Noninvertibility
| Evelyn Sander et al. (2009), Scholarpedia, 4(8):2242. | doi:10.4249/scholarpedia.2242 | revision #91584 [link to/cite this article] |
A dynamical system is noninvertible if the backward time evolution is either multivalued or undefined.
It is well known that for an ordinary differential equation (ODE), two different initial states display different dynamics both forwards and backwards in time. Thus looking at the initial state of a quantity modeled by an ODE, it is possible to fully and uniquely describe both the future and the history of its time evolution. This non-overlapping nature of solutions is an important property, as it affects not only the predictability, but also the qualitative behavior of topological structures used for dynamical analysis of the model, such as global stable and unstable manifolds, synchronization manifolds, and basins of attraction.
Unlike the solutions to ordinary differential equations, natural processes can exhibit noninvertibility. This has important dynamical ramifications. Many of the major dynamical results from ODEs and invertible iterated maps carry over to systems with noninvertibility, including the shadowing lemma and the existence of both local invariant manifolds and Poincaré sections. However, the presence of noninvertibility can give rise to fundamentally different dynamical properties than those seen in the ODE case. Examples occur in adaptive control systems [1-3], neural networks [4], numerical methods [5,6], and synchronization [7-12]. Further theoretical treatment of noninvertibility in finite dimensions can be found in many works. See for example, the Scholarpedia entry on noninvertible maps, citations [13] and [14], and references therein. Noninvertibility also occurs for infinite-dimensional systems, such as solutions to both delay equations and partial differential equations.
Noninvertibility in bifurcations and chaos
Lorenz's work and the followup paper of Frouzakis, Kevrekidis, and Peckham [5,6] describe the connection between the onset of noninvertibility and chaotic behavior within an attractor in a specific planar map. When the attractor intersects the singular set of the map (i.e., the set of points for which the inverse does not exist), there is an increase in the attractor's complexity. Since the singular set of a finite dimensional map is typically codimension one, this type of bifurcation is expected in an open set of one-parameter families. Krauskopf, Osinga, and Peckham [21] give an example of a local bifurcation which fundamentally relies on noninvertibility. Namely, they give an unfolding of the codimension two cusp-cusp bifurcation for which the singular set and its image intersect a forward invariant curve. This is for example the bifurcation between a smooth invariant curve and an invariant curve with a loop described in reference [11].
The onset of noninvertibility appears to be fundamental for the existence of chaos in delay equations. For delay equations, monotonicity generically implies the invertibility of solutions, i.e., that a backwards orbit is unique (though it may not exist). Thus, one does not typically expect to find chaotic solutions to monotone delay differential equations. In this regard, Mallet-Paret and Sell have shown that if the nonlinearity F is monotone for a delay differential equation of the form
\[\frac{dx}{dt}(t)=F(x(t-\tau))-\gamma x(t) \],
then the dynamics are very simple and non-chaotic [16,17].
In contrast, the Mackey-Glass equation [15], where
\[F (x) = \frac{ax \theta^k}{\theta^k + x^k}\] with \(a>0\), \(\theta>0\), and \(k>1\)
is not monotonic, does have chaotic solutions for appropriate parameter values. Hale and Lin and Lani-Wayda have also each given examples of delay equations that exhibit chaos [18,19]. Specifically, they both construct examples with transverse homoclinic orbits, implying chaotic behavior. Hale and Verduyn-Lunel discuss a series of examples of simple delay equations such that there are infinitely many possible preimages [20].
References
[1] R. Adomaitis and I. Kevrekidis, Noninvertibility and the structure of basins of attraction in a model adaptive control system, Journal of Nonlinear Science, 1 (1991), 95–105.
[2] C. Frouzakis, R. Adomaitis, and I. Kevrekidis, An experimental and computational study of subcriticality, hysteresis, and global dynamics for a model adaptive control system, Computers and Chemical Engineering, 20 (1996), 1029–1034.
[3] C. Frouzakis, R. Adomaitis, I. Kevrekidis, M. Golden, and B. Ydstie, The structure of basin boundaries in a simple adaptive control system, In T. Bountis, editor, Chaotic Dynamics: Theory and Practice, Plenum Press, 1992, 195–210.
[4] R. Rico-Martínez, I. Kevrekidis, and R. Adomaitis, Noninvertibility in neural networks, In 1993 IEEE International Conference on Neural Networks, (1993),382–386.
[5] E. N. Lorenz, Computational chaos - a prelude to computational instability, Physica D, 25 (1989), 299–317.
[6] C. E. Frouzakis, I. G. Kevrekidis, and B. Peckham, A route to computational chaos revisited: noninvertibility and the breakup of an invariant circle, Physica D, 177 (2003), 101–121.
[7] V. Afraimovich, A. Cordonet and N. Rulkov, Generalized synchronization of chaos in noninvertible maps, Physical Review E, 66:016208, 2002.
[8] E. Barreto, K. Josíc, C. Morales, E. Sander, and P. So, The geometry of chaos synchronization, Chaos, 13 (2003) no. 1, 151–164.
[9] E. Barreto, P. So, B.J. Gluckman, and S. Schiff, From generalized synchrony to topological decoherence: Emergent sets in coupled chaotic systems, Physical Review Letters, 84 (2000) no. 8, 1689–1692.
[10] J. Chubb, E. Barreto, P. So, and B. J. Gluckman, The breakdown of synchronization in systems of non-identical chaotic oscillators: Theory and experiment, International Journal of Bifurcation and Chaos, 11 (2001) no. 10, 2705–2713.
[11] K. Josíc and E. Sander, The structure of synchronization sets for noninvertible systems, Chaos, 14 (2004) no.2, 249–262.
[12] P. So, E. Barreto, K. Josíc, E. Sander, and S.J. Schiff, Limits to the Experimental Detection of Nonlinear Synchrony, Physical Review E, 65 (2002) 046225.
[13] C. Frouzakis, L. Gardini, I. Kevrekidis, G. Millerioux, and C. Mira, On some properties of invariant sets of two-dimensional noninvertible maps, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 7 (1997). no 6, 1167–1194.
[14] E. Sander, Hyperbolic sets for noninvertible maps and relations, Discrete and Continuous Dynamical Systems, 5 (1999) no. 2, 339–358.
[15] M. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 15 (1997), 287–289.
[16] J. Mallet-Paret and G. R. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, Journal of Differential Equations, 125 (1996) no.2, 441–489.
[17] J. Mallet-Paret and G. R. Sell, Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions Journal of Differential Equations, 125 (1996) no.2, 385–440.
[18] J.K. Hale and X-B. Lin, Examples of transverse homoclinic orbits in delay equations, Nonlinear Anal. 10 (1986) no. 7, 693-709.
[19] B. Lani-Wayda, Erratic solutions of simple delay equations, Trans. Amer. Math. Soc., 351 (1999) no.3, 901–945.
[20] J. Hale and S. V. Lunel, Introduction to Functional Differential Equations. Springer-Verlag, Berlin, 1993.
[21] B. Krauskopf, H.M. Osinga & B.B. Peckham, Unfolding the cusp-cusp bifurcation of planar endomorphisms SIAM Journal on Applied Dynamical Systems 6(2): 403-440, 2007.
