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Evelyn Sander et al. (2009), Scholarpedia, 4(8):2242. doi:10.4249/scholarpedia.2242 revision #91583 [link to/cite this article]
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Curator: Paul So

A dynamical system is noninvertible if the backward time evolution is either multivalued or undefined.

The best known examples of noninvertible dynamical systems are quadratic maps in one dimension, and the complex quadratic family \[z \mapsto z^2 + c.\] For example, when z is real, and c=0, each positive number z has the two preimages \(\sqrt{z}\) and \(-\sqrt{z}\ ,\) whereas backwards time evolution is undefined for every negative value.

It is well known that when considering the flow generated by an ordinary differential equation (ODE), two different initial states display different states 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, solutions to discrete dynamical systems and infinite dimensional sytems, such as delay equations and partial differential equations, can exhibit noninvertibility. Many of the major dynamical results from ODEs and invertible iterated maps carry over to noninvertible systems, including the shadowing lemma and the existence of both local invariant manifolds and Poincaré sections. In addition, all the standard codimension one bifurcations and routes to chaos occur for noninvertible dynamical systems. Noninvertibility can, however, have important dynamical ramifications, including chaotic dynamics.

Theoretical treatment of noninvertibility in finite dimensions can be found in many works. One-dimensional examples such as the logistic map (quadratic map, Myrberg map) have been especially well-studied. See [Collet], or more introductory level textbooks such as [Devaney1]. In the complex setting, the quadratic map is also very well known. See [Benedicks] or [Devaney2]. Less is known about two-dimensional noninvertible real maps, but the work of Mira and Gumowski has uncovered many features unique to maps which are noninvertible [Gumowski]. See, for example, the Scholarpedia entry on noninvertible maps, citation [Frouzakis3], and references therein. More recent work addresses some general theoretical considerations. Citation [Sander1], for example, extends the theory of hyperbolic sets from invertible maps to relations, which includes both noninvertible and multivalued maps, including proofs of shadowing and the stable manifold theorem. Citation [Sander2] shows that all properties of noninvertible maps do not apply to noninvertible maps by giving an example with a transverse intersection of a stable and unstable "manifold," but with no chaotic dynamics.

Examples of noninvertibility occur in adaptive control systems [Adomaitis, Frouzakis1, Frouzakis2], neural networks [RicoMartinez], numerical methods [Lorenz, Frouzakis3], and synchronization [Afraimovich, Barreto1, Barreto2, Chubb, Josic, So].


Noninvertibility in bifurcations and chaos

Both invertible and noninvertible systems undergo a common period-doubling route to chaos, but other bifurcations, such as the creation of homoclinic orbits which are repelled from a periodic orbit but later land on that periodic orbit, are a feature of noninvertibility. Similarly, one-dimensional complex maps exhibit chaotic behavior on Julia sets, and standard bifurcations within the Mandelbrot or polynomial-like mappings [Douady]. In both cases, the orbit(s) of the critical point(s) play a significant role.

In two dimensions, the analogue of the critical point is the "critical line" \(J_0\ ,\) where the Jacobian determinant of the map equals zero. The most common type of noninvertibility corresponds geometrically to a "folding" of the phase space along \(J_0\ .\) The effects of noninvertibility come into play when various dynamical objects interact (as parameters are varied) with \(J_0\ .\) A fixed or periodic orbit interacting with \(J_0\) results in an eigenvalue of zero. The stable manifold theorem still holds in this case, so the corresponding local dynamical behavior is similar to the invertible case. The interaction of an invariant curve such as an unstable manifold or closed curve with \(J_0\) can result in the creation of a loop on the manifold (so that it is no longer technically a manifold). This is a truly noninvertible phenomenon. Loop creation has been discovered in many contexts. See for example, [Lorenz,Frouzakis4, Josic, Frouzakis5, Mira] and references therein. Loop creation on a closed curve seems to be especially significant because it appears to be the first step in a typical two-dimensional noninvertible map setting for a route from "tame" dynamics to "chaotic" dynamics. In particular, Lorenz' work [Lorenz] and the followup paper of Frouzakis, Kevrekidis, and Peckham [Frouzakis4] describe the connection between the onset of noninvertibility and chaotic behavior within an attractor in a specific planar map. When the attractor intersects \(J_0\ ,\) 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 [Krauskopf] give an example of a codimension-two local bifurcation which is further degenerate than a simple loop-creation bifurcation. In the process of describing the unfolding of this "cusp-cusp" bifurcation, they identify four other codimension-one bifurcations -- where equivalence is defined by the topological equivalence of the image of the invariant manifold and the image \(J_1\) of the singular curve \(J_0\ .\) All of these codimension-one bifurcations intersect at the cusp-cusp point in a two-dimensional parameter space.

Mira and coworkers describe noninvertible features such as absorbing regions, multiply connected basins of attraction, chaotic attractors which look like sheets which have been folded and projected to the phase plane, and "contact" bifurcations in which an attractor can also make contact with its basin boundary. See Figure 6 in Mira's Scholarpedia entry on noninvertible maps. The current article concentrates on topics which complement Mira's article.

Infinite dimensional systems

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 [MalletParet1, MalletParet2].

In contrast, the Mackey-Glass equation [Mackey], 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 [Hale1, LaniWayda]. 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 [Hale2].

Solutions for evolutionary partial differential equations standardly exhibit noninvertibility. For example solutions for evolutionary parabolic equations (with certain technical properties) such as the Cahn-Hilliard equation typically generate a semiflow rather than a flow. Ramifications of noninvertibility in this context are described in reference [Sell].


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[Afraimovich] V. Afraimovich, A. Cordonet and N. Rulkov, Generalized synchronization of chaos in noninvertible maps, Physical Review E, 66:016208, 2002.

[Barreto1] E. Barreto, K. Josíc, C. Morales, E. Sander, and P. So, The geometry of chaos synchronization, Chaos, 13 (2003) no. 1, 151–164.

[Barreto2] 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.

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[Frouzakis3] 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.

[Frouzakis4] 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.

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[Sander2] Sander E, "Homoclinic tangles for noninvertible maps," Nonlinear Analysis, 41(1-2):259-276, 2000.

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Internal references

  • John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.
  • John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
  • Olaf Sporns (2007) Complexity. Scholarpedia, 2(10):1623.
  • Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
  • Ernest Barreto (2008) Shadowing. Scholarpedia, 3(1):2243.
  • Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
  • David H. Terman and Eugene M. Izhikevich (2008) State space. Scholarpedia, 3(3):1924.
  • Arkady Pikovsky and Michael Rosenblum (2007) Synchronization. Scholarpedia, 2(12):1459.
  • James Murdock (2006) Unfoldings. Scholarpedia, 1(12):1904.

See also

Noninvertible maps

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