Into higher dimensions for nonsmooth dynamical systems

Jeffrey, Mike R.; Seidman, Thomas I.; Teixeira, Marco Antônio; Utkin, Vadim I.

doi:10.1016/j.physd.2022.133222

Cited by 7 publications

(11 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The approximation (32) (and hence (33)) captures the slow arcs of the exact flow (22) (or (26)) increasingly well as x increases, as shown in Fig. 11.…”

Section: Lemma 2 (Slow Dynamics) the Dynamics Onmentioning

confidence: 79%

See 1 more Smart Citation

Novel Slow-Fast Behaviour in an Oscillator Driven by a Frequency-Switching Force

Jeffrey

Bonet²,

Martín³

et al. 2022

SSRN Journal

View full text Add to dashboard Cite

“…The approximation (32) (and hence (33)) captures the slow arcs of the exact flow (22) (or (26)) increasingly well as x increases, as shown in Fig. 11.…”

Section: Lemma 2 (Slow Dynamics) the Dynamics Onmentioning

confidence: 79%

“…For large x in (22), it becomes impossible to separate out the fast oscillation of the sin(π x(1 + 1 2 u)) term from the slow-fast timescale separation created by small ε. Hence the slow dynamics depends not just on the small parameter ε, but also crucially on the largeness of the time x.…”

Section: Slow Dynamics For Large X: Large and Small Arcsmentioning

confidence: 99%

Novel Slow-Fast Behaviour in an Oscillator Driven by a Frequency-Switching Force

Jeffrey

Bonet²,

Martín³

et al. 2022

SSRN Journal

View full text Add to dashboard Cite

“…Based on different arguments methods may result in different sliding mode equations. Methods 1, 2 and 6 are compared in (Utkin, 1971(Utkin, , 1972, Methods 1,4 and 6 in (Jeffrey et al, 2022;Polyakov & Fridman, 2014). The question, which is correct, is not legitimate-all equations were postulated.…”

Section: Equivalent Control Methodsmentioning

confidence: 99%

Brief Comments for Doubts in Filippov Method

Utkin

2022

J Control Autom Electr Syst

View full text Add to dashboard Cite

Different methods of solution continuation for the system with discontinuous control and sliding modes are discussed in the paper. First, the systems with one discontinuity surface are discussed when the right-hand side of differential equation can take two values only for the selected state x and time t. According to Filippov method the right-hand side of sliding mode equation can be found in convex hull of these two values. Other five methods of solution continuation are developed under additional assumption about values of control on discontinuity surface. They led to sliding mode equations different from Filippov's equations both for affine systems and systems nonlinear with respect to control. However, if we apply Filippov method based on the procedure of passing over to a convex hull considering additional assumptions of each method then the sliding mode equation is the same as for this method. Keywords Discontinuous control • Sliding mode • Convex hullPublisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

show abstract

“…In the literature, authors have demonstrated that there exist conditions for which the use of smoothing functions is admissible and produce commensurate and acceptable dynamic responses [53,54]; nonetheless, more recent studies have highlighted how such regularisations can create ambiguities [55]. Jeffrey et al [56] suggested that the nature of such ambiguities is precisely the nonuniqueness of the solution which induces the generation of a set of solutions at the discontinuity border. Each of these solutions is equally valid but represents only a portion of the system dynamic response.…”

Section: Introductionmentioning

confidence: 99%

“…Each of these solutions is equally valid but represents only a portion of the system dynamic response. Nevertheless, in some cases the Hidden Dynamics [57] of the sliding solutions 3 could be missed: in fact, the adopted regularisation (see Examples 1 and 2 of [56]) and solution method (see Example 1 of [57]) can deeply affect the solution of the system. If the solution method can only miss the hidden attractors, the regularisation can affect their existence by preserving the nonlinear properties of discontinuity boundary [57].…”

Section: Introductionmentioning

confidence: 99%

Approximating piecewise nonlinearities in dynamic systems with sigmoid functions: advantages and limitations

2023

View full text Add to dashboard Cite

In the industry field, the increasingly stringent requirements of lightweight structures are exposing the ultimately nonlinear nature of mechanical systems. This is extremely true for systems with moving parts and loose fixtures which show piecewise stiffness behaviours. Nevertheless, the numerical solution of systems with ideal piecewise mathematical characteristics is associated with time-consuming procedures and a high computational burden. Smoothing functions can conveniently simplify the mathematical form of such systems, but little research has been carried out to evaluate their effect on the mechanical response of multi-degree-of-freedom systems. To investigate this problem, a slightly damped mechanical two-degree-of-freedom system with soft piecewise constraints is studied via numerical continuation and numerical integration procedures. Sigmoid functions are adopted to approximate the constraints, and the effect of such approximation is explored by comparing the results of the approximate system with the ones of the ideal piecewise counter-part. The numerical results show that the sigmoid functions can correctly catch the very complex dynamics of the proposed system when both the above-mentioned techniques are adopted. Moreover, a reduction in the computational burden, as well as an increase in numerical robustness, is observed in the approximate case.

show abstract

Into higher dimensions for nonsmooth dynamical systems

Cited by 7 publications

References 37 publications

Novel Slow-Fast Behaviour in an Oscillator Driven by a Frequency-Switching Force

Novel Slow-Fast Behaviour in an Oscillator Driven by a Frequency-Switching Force

Brief Comments for Doubts in Filippov Method

Approximating piecewise nonlinearities in dynamic systems with sigmoid functions: advantages and limitations

Contact Info

Product

Resources

About