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We study a class of discontinuous vector fields brought to our attention by multi-legged animal locomotion.
“Beweis der invarianz desn-dimensionalen gebiets”
L.E.J. Brouwer · 1911
Earlier work this paper cites.
“Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales”
S. Banach · 1922
Earlier work this paper cites.
“Determination of Stability by Linear Approximation of a Periodic Solution of a System of Differential Equations with Discontinuous Right–Hand Sides”
M.˜A. Aizerman and F.˜R. Gantmacher · 1958
Earlier work this paper cites.
“Differential equations, dynamical systems, and linear algebra”
M.˜W. Hirsch and S. Smale · 1974
Earlier work this paper cites.
“Differential topology”
M.W. Hirsch · 1976
Earlier work this paper cites.
“Variable structure systems with sliding modes”
V. Utkin · 1977
Earlier work this paper cites.
“Integrate–and–Fire Models of Nerve Membrane Response to Oscillatory Input”
J. Keener, F. Hoppensteadt and J. Rinzel · 1981
Earlier work this paper cites.
“Stabilization with relaxed controls”
Zvi Artstein · 1983
Earlier work this paper cites.
“Nonlinear oscillations, dynamical systems, and bifurcations of vector fields”
J. Guckenheimer and P. Holmes · 1983
Earlier work this paper cites.
“The Gaits of Bipedal and Quadrupedal Animals”
R.˜M. Alexander · 1984
Earlier work this paper cites.
“Local structure of feasible sets in nonlinear programming, Part III: Stability and sensitivity”
Stephen˜M. Robinson · 1987
Earlier work this paper cites.
“Manifolds, tensor analysis, and applications”
R. Abraham, J.˜E. Marsden and T.˜S. Ratiu · 1988
Earlier work this paper cites.
“Differential equations with discontinuous righthand sides”
A.˜F. Filippov · 1988
Earlier work this paper cites.
“Dynamic pattern generation in behavioral and neural systems”
G Schoner and J˜A Kelso · 1988
Earlier work this paper cites.
“A ‘universal’ construction of Artstein’s theorem on nonlinear stabilization”
Eduardo˜D. Sontag · 1989
Earlier work this paper cites.
“Optimization and nonsmooth analysis”
F.˜H. Clarke · 1990
Earlier work this paper cites.
“Synchronization of Pulse–Coupled Biological Oscillators”
Renato˜E. Mirollo and Steven˜H. Strogatz · 1990
Earlier work this paper cites.
“Analysis of a Simplified Hopping Robot”
Daniel˜E. Koditschek and Martin B\"uhler · 1991
Cited alongside, same era.
“Voltage collapse precipitated by the immediate change in stability when generator reactive power limits are encountered”
I. Dobson and L. Lu · 1992
Cited alongside, same era.
“Control of nonlinear dynamical systems using neural networks: controllability and stabilization”
A.˜U. Levin and K.˜S. Narendra · 1993
Cited alongside, same era.
“Rapid local synchronization of action potentials: toward computation with coupled integrate–and–fire neurons”
J˜J Hopfield and A˜V Herz · 1995
Cited alongside, same era.
“Analysis tools for power systems–contending with nonlinearities”
I.˜A. Hiskens · 1995
Cited alongside, same era.
“Optimization: algorithms and consistent approximations”
“Towards a geometric theory of hybrid systems”
S.˜N. Simic, K.˜H. Johansson, J. Lygeros and S.˜S. Sastry · 2005
Later among the works it cites.
“The Dynamics of Legged Locomotion: Models, Analyses, and Challenges”
P. Holmes, R.˜J. Full, D.˜E. Koditschek and J. Guckenheimer · 2006
Later among the works it cites.
“Piecewise–smooth dynamical systems: theory and applications”
M. Di˜Bernardo, C.˜J. Budd, P. Kowalczyk and A.˜R. Champneys · 2008
Later among the works it cites.
“Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results”
Hai Lin and P.˜J. Antsaklis · 2008
Later among the works it cites.
“Lateral stability of the spring–mass hopper suggests a two–step control strategy for running”
S.˜G. Carver, N.˜J. Cowan and J.˜M. Guckenheimer · 2009
Later among the works it cites.
“Fundamental matrix solutions of piecewise smooth differential systems”
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Cited alongside, same era.
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Cited alongside, same era.
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A.P. Ivanov · 1998
Cited alongside, same era.
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Winfried Lohmiller and Jean-Jacques Slotine · 1998
Cited alongside, same era.
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Martin Golubitsky, Ian Stewart, Pietro-Luciano Buono and J.˜J. Collins · 1999
Cited alongside, same era.
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T.˜M. Kubow and R.˜J. Full · 1999
Cited alongside, same era.
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S.˜S. Sastry · 1999
Cited alongside, same era.
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Later among the works it cites.
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Later among the works it cites.
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Later among the works it cites.
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Later among the works it cites.
“Introduction to piecewise differentiable equations”
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Later among the works it cites.
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