Non-Smooth Deterministic or Stochastic Discrete Dynamical Systems

Jérôme BASTIEN is Assistant Professor at the University Lyon 1 (Centre de recherche et d'Innovation sur le sport) in France.Frédéric BERNARDIN is a Research Engineer at Département Laboratoire de Clermont-Ferrand (DLCF), Centre d'Etudes Techniques de l'Equipement (CETE), Lyon, France.Claude-Henri LAMARQUE is Head of Laboratoire Géomatériaux et Génie Civil (LGCB) and Professor at Ecole des Travaux Publics de l'Etat (ENTPE), Vaulx-en-Velin, France.

• Introduction xiChapter 1. Some Simple Examples 11.1. Introduction 11.2. Frictions 11.2.1. Coulomb’s law 11.2.2. Differential equation with univalued operator and usual sign 31.2.3. Differential equation with multivalued term: differential inclusion 111.2.4. Other friction laws 121.3. Impact 161.3.1. Difficulties with writing the differential equation 161.3.2. Ill-posed problems 191.4. Probabilistic context 22Chapter 2. Theoretical Deterministic Context 272.1. Introduction 272.2. Maximal monotone operators and first result on differential inclusions (in R) 272.2.1. Graphs (operators) definitions 282.2.2. Maximal monotone operators 292.2.3. Convex function, subdifferentials and operators 332.2.4. Resolvent and regularization 382.2.5. Taking the limit 402.2.6. First result of existence and uniqueness for a differential inclusion 402.3. Extension to any Hilbert space 452.4. Existence and uniqueness results in Hilbert space 572.5. Numerical scheme in a Hilbert space 592.5.1. The numerical scheme 592.5.2. State of the art summary and results shown in this publication 602.5.3. Convergence (general results and order 1/2) 612.5.4. Convergence (order one) 672.5.5. Change of scalar product 722.5.6. Resolvent calculation 742.5.7. More regular schemes 76Chapter 3. Stochastic Theoretical Context 793.1. Introduction 793.2. Stochastic integral 793.2.1. The stochastic processes background 803.2.2. Stochastic integral 843.3. Stochastic differential equations 903.3.1. Existence and uniqueness of strong solution 913.3.2. Existence and uniqueness of weak solution 923.3.3. Kolmogorov and Fokker–Planck equations 953.4. Multivalued stochastic differential equations 1013.4.1. Problem statement 1013.4.2. Uniqueness and existence results 1033.5. Numerical scheme 1043.5.1. Which convergence: weak or strong? 1063.5.2. Strong convergence results 1083.5.3. Weak convergence results 122Chapter 4. Riemannian Theoretical Context 1294.1. Introduction 1294.2. First or second order 1294.3. Differential geometry 1314.3.1. Sphere case 1314.3.2. General case 1324.4. Dynamics of the mechanical systems 1394.4.1. Definition of mechanical system 1394.4.2. Equation of the dynamics 1414.5. Connection, covariant derivative, geodesics and parallel transport 1444.6. Maximal monotone term 1484.7. Stochastic term 1494.8. Results on the existence and uniqueness of a solution 151Chapter 5. Systems with Friction 1555.1. Introduction 1555.2. Examples of frictional systems with a finite number of degrees of freedom 1555.2.1. General framework 1555.2.2. Two elementary models 1565.2.3. Assembly and results in finite dimensions 1655.2.4. Conclusion 1935.2.5. Examples of numerical simulation 1945.2.6. Identification of the generalized Prandtl model (principles and simulation) 2055.3. Another example: the case of a pendulum with friction 2155.3.1. Formulation of the problem, existence and uniqueness 2155.3.2. Numerical scheme 2185.3.3. Numerical estimation of the order 2195.3.4. Example of numerical simulations 2215.3.5. Free oscillations 2215.3.6. Forced oscillations 2215.3.7. Transition matrix and calculation of the Lyapunov exponents 2225.3.8. Melnikov’s method, transitory chaos and Lyapunov exponents 2305.4. Elastoplastic oscillator under a stochastic forcing 2315.4.1. Introduction 2315.4.2. Modeling 2325.4.3. Numerical scheme 2365.4.4. Numerical results 2385.5. Spherical pendulum under a stochastic external force 2435.5.1. Establishment of the model 2435.5.2. Numerical aspects 2485.6. Gephyroidal model 2555.6.1. Introduction 2555.6.2. Description and transformation of the model 2565.6.3. Quasi-static problems 2635.6.4. Numerical simulations 2655.6.5. Conclusion 2675.7. Chain 2685.7.1. Introduction 2685.7.2. Description of the model 2705.7.3. Transformation of the equations 2715.7.4. Conclusion 2835.8. An infinity of internal variables: continuous generalized Prandtl model 2835.8.1. Introduction 2835.8.2. Description of the continuous model 2845.8.3. Existence, uniqueness and regularity results 2875.8.4. Application to the discrete case, and convergence of the discrete model to the continuous model  2895.8.5. Numerical scheme 2915.8.6. Study of hysteresis loops 2935.8.7. Numerical simulations 3015.9. Locally Lipschitz continuous spring 3015.9.1. Introduction 3015.9.2. The studied model 3015.9.3. Results for the existence and uniqueness of the solutions 3035.9.4. Convergence results for the numerical schemes 3115.9.5. The locally Lipschitz continuous case 3135.9.6. Identification of the parameters from the hysteresis loops 3145.9.7. Numerical simulations 320Chapter 6. Impact Systems 3256.1. Existence and uniqueness for simple problems (one degree of freedom) 3266.1.1. The work of Schatzman–Paoli 3266.1.2. Simple case with one degree of freedom, forcing and impact: piecewise analytical solutions 3276.1.3. Adaptation of some classical methods 3296.1.4. Movement with the accumulation of impacts and a sticking phase 3336.1.5. Behavior of the numerical methods 3376.1.6. Convergence and order of one-step numerical methods applied to non-smooth differential systems 3386.1.7. Results of numerical experiments 3436.2. A particular behavior: grazing bifurcation 3486.2.1. Approximation of the map in the general case 3496.2.2. Particular case 3506.2.3. Stability of the non-differentiable fixed point 3516.2.4. Numerical example 353Chapter 7. Applications–Extensions  3557.1. Oscillators with piecewise linear coupling and passive control 3557.1.1. Description of the model 3567.1.2. Free oscillations of the system 3567.1.3. Order 1 3627.1.4. Case of periodic forcing 3667.1.5. Conclusion 3777.2. Friction and passive control 3787.2.1. Introduction 3787.2.2. Introduction to the models: smooth and non-smooth systems 3797.3. The billiard ball 3867.3.1. Maximal monotone framework 3867.3.2. More realistic but non-maximal monotone framework 3897.4. An industrial application: the case of a belt tensioner 3907.4.1. The theory 3907.4.2. The tensioner used 3927.4.3. Identification of the parameters 3927.4.4. Validation 3937.5. Problems with delay and memory 3967.5.1. Theory 3967.5.2. Applications 3997.6. Other friction forces 4007.6.1. More general forms (variable dynamical coefficient) 4017.6.2. With a variable static coefficient 4197.6.3. With variable static and dynamical coefficients 4217.7. With the viscous dissipation term 4237.8. Ill-posed problems 4247.8.1. First model: limit of a well-posed friction law 4267.8.2. Second model: a differential inclusion without uniqueness 4277.8.3. Conclusion 429Appendix 1. Mathematical Reminders 431A1.1. Two Gronwall’s lemmas 431A1.2. Norms, scalar products, normed vector space, Banach and Hilbert space 432A1.2.1. Scalar products, norms 432A1.2.2. Banach and Hilbert space, separable space 433A1.3. Symmetric positive definite matrices 435A1.4. Differentiable function 435A1.5. Weak limit 436A1.6. Continuous function spaces 436A1.7. Lp space of integrable functions 437A1.7.1. Lp(Ω) space 437A1.7.2. Lp(Ω, Rq ) space 438A1.7.3. Lp(Ω; H) spaces 438A1.8. Distributions 439A1.8.1. Real values distributions 439A1.8.2. Distributions with values in Rq 440A1.8.3. Distributions with values in Hilbert space 440A1.9. Sobolev space definition 441A1.9.1. Functions with real values 441A1.9.2. Functions with values in Hilbert space 441Appendix 2. Convex Functions 443A2.1. Functions defined on R 443A2.2. Functions defined on Hilbert space 446A2.2.1. Any Hilbert space 446A2.2.2. Particular case of the finite dimension 446Appendix 3. Proof of Theorem 2.20 447Appendix 4. Proof of Theorem 3.18 455Appendix 5. Research of Convex Potential 467A5.1. Method used 467A5.2. Lemma 5.1 468A5.3. Lemma 5.4 473A5.4. Lemma 7.1 476Bibliography 477Index 495