# Blue-sky catastrophe

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 Fig.<ref>fig0</ref>) 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].

## 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 Fig.<ref>fig0</ref> 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]:

- <math e12.4.11>

\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} </math>.

In this example, an early development of the catastrophe begins with a homoclinic connection to an equilibrium state having the characteristic exponents equal to (\(0,\pm i \omega\)), a local cod-2 bifurcation named after Gavrilov-Guckenheimer.

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 leech heart interneuron model [A. Shilnikov and Cymbalyuk, 2005]:

- <math biot> \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]</math>, \(\mathrm{\dot h_{Na}} = \mathrm{[f_{\infty}(500,\,0.0325,\,V)-h_{Na}]/0.0406}\), \[\mathrm{\dot m_{K2}} =\mathrm{[f_{\infty}(-83,V_{\frac{1}{2}}+V_{K2}^{shift},V)-m_{K2}]/0.9}\],

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 \(\mathrm{V^{shift}_{K2}}=-0.02425\) (Fig.<ref>fig3</ref>).

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 (<ref>fig4</ref>) 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. Parts I and II, World Scientific, Singapore, 1998 and 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.

## External Links

see Blue Sky Catastrophe in action