Abstract
In about the last two decades, global nonlinear dynamics has been evolving in a revolutionary way, with the development of sophisticated techniques employing concepts/tools of dynamical systems, bifurcation, and chaos theory, and applications to a wide variety of mechanical/structural systems. The relevant achievements entail a substantial change of perspective in dealing with vibration problems and are ready to meaningfully affect the analysis, control, and design of systems at different scales, in multiphysics contexts too. After properly framing the subject within some main stages of developments of nonlinear dynamics in solid/structural mechanics, as occurred over the last 40 years, the article focuses on highlighting the role played by global analysis in unveiling the nonlinear response and actual safety of engineering systems in different environments. Reduced order models of macro-/micro-structures are considered. Global dynamics of a laminated plate with von Kármán nonlinearities, shear deformability, and full thermomechanical coupling allows to highlight the meaningful effects entailed by the slow transient thermal dynamics on the fast steady mechanical responses. An atomic force microcantilever is referred to for highlighting the severe worsening of overall stability associated with the application of an external feedback control and the importance of global dynamics for conceiving and effectively implementing a control procedure aimed at enhancing engineering safety. The last part of the article dwells on the general role that a global dynamics perspective is expected to play in the safe design of real systems, in the near future, focusing on how properly exploiting concepts and tools of dynamical integrity to evaluate response robustness in the presence of unavoidable imperfections, and to improve the system’s actual load carrying capacity.
Similar content being viewed by others
Abbreviations
- AFM:
-
Atomic force microscope
- BC:
-
Boundary crisis
- BS:
-
Basin stability
- DI:
-
Dynamical integrity
- GIM:
-
Global integrity measure
- IF:
-
Integrity factor
- LIM:
-
Local integrity measure
- MEMS:
-
Micro-electro-mechanical system
- nD:
-
n-dimensional
- ODE:
-
Ordinary differential equation
- OGY:
-
Ott–Grebogi–Yorke
- PD:
-
Period doubling
- Pm:
-
m-period solution
- ROM:
-
Reduced order model
- sdof:
-
single-degree-of-freedom
- SN:
-
Saddle-node
References
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer, New York (1983)
Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990)
Thompson, J.M.T., Stewart, H.B.: Nonlinear Dynamics and Chaos. Wiley, Chichester (1986)
Strogatz, S.H.: Nonlinear Dynamics and Chaos. Addison-Wesley, New York (1994)
Hsu, C.S.: Cell to Cell Mapping: A Method of Global Analysis for Nonlinear Systems. Springer, New York (1987)
Ott, E., Sauer, T., Yorke, J.A.: Coping with Chaos. Wiley, New York (1994)
Rega, G.: Nonlinear dynamics in mechanics and engineering: 40 years of developments and Ali H. Nayfeh’s legacy. Nonlinear Dyn. 99(1), 11–34 (2020)
Settimi, V., Saetta, E., Rega, G.: Nonlinear dynamics of a third-order reduced model of thermomechanically coupled plate under different thermal excitations. Meccanica 55, 2451–2473 (2020)
Saetta, E., Rega, G.: Third-order thermomechanically coupled laminated plates: 2D nonlinear modelling, minimal reduction and transient/post-buckled dynamics under different thermal excitations. Compos. Struct. 174, 420–441 (2017)
Doedel, E., Oldeman, B.: AUTO-07p: Continuation and bifurcation Software for Ordinary Differential Equations. Concordia University Press, Montreal (2012)
Katz, A., Dowell, E.H.: From single well chaos to cross well chaos: a detailed explanation in terms of manifold intersections. Int. J. Bif. Chaos 4, 933–941 (1994)
Rega, G., Lenci, S., Thompson, J.M.T.: Controlling chaos: The OGY method, its use in mechanics, and an alternative unified framework for control of non-regular dynamics. In: Thiel, M., Kurths, J., Romano, C., Moura, A., Károlyi, G. (eds.) Nonlinear Dynamics and Chaos: Advances and Perspectives, pp. 211–269. Springer, Berlin (2010)
Lenci, S., Rega, G.: Optimal control of homoclinic bifurcation: Theoretical treatment and practical reduction of safe basin erosion in the Helmholtz oscillator. J. Vibration Control 9, 281–315 (2003)
Lenci, S., Rega, G.: Forced harmonic vibration in a system with negative linear stiffness and linear viscous damping. In: Kovacic, I., Brennan, M. (eds.) The Duffing Equation. Non-linear Oscillators and Their Behaviour, pp. 219–276. Wiley, New York (2011)
Lenci, S., Rega, G.: A unified control framework of the nonregular dynamics of mechanical oscillators. J. Sound Vibr. 278, 1051–1080 (2004)
Lenci, S., Rega, G.: Heteroclinic bifurcations and optimal control in the nonlinear rocking dynamics of generic and slender rigid blocks. Int. J. Bif. Chaos 15(6), 1901–1918 (2005)
Orlando, D., Gonçalves, P.B., Rega, G., Lenci, S.: Influence of modal coupling on the nonlinear dynamics of Augusti’s model. J. Comput. Nonlinear Dyn. 6(4), 041014 (2011)
Gendelman, O.V.: Escape of a harmonically forced particle from an infinite-range potential well: a transient resonance. Nonlinear Dyn. 93, 79–88 (2018)
Gendelman, O.V., Karmi, G.: Basic mechanisms of escape of a harmonically forced classical particle from a potential well. Nonlinear Dyn. 98(4), 2775–2792 (2019)
Zhong, J., Virgin, L.N., Ross, S.D.: A tube dynamics perspective governing stability transitions: An example based on snap-through buckling. Int. J. Mech. Sci. 149, 413–428 (2018)
Zhong, J., Ross, S.D.: Geometry of escape and transition dynamics in the presence of dissipative and gyroscopic forces in two degree of freedom systems. Commun. Nonlinear Sci. Numer. Simul. 82, 105033 (2020)
Thompson, J.M.T.: Chaotic phenomena triggering the escape from a potential well. Proc. R. Soc. Lond. A 421, 195–225 (1989)
Lenci, S., Rega, G.: Load carrying capacity of systems within a global safety perspective. Part I. Robustness of stable equilibria under imperfections. Int. J. Nonlinear Mech. 46, 1232–1239 (2011)
Rega, G., Lenci, S., Ruzziconi, L.: Dynamical integrity: A novel paradigm for evaluating load carrying capacity. In: Lenci, S., Rega, G. (eds.) Global Nonlinear Dynamics for Engineering Design and System Safety, CISM Courses and Lectures 588, pp. 27–112. Springer, Berlin (2018)
Thompson, J.M.T.: Dynamical integrity: Three decades of progress from macro to nano mechanics. In: Lenci, S., Rega, G. (eds.) Global Nonlinear Dynamics for Engineering Design and System Safety, CISM Courses and Lectures 588, pp. 1–26. Springer, Berlin (2018)
Soliman, M.S., Thompson, J.M.T.: Integrity measures quantifying the erosion of smooth and fractal basins of attraction. J. Sound Vib. 135, 453–475 (1989)
Lenci, S., Rega, G.: Optimal control of nonregular dynamics in a Duffing oscillator. Nonlinear Dyn. 33, 71–86 (2003)
Ruzziconi, L., Younis, M.I., Lenci, S.: Multistability in an electrically actuated carbon nanotube: A dynamical integrity perspective. Nonlinear Dyn. 74(3), 533–549 (2013)
Belardinelli, P., Lenci, S., Rega, G.: Seamless variation of isometric and anisometric dynamical integrity measures in basins’ erosion. Commun. Nonlinear Sci. Numer. Simul. 56, 499–507 (2018)
Thompson, J.M.T., Ueda, Y.: Basin boundary metamorphoses in the canonical escape equation. Dyn. Stabil. Syst. 4(3–4), 285–294 (1989)
Younis, M.I.: MEMS Linear and Nonlinear Statics and Dynamics. Springer, New York (2011)
Nandakumar, K., Wiercigroch, M., Chatterjee, A.: Optimum energy extraction from rotational motion in a parametrically excited pendulum. Mech. Res. Commun. 43, 7–14 (2012)
Lenci, S., Rega, G.: Experimental versus theoretical robustness of rotating solutions in a parametrically excited pendulum: A dynamical integrity perspective. Phys. D 240, 814–824 (2011)
Thompson, J.M.T., Rainey, R.C.T., Soliman, M.S.: Ship stability criteria based on chaotic transients from incursive fractals. Philos. Trans. R. Soc. Lond. A 332(1624), 149–167 (1990)
Thompson, J.M.T.: Designing against capsize in beam seas: Recent advances and new insights. Appl. Mech. Rev. 50, 307–325 (1997)
Rega, G., Lenci, S.: Dynamical integrity and control of nonlinear mechanical oscillators. J. Vibr. Control 14, 159–179 (2008)
Lenci, S., Rega, G.: Competing dynamic solutions in a parametrically excited pendulum: Attractor robustness and basin integrity. J. Comput. Nonlinear Dyn. 3, 041010-1–041010-9 (2008)
Gonçalves, P.B., Santee, D.M.: Influence of uncertainties on the dynamic buckling loads of structures liable to asymmetric post-buckling behavior. Mathematical Problems in Engineering, Article ID 490137 (2008)
Lenci, S., Rega, G.: Load carrying capacity of systems within a global safety perspective. Part II. Attractor/basin integrity under dynamic excitations. Int. J. Nonlinear Mech. 46, 1240–1251 (2011)
Silva, F.M.A., Gonçalves, P.B.: The influence of uncertainties and random noise on the dynamic integrity analysis of a system liable to unstable buckling. Nonlinear Dyn. 81, 707–724 (2015)
Orlando, D., Gonçalves, P.B., Rega, G., Lenci, S.: Influence of transient escape and added load noise on the dynamic integrity of multistable systems. Int. J. Non-Linear Mechanics 109, 140–154 (2019)
Lenci, S., Orlando, D., Rega, G., Gonçalves, P.B.: Controlling practical stability and safety of mechanical systems by exploiting chaos properties. Chaos 22(4), 047502-1–047502-15 (2012)
Lenci, S., Orlando, D., Rega, G., Gonçalves, P.B.: Controlling nonlinear dynamics of systems liable to unstable interactive buckling. Proc. IUTAM 5, 108–123 (2012)
Eason, R.P., Dick, A.J., Nagarajaiah, S.: Numerical investigation of coexisting high and low amplitude responses and safe basin erosion for a coupled linear oscillator and nonlinear absorber system. J. Sound Vib. 333, 3490–3504 (2014)
Piccirillo, V., do Prado, T.G., Tusset, A.M., Balthazar, J.M.: Dynamic integrity analysis on a non-ideal oscillator. Math. Eng. Sci. Aerosp 11(3), 1–7 (2020)
Benedetti, K.C.B., Gonçalves, P.B., Silva, F.M.A.: Nonlinear oscillations and bifurcations of a multistable truss and dynamic integrity assessment via a Monte Carlo approach. Meccanica 55, 2623–2657 (2020)
De Freitas, M.S.T., Viana, R.L., Grebogi, C.: Erosion of the safe basin for the transversal oscillations of a suspension bridge. Chaos, Solitons Fractals 18(4), 829–841 (2003)
Soliman, M.S., Gonçalves, P.B.: Chaotic behaviour resulting in transient and steady state instabilities of pressure-loaded shallow spherical shells. J. Sound Vib. 259, 497–512 (2003)
Gonçalves, P.B., Silva, F.M.A., Rega, G., Lenci, S.: Global dynamics and integrity of a two-dof model of a parametrically excited cylindrical shell. Nonlinear Dyn. 63, 61–82 (2011)
Silva, F.M.A., Gonçalves, P.B., Del Prado, Z.J.G.N.: An alternative procedure for the non-linear vibration analysis of fluid-filled cylindrical shells. Nonlinear Dyn. 66(3), 303–333 (2011)
Silva, F.M.A., Gonçalves, P.B., Del Prado, Z.J.G.N.: Influence of physical and geometrical system parameters uncertainties on the nonlinear oscillations of cylindrical shells. J. Braz. Soc. Mech. Sci. Eng. 34, 622–632 (2012)
Rodrigues, L., Silva, F.M.A., Gonçalves, P.B., Del Prado, Z.J.G.N.: Effects of modal coupling on the dynamics of parametrically and directly excited cylindrical shells. Thin Walled Struct. 81, 210–224 (2014)
Silva, F.M.A., Brazão, A.F., Gonçalves, P.B., Del Prado, Z.J.G.N.: Influence of physical and geometrical uncertainties in the parametric instability load of an axially excited cylindrical shell. Math. Probl. Eng., Article ID 758959 (2015)
Silva, F.M.A., Soares, R.M., Del Prado, Z.J.G.N., Gonçalves, P.B.: Intra-well and cross-well chaos in membranes and shells liable to buckling. Nonlinear Dyn. 102, 877–906 (2020)
Coaquira, J.C., Cardoso, D.C.T., Gonçalves, P.B., Orlando, D.: Parametric instability and nonlinear oscillations of an FRP channel section column under axial load. Nonlinear Dyn. (2020). https://doi.org/10.1007/s11071-020-05663-x
Lenci, S., Rega, G.: Control of pull-in dynamics in a nonlinear thermoelastic electrically actuated microbeam. J. Micromech. Microeng. 16(2), 390–400 (2006)
Alsaleem, F.M., Younis, M.I., Ruzziconi, L.: An experimental and theoretical investigation of dynamical pull-in in MEMS resonators actuated electrostatically. J. Microelectromech. Syst. 19(4), 794–806 (2010)
Alsaleem, F., Younis, M.I.: Integrity analysis of electrically actuated resonators with delayed feedback controller. J. Dyn. Syst. Meas. Control 133(3), 031011 (2011)
Ruzziconi, L., Younis, M.I., Lenci, S.: An electrically actuated imperfect microbeam: Dynamical integrity for interpreting and predicting the device response. Meccanica 48(7), 1761–1775 (2013)
Ruzziconi, L., Lenci, S., Younis, M.I.: An imperfect microbeam under an axial load and electric excitation: Nonlinear phenomena and dynamical integrity. Int. J. Bifurc. Chaos 23(2), 1350026-1–1350026–17 (2013)
Ruzziconi, L., Ramini, A., Younis, M., Lenci, S.: Theoretical prediction of experimental jump and pull-in dynamics in a MEMS sensor. Sensors 14, 17089–17111 (2014)
Belardinelli, P., Sajadi, B., Lenci, S., Alijani, F.: Global dynamics and integrity of a micro-plate pressure sensor. Commun. Nonlinear Sci. Numer. Simul. 69, 432–444 (2019)
Rega, G., Settimi, V.: Bifurcation, response scenarios and dynamic integrity in a singlemode model of noncontact atomic force microscopy. Nonlinear Dyn. 73(1–2), 101–123 (2013)
Settimi, V., Rega, G.: Global dynamics and integrity in noncontacting atomic force microscopy with feedback control. Nonlinear Dyn. 86(4), 2261–2277 (2016)
Settimi, V., Rega, G.: Exploiting global dynamics of a noncontact atomic force microcantilever to enhance its dynamical robustness via numerical control. Int. J. Bifurc. Chaos 26, 1630018-1–1630018-17 (2016)
Chandrashekar, A., Belardinelli, P., Staufer, U., Alijani, F.: Robustness of attractors in tapping mode atomic force microscopy. Nonlinear Dyn. 97, 1137–1158 (2019)
Lenci, S., Rega, G.: A procedure for reducing the chaotic response region in an impact mechanical system. Nonlinear Dyn. 15, 391–409 (1998)
Rega, G., Lenci, S.: Bifurcations and chaos in single-d.o.f. mechanical systems: Exploiting nonlinear dynamics properties for their control. In: Luongo, A. (ed.) Recent Research Developments in Structural Dynamics, pp. 331–369. Research Signpost, Kerala (2003)
Rega, G., Lenci, S.: Identifying, evaluating, and controlling dynamical integrity measures in nonlinear mechanical oscillators. Nonlinear Anal. Real World Appl. 63, 902–914 (2005)
Lenci, S., Rega, G.: Controlling nonlinear dynamics in a two-well impact system. I. Attractors and bifurcation scenario under symmetric excitations. Int. J. Bifurc. Chaos 8, 2387–2408 (1998)
Lenci, S., Rega, G.: Controlling nonlinear dynamics in a two-well impact system. II. Attractors and bifurcation scenario under unsymmetric optimal excitations. Int. J. Bifurc. Chaos 8, 2409–2424 (1998)
Lenci, S., Rega, G.: Optimal numerical control of single-well to cross-well chaos transition in mechanical systems. Chaos Solitons Fractals 15, 173–186 (2003)
Lenci, S., Rega, G.: Global optimal control and system-dependent solutions in the hardening Helmholtz–Duffing oscillator. Chaos Solitons Fractals 21, 1031–1046 (2004)
Lenci, G., Rega, S.: Optimal control and anti-control of the nonlinear dynamics of a rigid block. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 364, 2353–2381 (2006)
Lenci, S., Rega, G.: Control of the homoclinic bifurcation in buckled beams: infinite-dimensional vs reduced-order modeling. Int. J. Non-Linear Mech. 43, 474–489 (2008)
Orlando, D., Gonçalves, P.B., Lenci, S., Rega, G.: Increasing practical safety of Von Mises truss via control of dynamic escape. Appl. Mech. Mater. 849, 46–56 (2016)
Rega, G., Lenci, S.: A global dynamics perspective for system safety from macro- to nanomechanics: Analysis, control, and design engineering. Appl. Mech. Rev. 67, 050802-1–050802-19 (2015)
Gonçalves, P.B., Orlando, D., Lenci, S., Rega, G.: Nonlinear dynamics, safety and control of structures liable to interactive unstable buckling. In: Lenci, S., Rega, G. (eds.) Global Nonlinear Dynamics for Engineering Design and System Safety. CISM Courses and Lectures 588, pp. 167–228. Springer, Berlin (2018)
Settimi, V., Rega, G.: Local versus global dynamics and control of an AFM model in a safety perspective. In: Lenci, S., Rega, G. (eds.) Global Nonlinear Dynamics for Engineering Design and System Safety. CISM Courses and Lectures 588, pp. 229–286. Springer, Berlin (2018)
Dudkowski, P., Jafari, S., Kapitaniak, T., Kuznetsov, N.V., Leonov, G.A., Prasad, A.: Hidden attractors in dynamical systems. Phys. Rep. 637, 1–50 (2016)
Menck, P.J., Heitzig, J., Marwan, N., Kurths, J.: How basin stability complements the linear-stability paradigm. Nat. Phys. 9(2), 89–92 (2013)
Hellmann, F., Schultz, P., Grabow, C., Heitzig, J., Kurths, J.: Survivability of deterministic dynamical systems. Sci. Rep. 6, 29654 (2016)
Daza, A., Wagemakers, A., Georgeot, B., Guery-Odelin, D., Sanjuan, M.A.F.: Basin entropy: a new tool to analyze uncertainty in dynamical systems. Sci. Rep. 6, 31416 (2016)
Brzeski, P., Lazarek, M., Kapitaniak, T., Kurths, J., Perlikowski, P.: Basin stability approach for quantifying responses of multistable systems with parameters mismatch. Meccanica 51(11), 2713–2726 (2016)
Brzeski, P., Wojewoda, J., Kapitaniak, T., Kurths, J., Perlikowski, P.: Sample-based approach can outperform the classical dynamical analysis—experimental confirmation of the basin stability method. Sci. Rep. 7, 6121 (2017)
Brzeski, P., Belardinelli, P., Lenci, S., Perlikowski, P.: Revealing compactness of basins of attraction of multi-DoF dynamical systems. Mech. Syst. Signal Process. 111, 348–361 (2018)
Brzeski, P., Perlikowski, P.: Sample-based methods of analysis for multistable dynamical systems. Arch. Comput. Methods Eng. 26, 1515–1545 (2019)
Brzeski, P., Kurths, J., Perlikowski, P.: Time dependent stability margin in multistable systems. Chaos 28, 093104 (2018)
Settimi, V., Gottlieb, O., Rega, G.: Asymptotic analysis of a noncontact AFM microcantilever sensor with external feedback control. Nonlinear Dyn. 79(4), 2675–2698 (2015)
Settimi, V., Rega, G.: Influence of a locally-tailored external feedback control on the overall dynamics of a noncontact AFM model. Int. J. Non-Linear Mechanics 80, 144–159 (2016)
Settimi, V., Rega, G., Lenci, S.: Analytical control of homoclinic bifurcation of the hilltop saddle in a noncontact atomic force microcantilever. Proc. IUTAM 19, 19–26 (2016)
Lenci, S., Brocchini, M., Lorenzoni, C.: Experimental rotations of a pendulum on water waves. J. Comput. Nonlinear Dyn. 7, 011007 (2012)
Lenci, S., Rega, G., Ruzziconi, L.: Dynamical integrity as a conceptual and operating tool for interpreting/predicting experimental behavior. Philos. Trans. R. Soc. Lond. A 371(1993), 20120423-1–20120423-19 (2013)
Ruzziconi, L., Younis, M.I., Lenci, S.: Dynamical integrity for interpreting experimental data and ensuring safety in electrostatic MEMS. In: Wiercigroch, M., Rega, G. (eds.) IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design. IUTAM Bookseries 32, pp. 249–261. Springer, Berlin (2013)
Ruzziconi, L., Lenci, S., Younis, M.I.: Interpreting and predicting experimental responses of micro and nano devices via dynamical integrity. In: Lenci, S., Rega, G. (eds.) Global Nonlinear Dynamics for Engineering Design and System Safety. CISM Courses and Lectures 588, pp. 113–166. Springer, Berlin (2018)
van Campen, D.H., van de Vorst, E.L.B., van der Spek, J.A.W., de Kraker, A.: Dynamics of a multi-dof beam system with discontinuous support. Nonlinear Dyn. 8(4), 453–466 (1995)
Kreuzer, E., Lagemann, B.: Cell mapping for multi-degree-of-freedom-systems parallel computing in nonlinear dynamics. Chaos, Solitons Fractals 7(10), 1683–1691 (1996)
Eason, R., Dick, A.J.: A parallelized multi-degrees-of-freedom cell map method. Nonlinear Dyn. 77(3), 467–479 (2014)
Xiong, F.R., Qin, Z.C., Ding, Q., Hernández, C., Fernandez, J., Schütze, O., Sun, J.Q.: Parallel cell mapping method for global analysis of high-dimensional nonlinear dynamical systems. J. Appl. Mech. 82(11), 111010 (2015)
Xiong, F.R., Han, Q., Hong, L., Sun, J.Q.: Global analysis of nonlinear dynamical systems. In: Lenci, S., Rega, G. (eds.) Global Nonlinear Dynamics for Engineering Design and System Safety. CISM Courses and Lectures 588, pp. 287–318. Springer, Berlin (2018)
Belardinelli, P., Lenci, S.: A first parallel programming approach in basins of attraction computation. Int. J. Non Linear Mech. 80, 76–81 (2016)
Belardinelli, P., Lenci, S.: An efficient parallel implementation of cell mapping methods for mdof systems. Nonlinear Dyn. 86(4), 2279–2290 (2016)
Marszal, M., Jankowski, K., Perlikowski, P., Kapitaniak, T.: Bifurcations of oscillatory and rotational solutions of double pendulum with parametric vertical excitation. Math. Probl. Eng. 2014, 892793 (2014)
Carvalho, E.C., Goncalves, P.B., Rega, G., Del Prado, Z.J.G.N.: Influence of axial loads on the nonplanar vibrations of cantilever beams. Shock Vib. 20(6), 1073–1092 (2013)
Carvalho, E.C., Goncalves, P.B., Rega, G., Del Prado, Z.J.G.N.: Nonlinear nonplanar vibration of a functionally graded box beam. Meccanica 49(8), 1795–1819 (2014)
Goncalves, P.B., Silva, F.M.A., Del Prado, Z.J.G.N.: Global stability analysis of parametrically excited cylindrical shells through the evolution of basin boundaries. Nonlinear Dyn. 50, 121–145 (2007)
Schultz, P., Menck, P.J., Heitzig, J., Kurths, J.: Potentials and limits to basin stability estimation. New J. Phys. 19, 023005 (2017)
Agarwal, V., Yorke, J.A., Balachandran, B.: Noise-induced chaotic-attractor escape route. Nonlinear Dyn. 65, 1–11 (2020)
Benedetti, K.C.B., Gonçalves, P.B.: Nonlinear response of an imperfect microcantilever static and dynamically actuated considering uncertainties and noise. Nonlinear Dyn. (2021) (submitted)
Wiercigroch, M., Pavlovskaia, E.: Non-linear dynamics of engineering systems. Int. J. Non-Linear Mech. 43(6), 459–461 (2008)
Wiercigroch, M., Rega, G.: Introduction to NDATED. In: Wiercigroch, M., Rega, G. (eds.) IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design. IUTAM Bookseries 32, pp. 5–8. Springer, Berlin (2013)
Szemplinska-Stupnicka, W.: The analytical predictive criteria for chaos and escape in nonlinear oscillators: a survey. Nonlinear Dyn. 7(2), 129–147 (1995)
Acknowledgements
Contributions of Stefano Lenci and Paulo Gonçalves to the research background of this feature article are acknowledged.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Rega, G., Settimi, V. Global dynamics perspective on macro- to nano-mechanics. Nonlinear Dyn 103, 1259–1303 (2021). https://doi.org/10.1007/s11071-020-06198-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-020-06198-x