Lyapunov's first method for strongly non-linear systems

Козлов, Валерий Васильевич; Furta, Stanislav D.

doi:10.1016/0021-8928(96)00003-2

Cited by 15 publications

(24 citation statements)

References 3 publications

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“…In accordance with the major results from [2][3][4][5][6][7], referring to the generalization of Lyapunov's first method to the strongly nonlinear systems of differential equations, further considerations are based on the following statement: if differential equations (2) and (5) allow the existence of the solution q = q(t) with the property q (t) → 0 when t → −∞, then the equilibrium position q = 0,q = 0 is unstable.…”

Section: Introductionsupporting

confidence: 90%

“…When it is taken into account that (17) represents the condition for the extremum of a function Π * (m) = Π * (m) (q ) on the ellipsoid a * αβ (0)q α q β = 1 and that the negative minimum, if it exists, has the corresponding κ > 0, with respect to the relation (see (17)) κ = −mΠ * (m) , from (19) there follows uniquely λ > 0. In this way, the existence of the first term of the series (13) was established in the form q * = λe * (−t) −2 m−2 , which also entails [3][4][5][6] the existence of the series (13) and the instability of the position of equilibrium q = 0,q = 0. Now we can formulate the following theorem.…”

Section: The Case Of the Inversion Of The Lagrange-dirichlet Theoremmentioning

confidence: 99%

“…In the case with m > 2 (see (7)), the solution q = q (t) with the property q (t) → 0 when t → −∞ is sought in the form of the infinite series, possibly also divergent [5][6] ,…”

Section: Introductionmentioning

confidence: 99%

“…To find the conditions for the existence of the series given above, we take [5][6] a 1 = λe, where λ > 0, e = (e 1 , · · · , e n ). The series (13), for the case with μ = 2 m−2 , incorporated in (11) and (12), yields the following relations (only terms relevant to further analysis are shown),…”

Section: Introductionmentioning

confidence: 99%

“…If at the point q = q 0 , the conditions ∂Π ∂q i (q =q 0 ) = 0, ψ ν q 0 ,q = 0 = 0 (6) are fulfilled, then that point is the equilibrium position of the system (II kind). Further, without loss of generality of consideration, it will be considered that q 0 = 0.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On the stability of equilibria of nonholonomic systems with nonlinear constraints

Čović

Vesković

Djuric

et al. 2010

Appl. Math. Mech.-Engl. Ed.

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Lyapunov's first method, extended by V. V. Kozlov to nonlinear mechanical systems, is applied to the study of the instability of the position of equilibrium of a mechanical system moving in the field of conservative and dissipative forces. The motion of the system is limited by ideal nonlinear nonholonomic constraints. Five cases determined by the relationship between the degree of the first nontrivial polynomials in Maclaurin's series for the potential energy and the functions that can be generated from the equations of nonlinear nonholonomic constraints are analyzed. In the three cases, the theorem on the instability of the position of equilibrium of nonholonomic systems with linear homogeneous constraints ) is generalized to the case of nonlinear nonhomogeneous constraints. In the other two cases, new theorems are set extending the result from V. V. Kozlov (1994) to nonholonomic systems with nonlinear constraints.

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Section: Introductionsupporting

confidence: 90%

Section: The Case Of the Inversion Of The Lagrange-dirichlet Theoremmentioning

confidence: 99%

“…In the case with m > 2 (see (7)), the solution q = q (t) with the property q (t) → 0 when t → −∞ is sought in the form of the infinite series, possibly also divergent [5][6] ,…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the stability of equilibria of nonholonomic systems with nonlinear constraints

Čović

Vesković

Djuric

et al. 2010

Appl. Math. Mech.-Engl. Ed.

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Critical stable sectional area of downstream surge tank of hydropower plant with sloping ceiling tailrace tunnel

Guo

Zhu

2021

Energy Science & Engineering

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This paper investigates the critical stable sectional area (CSSA) of downstream surge tank (DST) of hydropower plant with sloping ceiling tailrace tunnel (SCTT). Firstly, two types of nonlinear mathematical model of hydropower plant with SCTT during transient process are established. Then, the criterion of operation stability of hydropower plant is obtained based on nonlinear stability theory. Formulae for CSSAs of DST under two types of mathematical model are derived. Finally, the similarities, differences, and relationships of different CSSAs are clarified. The effect mechanisms of the assumption of steady power output of hydro‐turbine and the SCTT on CSSA of DST are revealed. The recommended use conditions of different CSSAs are explained. The results indicate that, under the assumption of steady power output of hydro‐turbine, the formula for CSSA of DST of hydropower plant with SCTT is analytical and has a simple expression. Under the complete model, the formula for CSSA of DST of hydropower plant with SCTT is not analytical and has a complicated expression. The former formula is suggested to be given the priority of use because of good precision and concise expression. SCTT can improve the stability of hydropower plant and decrease the CSSA of DST.

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On the instability of equilibrium position of a mechanical system with singular constraints

et al. 2013

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Key words Lyapunov first method, nonholonomic system, singular constraints, instability of equilibrium.The Lyapunov first method generalized to the case of nonlinear differential equations is applied to the study of the instability of the equilibrium position of a mechanical system, whose motion is constrained by singular nonholonomic constraints. Starting from the results of S. D. Furta (On the instability of equilibrium position of constrained mechanical systems) three theorems on the instability are formulated. The first theorem considers the case of nonholonomic constraints that do not satisfy the condition of weak nonholonomity. The other two theorems are related to the case of weakly nonholonomic systems. In each of the formulated theorems it is shown that the minimum form of Maclaurin series for the potential energy has not a local minimum. Thus, a contribution has been made to the inversion of Lagrange's theorem.

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Lyapunov's first method for strongly non-linear systems

Cited by 15 publications

References 3 publications

On the stability of equilibria of nonholonomic systems with nonlinear constraints

On the stability of equilibria of nonholonomic systems with nonlinear constraints

Critical stable sectional area of downstream surge tank of hydropower plant with sloping ceiling tailrace tunnel

On the instability of equilibrium position of a mechanical system with singular constraints

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