Blue-sky catastrophe

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Andrey Shilnikov and Dmitry Turaev (2007), Scholarpedia, 2(8):1889. doi:10.4249/scholarpedia.1889 revision #17444 [link to/cite this article]
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Figure 1: Blue sky bifurcation in action: the unstable manifold \(\mathrm{W^u}\) comes back to the saddle-node periodic orbit while making infinitely many revolutions in the stable, node, region separated from the saddle one by the strongly stable manifold \(\mathrm{W^{ss}}\). The strong transverse contraction transforms the homoclinic connection into a stable periodic orbit slowing down near the "phantom" of the saddle-node orbit.

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].



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

Figure 2: Supercritical Andronov-Hopf bifurcation in \(\mathbb{R}^{3}\). A stable periodic orbit emerges from the equilibrium state making it a saddle-focus with 2D unstable and 1D stable manifolds. The period of the new orbit is determined by the magnitude of the imaginary part of the characteristic exponents of the bifurcating equilibrium state.
Figure 3: Local saddle-node bifurcation in \(\mathbb{R}^{3}\): a stable periodic orbit merges with a saddle one at \(\mu=\mu_0\) and both vanish when \(\mu>\mu_0\). The flight time of the phase point through the phantom of the saddle-node is estimated as \(1/\sqrt{\mu-\mu_0}\).
Figure 4: A stable periodic orbit disappears at a homoclinic loop of a saddle equilibrium state in \(\mathbb{R}^{3}\). The saddle value or quantity of the saddle is negative here. The period of the bifurcation orbit increases logarithmically fast as it approaches the saddle.
Figure 5: A stable periodic orbit becomes a homoclinic loop of a saddle-node equilibrium state. On the current jargon this bifurcation is also known as SNIC. The growth of the period of the orbit obeys the general saddle-node law.
Figure 6: Period-doubling bifurcation: a stable periodic orbit looses stability and becomes of the saddle type with 2D stable and unstable manifolds homeomorphic to a Mobius band. The stability is inherited by a periodic orbit of a doubled period. Both orbits are linked in \(\mathbb{R}^{3}\).
Figure 7: Non-resonant torus bifurcations: a stable spiraling periodic orbit becomes repelling surrounded by a stable 2D torus.
  • 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.


Figure 8: Torus is the closure of the unstable manifold \(W^u\) of the saddle-node periodic orbit. This bifurcation occurs on the boundaries of resonant zones.
Figure 9: Saddle-node periodic orbit on a Klein bottle which is the closure of its unstable manifold.
Figure 10: The unstable manifold \(W^u\) of the saddle-node orbit comes back along the noncentral (strongly stable) manifold \(W^{ss}\). This noncentral homoclinic bifurcation leads to the emergence of a chaotic hyperbolic subset [Lukyanov and L. Shilnikov, 1978] as the bifurcating orbit vanishes.


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].

Figure 11: Blue sky orbit in the Gavrilov-A. Shilnikov model near \(\mu=0.456\) and \(\varepsilon=0.0357\)

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.

Figure 12: Saddle-node periodic orbit bifurcation on the slow motion manifold \(M_{po}\) comprised of limit cycles of the fast subsystem. This manifold terminates through the homoclinic bifurcation discussed above.

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>).


Figure 13: Long bursting orbit switching repeatedly between the spiking manifold \(\mathrm{M_{lc}}\) and the hyperpolarized branch of the quiescent manifold \(\mathrm{M_{eq}}\) in the phase space of the interneuron model.
Figure 14: Shown in light blue is the saddle-node periodic orbit that makes the spiking phase of bursting infinite. In the heart interneuron model, periodic orbits are sought on the spiking manifold using the proposed averaged nullclines: their tangency yields the saddle-node orbit. After the saddle-node orbit decouples, the model exhibits a periodic tonic-spiking activity.





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.

Figure 15: Waveforms generated by the heart interneuron model (<ref>biot</ref>) for decreasing values of the bifurcation parameter \(\mathrm{V^{shift}_{K2}}\). The bursting regime (three top traces) is continuously transformed into tonic spiking (bottom trace).
Figure 16: The burst period increases as \(0.31/\sqrt{|(\mathrm{V^{shift}_{K2}}+24.25)|}\), where \(0.02425V\) is 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

A. Shilnikov's website

see Blue Sky Catastrophe in action

See Also

Bifurcation, Periodic Orbit

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