# Blue-sky catastrophe

Andrey Shilnikov and Dmitry Turaev (2007), Scholarpedia, 2(8):1889. | doi:10.4249/scholarpedia.1889 | revision #91070 [link to/cite this article] |

This stunning name has been given to the last, out of the seven known, main bifurcations of a
periodic orbit. While the first six bifurcations had been known for almost 70 years
[Andronov and Leontovich 1937, Andronov et al 1966], the **blue-sky catastrophe** (see Figure 1) has been discovered and studied quite recently [Turaev and L. Shilnikov 1995, 1996; Gavrilov and A. Shilnikov 2000; L. Shilnikov et al, 2001, A. Shilnikov et al 2005].

## Contents |

## Codimension-one Bifurcations

The loss of stability or disappearance of a periodic orbit corresponds to a certain bifurcation: the main stability boundaries correspond to bifurcations of codimension 1 (i.e. those that occur in one-parameter families of the general position). For systems on a plane, there are four such stability boundaries, all discovered and described by Leontovich and Andronov. These are also the *existence boundaries*, i.e. the periodic orbit disappears at the bifurcation moment or immediately after it. Namely, the periodic orbit either

- 1. collapses into an equilibrium state through a supercritical Andronov-Hopf bifurcation; or
- 2. collides with an unstable periodic orbit (acquiring a multiplier equal to +1) and vanishes; or
- 3. becomes a homoclinic loop to a saddle equilibrium state; or
- 4. transforms into a homoclinic loop of a saddle-node equilibrium state.

Higher-dimensional systems add two more possibilities, where the periodic orbit no longer disappears at the bifurcation but only loses its stability via:

- 5. period doubling or flip bifurcation where a multiplier of the orbit decreases through -1; the stability of the original orbit is inherited by an orbit of doubled period; or

- 6. the secondary Andronov-Hopf bifurcation where a pair of complex-conjugate multipliers \( e^{\pm i\phi}\ ,\) with \(\phi \neq 0, \pi/2, 2\pi/3, \pi\) of the periodic orbit, crosses a unit circle outwards, and the periodic orbit, as Andronov said, "loses its skin" that becomes a two-dimensional invariant torus.

One can also classify these cod-1 bifurcations by that how the period and length of the orbit depend on the control parameter, \(\mu\ ,\) approaching a finite bifurcation value \(\mu^+_0\ .\)

- Group I:
**finite Period & zero Length**First group consists of a single Andronov-Hopf Bifurcation bifurcation, at which a periodic orbit collapses into the equilibrium state with a pair of purely imaginary characteristic exponents \(\pm i \omega\ ,\) giving an estimate on its period \(T \sim 2 \pi /\omega\ .\)

- Group II:
**finite Period & Length**The second group includes the local saddle-node, the flip or period-doubling bifurcations, as well as the secondary Andronov-Hopf bifurcation (numbers 2,5 and 6 in the above list). It is worth noticing that the periodic orbit persists at \(\mu=\mu^{+}_{0}\) for the boundaries of Group II.

- Group III:
**\(\infty\) Period & finite Length**are the feature of homoclinic bifurcations of equilibria (cases 3 and 4 above). Moreover, the period of the orbit increases as \(1/\sqrt{\mu-\mu_0}\) before the former becomes a homoclinic orbit to a saddle-node equilibrium state (one zero exponent), or as \(-\ln(\mu-\mu_0)\) in case of a simple saddle.

- Group IV:
**\(\infty\) Period & \(\infty\) Length**- Blue Sky Catastrophe.

## Historical note

The question about the possibility for a periodic orbit to remain in a bounded region of the phase space while the period and length of the orbit increase with no bound as it approaches its existence boundary was raised by Palis and Pugh [1974]. The problem was code-named a "blue sky catastrophe" [Abraham, 1985] as the orbit, while getting longer and longer, would be virtually vanishing in the space. The first examples of such one-parameter families of periodic orbits were suggested by Medvedev [1980]. However, Medvedev families are not in general position. As the analysis in [Afraimovich and L. Shilnikov, 1982; Turaev and L. Shilnikov,1986; Li and Zhang, 1991] showed, the generic version of one of the Medvedev examples (on a Klein bottle) gave a new existence boundary for periodic orbits, approaching which the orbit changes its stability infinitely many times in a sequence of forward and backward flip bifurcations.

The question on the possibility for the periodic orbit to disappear in the blue sky
*without losing stability en route * remained open until it was
solved positively by L. Shilnikov and Turaev [1995; 2000] who found the following configuration in \(\mathbb{R}^3\) and higher: its core is the way
the two-dimensional unstable manifold of a saddle-node periodic orbit returns
to the orbit from the stable (node) region where it makes
infinitely many revolutions while approaching the saddle-node, as shown in Figure 1 above. The second component of this configuration is the strong transverse contraction along the homoclinic connection: that ensures that its closure becomes an arbitrarily long (of period and length both evaluated as \(1/\sqrt{\mu-\mu_0}\)) stable periodic orbit after the saddle-node orbit has vanished.

In other cases, the closure of the unstable manifold of the saddle-node periodic orbit can be a two-dimensional torus -- that corresponds to the border of a synchronization zone (Arnold tongue), or a Klein bottle (as in the Medvedev example), or the unstable manifold may come back crossing transversely the strongly stable manifold \(W^{ss}\) of the saddle-node orbit, this leads to chaotic shift dynamics [Lukyanov and L. Shilnikov, 1978; A. Shilnikov et al., 2005] (see Lukyanov-Shilnikov Bifurcation). Notably, a saddle-node bifurcation in \(R^4\) can even lead to the emergence of a hyperbolic strange attractor (the Smale-Williams solenoid) under the fulfillment of a few simple conditions on the shape of \(W^u\) as it returns to the node region [L. Shilnikov and Turaev, 2000].

## Applications

The first example of the specific equations undergoing the catastrophe was given by N. Gavrilov and A. Shilnikov [Gavrilov and Shilnikov, 2000; L. Shilnikov et al, 2001]: \[\tag{1} \begin{array}{rcl} \dot x &=& x(2+\mu -10(x^{2}+y^{2})) +z^{2}+y^{2}+2y,\\ \dot y &=& -z^{3}-(1+y)(z^{2}+y^{2}+2y) -4x +\mu y,\\ \dot z &=& (1+y)z^{2}+x^{2}-\varepsilon, \end{array} \ .\]

The early development of the blue sky catastrophe in this system begins with a homoclinic connection to an equilibrium state with the characteristic exponents (\(0,\pm i \omega\)); this is indeed a cod-2 bifurcation named after Gavrilov-Guckenheimer or aka the homoclinic Fold-Hopf.

The blue sky catastrophe has turned out to be a typical phenomenon in slow-fast systems [L. Shilnikov et al, 2001; A. Shilnikov et al, 2005]. The dynamics of such a system are known to center around the attracting segments of the slow motion manifolds, which are formed by the limit sets, such as equilibria (labeled \(M_{eq}\)) and periodic orbits (\(M_{po}\)), of its fast subsystem (see the corresponding sketch). The blue sky catastrophe occurs here when a saddle-node orbit emerges on the manifold \(M_{po}\) shutting the passage along it for the solutions of the system. The stability of the blue sky orbit is due to the contraction across the manifold \(M_{eq}\) that is comprised by the stable equilibrium states of the fast subsystem.

In slow-fast Hodgkin-Huxley models of computational neuroscience the blue sky catastrophe describes a continuous and reversible transition between periodic bursting and tonic spiking activities, for example, in a reduced oscillatory heart interneuron model [A. Shilnikov and Cymbalyuk, 2005]: \[\tag{2} \mathrm{\dot V} = \mathrm{-2\,[30\, m^2_{K2} (V+0.07)+8\,(V+0.046)}+ \mathrm{200\, f^3_{\infty}(-150,\,0.0305\,,V) h_{Na}\,(V-0.045)}+0.0060]\ ,\]

\(\mathrm{\dot h_{Na}} = \mathrm{[f_{\infty}(500,\,0.0325,\,V)-h_{Na}]/0.0406}\ ,\)

\[\mathrm{\dot m_{K2}} =\mathrm{[f_{\infty}(-83,V_{\frac{1}{2}}+V_{K2}^{shift},V)-m_{K2}]/0.9}\ ,\]

where \(\mathrm{V}\) is the membrane potential, \(\mathrm{h}_{\rm Na}\) is inactivation of the fast sodium current, and \(\mathrm{m}_{\rm K2}\) is activation of persistent potassium one; a Boltzmann function \(\mathrm{f_{\infty}(a,b,V)=1/(1+e^{a(b+V)})}\) describes kinetics of (in)activation of the currents. The bifurcation parameter \(\mathrm{V^{shift}_{K2}}\) is a deviation from the canonical value \(\mathrm{V_{\frac{1}{2}}}=0.018\)V corresponding to \(f_{\infty}=1/2\ ,\) i.e. to the semi-activated potassium channel. The blue sky catastrophe occurs in the model near \(\mathrm{V^{shift}_{K2}}=-0.02425\) (Figure 14).

It is worth noticing that since the blue sky catastrophe is locally based on the saddle-node bifurcation, the period of the bursting orbit obeys the law of \(1/\sqrt{\mu-\mu_0}\ .\) This means that the slow component of the phase point slows down near the phantom of the vanished saddle-node, thereby allowing the bursting orbit to absorb arbitrarily many new spikes one by one (Figure 16) as the bifurcation parameter approaches the transition value.

## References

A.A. Andronov, E.A. Leontovich, Some cases of dependence of limit cycles on a parameter, Uchenye zapiski Gorkovskogo Universiteta (Research notes of Gorky University) 6, 3-24, 1937.

A.A. Andronov, E.A. Leontovich, I.E. Gordon, A.G. Maier. The theory of bifurcations of dynamical systems on a plane, Wiley, New York, 1971.

J. Palis, C. Pugh, in Fifty problems in dynamical systems, Dynamical systems - Warwick, 1974, Springer Lecture Notes 468, 1975.

R.H. Abraham, Catastrophes, intermittency, and noise, in Chaos, Fractals, and Dynamics, Lect. Notes Pure Appl. Math. 98, 3-22, 1985.

V.S. Medvedev, The bifurcation of the “blue sky catastrophe” on two-dimensional manifolds, Mathematical Notes, 51(1), 76-81, 1992.

W. Li, C. Li and Z.F. Zhang, Unfolding critical homocllinic orbit of a class of degenerate equilibrium points, Symp. Special Year of ODE and Dyn. Systems in Nankai Univ. in 1990. World Sci. Publ, 99-110, 1992.

D.V. Turaev, L.P. Shilnikov, Blue sky catastrophes. Dokl. Math. 51, 404-407, 1995.

L. Shilnikov, D. Turaev, A new simple bifurcation of a
periodic orbit of blue sky catastrophe type, in ``Methods of
qualitative theory of differential equations and related topics*,*
AMS Transl. Series II, v.200, 165-188, 2000.

N. Gavrilov, A. Shilnikov, Example of a blue sky catastrophe,
*ibid*, 99-105, 2000.

L. Shilnikov, A. Shilnikov, D. Turaev, L. Chua, Methods of qualitative theory in nonlinear dynamics. Part I. World Scientific, Singapore, 1998.

L. Shilnikov, A. Shilnikov, D. Turaev, L. Chua, Methods of qualitative theory in nonlinear dynamics. Parts II, World Scientific, Singapore, 2001.

A. Shilnikov, L.P. Shilnikov, D. Turaev. Blue sky catastrophe
in singularly perturbed systems. Moscow Math. Journal 5(1),
205-218, 2005.

V. Lukyanov, L.P. Shilnikov, On some bifurcations of dynamical systems with homoclinic structures. Soviet Math. Dokl. 19(6), 1314-1318, 1978.

A. Shilnikov, G. Cymbalyuk, Transition between tonic-spiking and bursting in a neuron model via the blue-sky catastrophe, Phys Review Letters, 94, 048101, 2005.

**Internal references**

- Yuri A. Kuznetsov (2006) Andronov-Hopf bifurcation. Scholarpedia, 1(10):1858.
- John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
- Eugene M. Izhikevich (2006) Bursting. Scholarpedia, 1(3):1300.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
- Yuri A. Kuznetsov (2006) Saddle-node bifurcation. Scholarpedia, 1(10):1859.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
- Arkady Pikovsky and Michael Rosenblum (2007) Synchronization. Scholarpedia, 2(12):1459.
- Paul So (2007) Unstable periodic orbits. Scholarpedia, 2(2):1353.

## External Links

see Blue Sky Catastrophe in action

## See Also

Bifurcation, Periodic Orbit, saddle-node, Shilnikov saddle-node, homoclinic, homoclinic bifurcation, Andronov-Hopf, Shilnikov-Hopf, Gavrilov-Guckenheimer, saddle-node-Hopf, Fold-Hopf