Quasiperiodic, periodic, and slowing-down states of coupled heteroclinic cycles

Li, Dong; Cross, M. C.; Zhou, Changsong; Zheng, Zhigang

doi:10.1103/physreve.85.016215

Cited by 11 publications

(5 citation statements)

References 46 publications

(80 reference statements)

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“…Additionally, the techniques detailed in Shaw et al (2012b) may be able to be adapted to formally design deterministic systems that retain many of the responsive and variable dwell-time capabilities of the stochastic system described here. Future work will also include investigating how SHC cycles of various dimensions can be coupled (Li et al 2012), guidelines for network design, and the generation of more complex waveforms.…”

Section: Discussionmentioning

confidence: 99%

“…Guckenheimer and Holmes (1988) formally documented structurally stable heteroclinic cycles in a model of turbulent flow (Busse and Heikes 1980). This model, which has a mathematical structure that closely resembles the Lotka-Volterra system used in this paper, is also commonly used (McInnes and Brown 2010, Ashwin and Karabacak 2011, Li et al 2012.…”

Section: Stable Heteroclinic Channels (Shcs)mentioning

confidence: 99%

See 1 more Smart Citation

Designing responsive pattern generators: stable heteroclinic channel cycles for modeling and control

Horchler¹,

Daltorio²,

Chiel³

et al. 2015

Bioinspir. Biomim.

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A striking feature of biological pattern generators is their ability to respond immediately to multisensory perturbations by modulating the dwell time at a particular phase of oscillation, which can vary force output, range of motion, or other characteristics of a physical system. Stable heteroclinic channels (SHCs) are a dynamical architecture that can provide such responsiveness to artificial devices such as robots. SHCs are composed of sequences of saddle equilibrium points, which yields exquisite sensitivity. The strength of the vector fields in the neighborhood of these equilibria determines the responsiveness to perturbations and how long trajectories dwell in the vicinity of a saddle. For SHC cycles, the addition of stochastic noise results in oscillation with a regular mean period. In this paper, we parameterize noise-driven Lotka-Volterra SHC cycles such that each saddle can be independently designed to have a desired mean sub-period. The first step in the design process is an analytic approximation, which results in mean sub-periods that are within 2% of the specified sub-period for a typical parameter set. Further, after measuring the resultant sub-periods over sufficient numbers of cycles, the magnitude of the noise can be adjusted to control the mean period with accuracy close to that of the integration step size. With these relationships, SHCs can be more easily employed in engineering and modeling applications. For applications that require smooth state transitions, this parameterization permits each state's distribution of periods to be independently specified. Moreover, for modeling context-dependent behaviors, continuously varying inputs in each state dimension can rapidly precipitate transitions to alter frequency and phase.

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Section: Discussionmentioning

confidence: 99%

Section: Stable Heteroclinic Channels (Shcs)mentioning

confidence: 99%

Designing responsive pattern generators: stable heteroclinic channel cycles for modeling and control

Horchler¹,

Daltorio²,

Chiel³

et al. 2015

Bioinspir. Biomim.

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show abstract

“…The effect of noise on SHCs has been explored in other work, and for practical uses, we can establish a reasonable noise magnitude compared to the rest of the system [13,14,[52][53][54][55].…”

Section: System Modelmentioning

confidence: 99%

Stable Heteroclinic Channel-Based Movement Primitives: Tuning Trajectories Using Saddle Parameters

Rouse,

Daltorio

2024

Applied Sciences

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Dynamic systems which underlie controlled systems are expected to increase in complexity as robots, devices, and connected networks become more intelligent. While classical stable systems converge to a stable point (a sink), another type of stability is to consider a stable path rather than a single point. Such stable paths can be made of saddle points that draw in trajectories from certain regions, and then push the trajectory toward the next saddle point. These chains of saddles are called stable heteroclinic channels (SHCs) and can be used in robotic control to represent time sequences. While we have previously shown that each saddle is visualizable as a trajectory waypoint in phase space, how to increase the fidelity of the trajectory was unclear. In this paper, we hypothesized that the waypoints can be individually modified to locally vary fidelity. Specifically, we expected that increasing the saddle value (ratio of saddle eigenvalues) causes the trajectory to slow to more closely approach a particular saddle. Combined with other parameters that control speed and magnitude, a system expressed with an SHC can be modified locally, point by point, without disrupting the rest of the path, supporting their use in motion primitives. While some combinations can enable a trajectory to better reach into corners, other combinations can rotate, distort, and round the trajectory surrounding the modified saddle. Of the system parameters, the saddle value provides the most predictable tunability across 3 orders of magnitude.

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“…Noise is a critical factor in the use of LV kernels; noise, or external perturbation, z j , ensures that the system variable, x, passes close to the SHC saddle point, but not so close that the system remains in static equilibrium [6]. The effect of noise on SHCs has been explored in other work, and for practical uses, we can establish a reasonable noise magnitude compared to the rest of the system [4,5,[29][30][31][32].…”

Section: System Modelmentioning

confidence: 99%

Stable Heteroclinic Channel-based Movement Primitives: Tuning Trajectories using Saddle Parameters

Rouse,

Daltorio

2024

Preprint

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Dynamic systems which underly controlled systems are expected to increase in complexity as robots, devices, and connected networks become more intelligent. While classical stable systems converge to a stable point (a sink), another type of stability is to consider a stable path rather than a single point. Such stable paths can be made of saddles which draw in trajectories from certain regions, and then push the trajectory toward the next saddle point. These are chains of saddles are called stable heteroclinic channels (SHCs), and can be used in robotic control to represent time sequences. While we have previously shown that each saddle is visualizable as a trajectory waypoint in phase space, how to increase the fidelity of the trajectory was unclear. In this paper, we hypothesized that the waypoints can be individually modified to locally vary fidelity. Specifically, we expected that increasing the saddle value (ratio of saddle eigenvalues) causes the trajectory to slow to more closely approach a particular saddle. Combined with other parameters that control speed and magnitude, a system expressed with an SHC can be modified locally, point by point, without disrupting the rest of the path, supporting their use in motion primitives. However, even more complex trajectory shape modifications are possible. While some combinations can enable a trajectory to better reach into corners, other combinations can rotate, distort and round the trajectory in the region of a corner. In our example, modifying select saddle values produced a 32% decrease in the trajectory error of a complex trajectory produced using this system. This is an effect not visible in previous 1D studies, that can lead to different learnable and tunable representations of dynamic systems.

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Quasiperiodic, periodic, and slowing-down states of coupled heteroclinic cycles

Cited by 11 publications

References 46 publications

Designing responsive pattern generators: stable heteroclinic channel cycles for modeling and control

Designing responsive pattern generators: stable heteroclinic channel cycles for modeling and control

Stable Heteroclinic Channel-Based Movement Primitives: Tuning Trajectories Using Saddle Parameters

Stable Heteroclinic Channel-based Movement Primitives: Tuning Trajectories using Saddle Parameters

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