M. Di-bernardo, K. H. Johansson, and F. Vasca, SELF-OSCILLATIONS AND SLIDING IN RELAY FEEDBACK SYSTEMS: SYMMETRY AND BIFURCATIONS, International Journal of Bifurcation and Chaos, vol.4, issue.12, pp.1121-1140, 2001.
DOI : 10.1109/9.250509

A. Jenkins, Self-oscillation, Physics Reports, vol.525, issue.2, pp.167-222, 2013.
DOI : 10.1016/j.physrep.2012.10.007

S. Chatterjee, Self-excited oscillation under nonlinear feedback with time-delay, Journal of Sound and Vibration, vol.330, issue.9, pp.1860-1876, 2011.
DOI : 10.1016/j.jsv.2010.11.005

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 1983.

J. Sun and A. C. Luo, Bifurcation and Chaos in Complex Nonlinear Dynamical Systems, Advances in Nonlinear Science and ComplexityPhysics), 2006.

S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering Studies in Nonlinearity, 2014.

A. Malas and S. Chatterjee, Generating self-excited oscillation in a class of mechanical systems by relay-feedback, Nonlinear Dynamics, vol.82, issue.9, pp.1253-1269, 2014.
DOI : 10.1080/00207170802657363

A. Malas and S. Chatterjee, Analysis and synthesis of modal and non-modal self-excited oscillations in a class of mechanical systems with nonlinear velocity feedback, Journal of Sound and Vibration, vol.334, pp.296-318, 2015.
DOI : 10.1016/j.jsv.2014.09.011

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, of Texts in Applied Mathematics, 2003.

M. W. Spong, Underactuated mechanical systems, Control Problems in Robotics and Automation, pp.135-150, 1998.
DOI : 10.1007/BFb0015081

URL : http://robot0.ge.uiuc.edu/~spong/Conference_Proceedings/cdc97_motion.ps.Z

Y. Liu and H. Yu, A survey of underactuated mechanical systems, IET Control Theory & Applications, vol.7, issue.7, pp.921-935, 2013.
DOI : 10.1049/iet-cta.2012.0505

A. Choukchou-braham, B. Cherki, M. Djemai, and K. Busawon, Analysis and Control of Underactuated Mechanical Systems
DOI : 10.1007/978-3-319-02636-7

I. Fantoni and R. Lozano, Nonlinear Control for Underactuated Mechanical Systems, 2002.

S. Rudra, R. K. Barai, and M. Maitra, Block Backstepping Design of Nonlinear State Feedback Control Law for Underactuated Mechanical Systems, 2017.
DOI : 10.1007/978-981-10-1956-2

J. Moreno-valenzuela, C. Aguilar-avelar, S. Puga-guzmán, and V. Santibánez, model regressor, Intelligent Automation & Soft Computing, vol.9, issue.1, pp.1-11, 2015.
DOI : 10.1016/S0005-1098(01)00092-9

Z. Li and Y. Zhang, Robust adaptive motion/force control for wheeled inverted pendulums, Automatica, vol.46, issue.8, pp.1346-1353, 2010.
DOI : 10.1016/j.automatica.2010.05.015

M. W. Spong, P. Corke, and R. Lozano, Nonlinear control of the Reaction Wheel Pendulum, Automatica, vol.37, issue.11, pp.1845-1851, 2001.
DOI : 10.1016/S0005-1098(01)00145-5

D. J. Block, K. J. Astrom, and M. W. Spong, The Reaction Wheel Pendulum, Synthesis Lectures on Controls and Mechatronics, vol.15, issue.1, 2007.
DOI : 10.2200/S00085ED1V01Y200702CRM001

R. Olfati-saber, Nonlinear control of underactuated mechanical systems with application to robotics and aerospace vehicles, 2001.

R. Olfati-saber, Global stabilization of a flat underactuated system: the inertia wheel pendulum, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228), 2001.
DOI : 10.1109/CDC.2001.980449

S. Ramamoorthy and B. Kuipers, Qualitative Heterogeneous Control of Higher Order Systems, Hybrid Systems: Computation and Control, pp.417-434, 2003.
DOI : 10.1007/3-540-36580-X_31

V. Santibanez, R. Kelly, and J. Sandoval, Control of the Inertia Wheel Pendulum by Bounded Torques, Proceedings of the 44th IEEE Conference on Decision and Control, pp.8266-8270, 2005.
DOI : 10.1109/CDC.2005.1583500

R. Ortega, M. Spong, F. Gomez-estern, and G. Blankenstein, Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment, IEEE Transactions on Automatic Control, vol.47, issue.8, pp.1218-1233, 2002.
DOI : 10.1109/TAC.2002.800770

R. Ortega and E. Garcia-canseco, Interconnection and Damping Assignment Passivity-Based Control: A Survey, European Journal of Control, vol.10, issue.5, pp.432-450, 2004.
DOI : 10.3166/ejc.10.432-450

R. Ortega, A. Van-der-schaft, B. Maschke, and G. Escobar, Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems, Automatica, vol.38, issue.4, pp.585-596, 2002.
DOI : 10.1016/S0005-1098(01)00278-3

F. Gómez-estern and A. J. , Physical Damping in IDA-PBC Controlled Underactuated Mechanical Systems, European Journal of Control, vol.10, issue.5, pp.451-468, 2004.
DOI : 10.3166/ejc.10.451-468

C. Aguilar-ibañez, O. O. Gutiérrez-frias, and M. S. Suarez-castanon, Controlling the strongly damping inertia wheel pendulum via nested saturation function, Computación y Sistemas, vol.12, issue.4, pp.436-449, 2009.

N. Qaiser, N. Iqbal, A. Hussain, and N. Qaiser, EXPONENTIAL STABILIZATION OF THE INERTIA WHEEL PENDULUM USING DYNAMIC SURFACE CONTROL, Journal of Circuits, Systems and Computers, vol.16, issue.01, pp.81-92, 2007.
DOI : 10.1109/9.983365

N. Touati and A. Chemori, Predictive control for the stabilization of a fast mechatronic system : from simulation to real-time experiments, Proceedings of the International IFAC Symposium on Mechatronic Systems, pp.237-242, 2013.
DOI : 10.3182/20130410-3-CN-2034.00031

URL : https://hal.archives-ouvertes.fr/lirmm-00809710

M. Olivares and P. , On the linear control of underactuated systems: The flywheel inverted pendulum, 2013 10th IEEE International Conference on Control and Automation (ICCA), pp.27-32, 2013.
DOI : 10.1109/ICCA.2013.6564905

M. Olivares and P. , Linear control of the flywheel inverted pendulum, ISA Transactions, vol.53, issue.5, pp.1396-1403, 2014.
DOI : 10.1016/j.isatra.2013.12.030

N. H. Khraief, A. Chemori, and S. Belghith, External disturbance rejection in IDA-PBC controller for underactuated mechanical systems: From theory to real time experiments, Proceedings of the IEEE Conference on Control Applications, pp.1747-1752, 2014.
URL : https://hal.archives-ouvertes.fr/lirmm-01722715

N. H. Khraief, A. Chemori, J. J. Pena, and S. Belghith, Stabilization of inertia wheel inverted pendulum by model reference adaptive IDA-PBC: From simulation to real-time experiments, Proceedings of the 3rd International Conference onControl, Engineering Information Technology (CEIT), pp.1-6, 2015.

M. Ryalat and D. S. Laila, A simplified IDA-PBC design for underactuated mechanical systems with applications, European Journal of Control, vol.27, pp.1-16, 2016.
DOI : 10.1016/j.ejcon.2015.12.001

URL : https://curve.coventry.ac.uk/open/items/836f6b14-b9d4-4d63-a629-9200f255b6f3/1/dinaejccomb.pdf

Z. Guo, J. Xu, and T. H. Lee, Design and implementation of a new sliding mode controller on an underactuated wheeled inverted pendulum, Journal of the Franklin Institute, vol.351, issue.4, pp.2261-2282, 2014.
DOI : 10.1016/j.jfranklin.2013.02.002

S. Mobayen, Design of LMI-based sliding mode controller with an exponential policy for a class of underactuated systems, Complexity, vol.47, issue.5, pp.1-8, 2014.
DOI : 10.1007/s11071-014-1724-3

M. Yue, X. Sun, N. Li, and C. An, Dynamic Motion Planning and Adaptive Tracking Control for a Class of Two-Wheeled Autonomous Vehicle With an Underactuated Pendular Suspension, Journal of Dynamic Systems, Measurement, and Control, vol.137, issue.10, pp.2015-2030
DOI : 10.1115/1.4030785

A. Zhang, C. Yang, S. Gong, and J. Qiu, Nonlinear stabilizing control of underactuated inertia wheel pendulum based on coordinate transformation and time-reverse strategy, Nonlinear Dynamics, vol.40, issue.2, pp.2467-2476, 2016.
DOI : 10.1109/CDC.2001.980449

L. T. Aguilar, I. Boiko, L. Fridman, and R. Iriarte, Generating Self-Excited Oscillations via Two-Relay Controller, IEEE Transactions on Automatic Control, vol.54, issue.2, pp.416-420, 2009.
DOI : 10.1109/TAC.2008.2009615

L. T. Aguilar, I. Boiko, L. Fridman, and R. Iriarte, Generating self-excited oscillations for underactuated mechanical systems via two-relay controller, International Journal of Control, vol.5, issue.9, pp.1678-1691, 2009.
DOI : 10.1109/9.898696

L. T. Aguilar, I. Boiko, L. Fridman, and L. Freidovich, Generating oscillations in inertia wheel pendulum via two-relay controller, International Journal of Robust and Nonlinear Control, vol.41, issue.4, pp.318-330, 2012.
DOI : 10.1016/j.automatica.2004.11.002

R. Iriarte, L. T. Aguilar, and L. Fridman, Second order sliding mode tracking controller for inertia wheel pendulum, Journal of the Franklin Institute, vol.350, issue.1, pp.92-106, 2013.
DOI : 10.1016/j.jfranklin.2012.10.013

L. T. Aguilar, I. Boiko, L. Fridman, and R. Iriarte, Self-Oscillations in Dynamic Systems: A New Methodology via Two-Relay Controllers, 2015.
DOI : 10.1007/978-3-319-23303-1

A. Estrada, L. T. Aguilar, R. Iriarte, and L. Fridman, Two relay controller for real time trajectory generation and its application to inverted orbital stabilization of inertia wheel pendulum via quasi-continuous HOSM, Asian Journal of Control, vol.49, issue.4, pp.58-66, 2012.
DOI : 10.1109/19.863938

L. B. Freidovich, P. X. Hera, U. Mettin, A. Robertsson, A. S. Shiriaev et al., Shaping stable periodic motions of inertia wheel pendulum: theory and experiment, Asian Journal of Control, vol.53, issue.10
DOI : 10.1002/asjc.135

A. S. Shiriaev, J. W. Perram, and C. C. De-wit, Constructive tool for orbital stabilization of underactuated nonlinear systems: virtual constraints approach, IEEE Transactions on Automatic Control, vol.50, issue.8, pp.1164-1176, 2005.
DOI : 10.1109/TAC.2005.852568

URL : https://hal.archives-ouvertes.fr/hal-00395031

A. Shiriaev, J. Perram, A. Robertsson, and A. Sandberg, Periodic motion planning for virtually constrained Euler???Lagrange systems, Systems & Control Letters, vol.55, issue.11, pp.900-907, 2006.
DOI : 10.1016/j.sysconle.2006.06.007

A. S. Shiriaev, L. B. Freidovich, and I. R. Manchester, Can we make a robot ballerina perform a pirouette? Orbital stabilization of periodic motions of underactuated mechanical systems, Annual Reviews in Control, vol.32, issue.2, pp.200-211, 2008.
DOI : 10.1016/j.arcontrol.2008.07.001

C. Gruber and M. Hofbaur, Periodic Motion Control of the Reaction Wheel Pendulum, in: 13th Mechatronic Forum, pp.1-8, 2012.

S. Andary, A. Chemori, and S. Krut, Stable limit cycle generation for underactuated mechanical systems, application: Inertia wheel inverted pendulum, 2008 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp.526-531, 2008.
DOI : 10.1109/IROS.2008.4650994

URL : https://hal.archives-ouvertes.fr/lirmm-00305317

S. Andary, A. Chemori, and S. Krut, Control of the Underactuated Inertia Wheel Inverted Pendulum for Stable Limit Cycle Generation, Advanced Robotics, vol.23, issue.15, pp.1999-2014, 2009.
DOI : 10.1163/016918609X12529279062438

URL : https://hal.archives-ouvertes.fr/lirmm-00455577

S. Andary, A. Chemori, and S. Krut, Estimation-based disturbance rejection in control for limit cycle generation on inertia wheel inverted pendulum testbed, 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp.1302-1307, 2009.
DOI : 10.1109/IROS.2009.5354120

URL : https://hal.archives-ouvertes.fr/lirmm-00429808

C. Zayane-aissa, T. Laleg-kirati, and A. Chemori, Control of a perturbed under-actuated mechanical system, 2015 IEEE Conference on Control Applications (CCA), pp.294-299, 2015.
DOI : 10.1109/CCA.2015.7320644

URL : https://hal.archives-ouvertes.fr/lirmm-01343312

C. A. Anez, J. C. Martinez, J. De, J. Rubio, and M. S. Suarez-castanon, Inducing sustained oscillations in feedback-linearizable single-input nonlinear systems, ISA Transactions, pp.54-117, 2015.

D. M. Alonso, E. E. Paolini, and J. L. Moiola, Global Bifurcation Analysis of a Controlled Underactuated Mechanical System, Nonlinear Dynamics, vol.197, issue.3?4, pp.205-225, 2005.
DOI : 10.1007/978-1-4757-2421-9

D. M. Alonso, E. E. Paolini, and J. L. Moiola, Controlling an Inverted Pendulum with Bounded Controls, Dynamics, Bifurcations, and Control, pp.3-16, 2002.
DOI : 10.1007/3-540-45606-6_1

D. Pagano, L. Pizarro, and J. , LOCAL BIFURCATION ANALYSIS IN THE FURUTA PENDULUM VIA NORMAL FORMS, International Journal of Bifurcation and Chaos, vol.10, issue.05, pp.981-995, 2000.
DOI : 10.1243/PIME_PROC_1992_206_043_02

S. Nikolov and V. Nedev, Abstract, Journal of Theoretical and Applied Mechanics, vol.46, issue.1, pp.17-32, 2016.
DOI : 10.1515/jtam-2016-0002

S. Boyd, L. El-ghaoui, E. Feron, and V. Balakrishnan, Linear matrix inequalities in system and control theory, Studies in Applied and Numerical Mathematics, vol.15, 1994.
DOI : 10.1137/1.9781611970777

I. R. Petersen and R. Tempo, Robust control of uncertain systems: Classical results and recent developments, Automatica, vol.50, issue.5, pp.1315-1335, 2014.
DOI : 10.1016/j.automatica.2014.02.042

R. C. Oliveira, M. C. De-oliveira, and P. L. Peres, Robust state feedback LMI methods for continuous-time linear systems: Discussions, extensions and numerical comparisons, 2011 IEEE International Symposium on Computer-Aided Control System Design (CACSD), pp.2011-1038, 2011.
DOI : 10.1109/CACSD.2011.6044553

L. A. Rodrigues, R. C. Oliveira, and J. F. Camino, New extended LMI characterization for state feedback control of continuous-time uncertain linear systems, 2015 European Control Conference (ECC), pp.1992-1997, 2015.
DOI : 10.1109/ECC.2015.7330831

F. Pfeiffer and C. Glocker, Multibody Dynamics with Unilateral Contacts, Wiley Series in Nonlinear Science, 2004.

A. Huang and Y. Chen, Adaptive Sliding Control for Single-Link Flexible-Joint Robot With Mismatched Uncertainties, IEEE Transactions on Control Systems Technology, vol.12, issue.5, pp.770-775, 2004.
DOI : 10.1109/TCST.2004.826968

M. Hosseinpour, P. Nikdel, M. Badamchizadeh, and M. Akbari, Modelling and control of flexible joint robot based on Takagi???Sugeno fuzzy approach and its stability analysis via sum of squares, Mathematical and Computer Modelling of Dynamical Systems, vol.3, issue.3, pp.250-262, 2013.
DOI : 10.1016/j.jpaa.2003.12.011

K. Merat, H. Salarieh, A. Alasty, and A. Meghdari, Stochastic piecewise affine control with application to pitch control of helicopter, Nonlinear Analysis: Hybrid Systems, vol.15, pp.86-97, 2015.
DOI : 10.1016/j.nahs.2014.08.001

H. Razavi, K. Merat, H. Salarieh, A. Alasty, and A. Meghdari, Observer based minimum variance control of uncertain piecewise affine systems subject to additive noise, Nonlinear Analysis: Hybrid Systems, vol.19, pp.153-167, 2016.
DOI : 10.1016/j.nahs.2015.09.002

R. C. Luo, J. Sheng, C. C. Chen, P. H. Chang, and C. I. Lin, Biped robot push and recovery using flywheel model based walking perturbation counteraction, 2013 13th IEEE-RAS International Conference on Humanoid Robots (Humanoids), pp.50-55, 2013.
DOI : 10.1109/HUMANOIDS.2013.7029954

R. C. Luo and C. W. Huang, A push-recovery method for walking biped robot based on 3-D flywheel model, IECON 2015, 41st Annual Conference of the IEEE Industrial Electronics Society, pp.2685-2690, 2015.
DOI : 10.1109/IECON.2015.7392507

M. Shafiee-ashtiani, A. Yousefi-koma, M. S. Panahi, and M. Khadiv, Push recovery of a humanoid robot based on model predictive control and capture point, 2016 4th International Conference on Robotics and Mechatronics (ICROM)
DOI : 10.1109/ICRoM.2016.7886777

J. Pratt, J. Carff, S. Drakunov, and A. Goswami, Capture Point: A Step toward Humanoid Push Recovery, 2006 6th IEEE-RAS International Conference on Humanoid Robots, pp.200-207, 2006.
DOI : 10.1109/ICHR.2006.321385

URL : http://www.cs.cmu.edu/~cga/legs/Pratt_Goswami_Humanoids2006.pdf

E. R. Westervelt, J. W. Grizzle, C. Chevallereau, J. Choi, and B. Morris, Feedback control of dynamic bipedal robot locomotion, 2007.
DOI : 10.1201/9781420053739

URL : https://hal.archives-ouvertes.fr/hal-01702841

C. E. De-brito-novaes, P. S. Da-silva, and P. Rouchon, Trajectory control of a bipedal walking robot with inertial disc, IFAC World Congress, pp.4843-4848, 2014.
DOI : 10.3182/20140824-6-ZA-1003.01688

C. E. De-brito-novaes and P. S. Da-silva, Numerical Estimation of Stability Region of Self-Clocked Bipedal Robots with Inertial Disk, Journal of Control, Automation and Electrical Systems, vol.23, issue.6, pp.634-645, 2016.
DOI : 10.1201/9781420053739

B. M. Fard, A. Bagheri, and N. Nariman-zadeh, Limit cycle walker push recovery based on a receding horizon control scheme, Proceedings of the Institution of Mechanical Engineers, pp.914-926, 2012.
DOI : 10.1109/9.898695

Y. Zhu, Y. Gao, C. Xu, J. Zhao, H. Jin et al., Adaptive Control of a Gyroscopically Stabilized Pendulum and Its Application to a Single-Wheel Pendulum Robot, IEEE/ASME Transactions on Mechatronics, vol.20, issue.5, pp.2095-2106, 2015.
DOI : 10.1109/TMECH.2014.2363090

M. W. Spong and M. Vidyasagar, Robot Dynamics and Control, 1989.

T. S. Parker and L. O. Chua, Practical numerical algorithms for chaotic systems, 1989.
DOI : 10.1007/978-1-4612-3486-9

C. Scherer and S. Weiland, The Netherlands, Linear Matrix Inequalities in Control, Dutch Institute of Systems and Control (DISC), 2005.

M. M. Asheghan and M. T. Beheshti, An LMI approach to robust synchronization of a class of chaotic systems with gain variations, Chaos, Solitons & Fractals, vol.42, issue.2, pp.1106-1111, 2009.
DOI : 10.1016/j.chaos.2009.03.152

S. Mobayen, An LMI-based robust controller design using global nonlinear sliding surfaces and application to chaotic systems, Nonlinear Dynamics, vol.12, issue.2, pp.1075-1084, 2015.
DOI : 10.1142/S021812740200631X

T. Kailath, Linear Systems, 1989.

N. J. Higham, Accuracy and Stability of Numerical Algorithms, Society for Industrial and Applied Mathematics, 2002.
DOI : 10.1137/1.9780898718027

J. G. Vanantwerp and R. D. Braatz, A tutorial on linear and bilinear matrix inequalities, Journal of Process Control, vol.10, issue.4, pp.363-385, 2000.
DOI : 10.1016/S0959-1524(99)00056-6

H. Gritli and S. Belghith, Walking dynamics of the passive compass-gait model under OGY-based control: Emergence of bifurcations and chaos, Communications in Nonlinear Science and Numerical Simulation, vol.47, pp.308-327, 2017.
DOI : 10.1016/j.cnsns.2016.11.022

H. Gritli, S. Belghith, and N. Khraeif, OGY-based control of chaos in semi-passive dynamic walking of a torso-driven biped robot, Nonlinear Dynamics, vol.47, issue.2, pp.1363-1384, 2015.
DOI : 10.1109/81.828574

H. Gritli and S. Belghith, Bifurcations and chaos in the semi-passive bipedal dynamic walking model under a modified OGY-based control approach, Nonlinear Dynamics, vol.10, issue.4, pp.1955-1973, 2016.
DOI : 10.1007/s11071-013-0918-4

Z. Wang, L. Zhou, Z. Chen, and J. Wang, Local bifurcation analysis and topological horseshoe of a 4D hyper-chaotic system, Nonlinear Dynamics, vol.36, issue.11, pp.2055-2066, 2016.
DOI : 10.1016/j.apm.2011.12.049

L. Zhou, Z. Chen, Z. Wang, and J. Wang, On the analysis of local bifurcation and topological horseshoe of a new 4D hyper-chaotic system, Chaos, Solitons & Fractals, vol.91, pp.148-156, 2016.
DOI : 10.1016/j.chaos.2016.05.017

W. Wu and Z. Chen, Hopf bifurcation and intermittent transition to hyperchaos in??a??novel strong four-dimensional hyperchaotic system, Nonlinear Dynamics, vol.74, issue.4, pp.615-630, 2010.
DOI : 10.1007/978-1-4757-4067-7

X. Li and P. Wang, Hopf bifurcation and heteroclinic orbit in a 3D autonomous chaotic system, Nonlinear Dynamics, vol.65, issue.1-2, pp.621-632, 2013.
DOI : 10.1007/s11071-010-9887-z

A. Algaba, M. C. Domínguez-moreno, M. Merino, and A. J. , Study of the Hopf bifurcation in the Lorenz, Chen and L?? systems, Nonlinear Dynamics, vol.375, issue.2, pp.885-902, 2015.
DOI : 10.1016/j.physleta.2011.08.022

K. Deng and S. Yu, Hopf bifurcation analysis of a new modified hyperchaotic L?? system, Optik - International Journal for Light and Electron Optics, vol.124, issue.23, pp.6265-6269, 2013.
DOI : 10.1016/j.ijleo.2013.05.011

W. Govaerts, Numerical bifurcation analysis for ODEs, Journal of Computational and Applied Mathematics, vol.125, issue.1-2, pp.57-68, 2000.
DOI : 10.1016/S0377-0427(00)00458-1

URL : https://doi.org/10.1016/s0377-0427(00)00458-1

A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, MATCONT, ACM Transactions on Mathematical Software, vol.29, issue.2, pp.141-164, 2003.
DOI : 10.1145/779359.779362