing dynamics plays an important role in the geometrical characteristics of the Basin of Attraction; this contributes to our understanding of the stability mechanism of bipedal walking.

  • Common formation mechanism of Basin of Attraction for bipedal walking models by saddle hyperbolicity and hybrid dynamics
    Japan Journal of Industrial and Applied Mathematics, 2015
    Co-Authors: Ippei Obayashi, Kazuo Tsuchiya, Hiroshi Kokubu
    Abstract:

    In this paper, we investigate the mathematical structures and mechanisms of bipedal walking from a dynamical viewpoint. Especially, we focus on the Basin of Attraction since it determines the stability of bipedal walking. We treat two similar but different bipedal walking models (passive and active dynamic walking models) and examine common mathematical structure between these models. We find that the saddle hyperbolicity and hybrid system play important roles for the shape of the Basin of Attraction in both models, which are quite common for more general bipedal models and important for understanding the stability mechanism of bipedal walking.

Juergen Kurths - One of the best experts on this subject based on the ideXlab platform.

  • Basin of Attraction determines hysteresis in explosive synchronization
    Physical Review Letters, 2014
    Co-Authors: Yong Zou, Tiago Pereira, Michael Small, Zonghua Liu, Juergen Kurths
    Abstract:

    Spontaneous explosive emergent behavior takes place in heterogeneous networks when the frequencies of the nodes are positively correlated to the node degree. A central feature of such explosive transitions is a hysteretic behavior at the transition to synchronization. We unravel the underlying mechanisms and show that the dynamical origin of the hysteresis is a change of Basin of Attraction of the synchronization state. Our findings hold for heterogeneous networks with star graph motifs such as scale-free networks, and hence, reveal how microscopic network parameters such as node degree and frequency affect the global network properties and can be used for network design and control.