Periodic orbit

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Figure 1: A periodic orbit shown in phase space and as a timeseries for a vector field.

A periodic orbit corresponds to a special type of solution for a dynamical system, namely one which repeats itself in time. A dynamical system exhibiting a stable periodic orbit is often called an oscillator.

Contents

Definition

Periodic Orbit for a Vector Field

Consider a system of ordinary differential equations \[ \frac{d x}{dt} = f(x), \qquad x \in \mathbb{R}^n \qquad (n \ge 2) \] or \[ \frac{d x}{dt} = f(x,t), \qquad x \in \mathbb{R}^n \qquad(n \ge 1), \] corresponding to an autonomous or non-autonomous vector field, respectively. A non-constant solution to such a system, \(x(t)\ ,\) is periodic if there exists a constant \(T>0\) such that

\[ x(t) = x(t+T) \]

for all \(t\ .\) The period of this solution is defined to be the minimum such \(T\ .\) The image of the periodicity interval \( [0,T] \) under \( x \) in the state space \( \mathbb{R}^n \) is called the periodic orbit or cycle.

Limit Cycle

A periodic orbit \(\Gamma\) on a plane (or on a two-dimensional manifold) is called a limit cycle if it is the \(\alpha\)-limit set or \(\omega\)-limit set of some point \(z\) not on the periodic orbit, that is, the set of accumulation points of either the forward or backward trajectory through \(z\ ,\) respectively, is exactly \(\Gamma\ .\) Asymptotically stable and unstable periodic orbits are examples of limit cycles.

Example (Guckenheimer and Holmes, 1983; Strogatz 1994)

The figure shows the periodic orbit which exists for the vector field \[ \begin{matrix} \frac{d x}{dt} = \alpha x-y-\alpha x(x^2+y^2) \\ \frac{d y}{dt} = x+\alpha y-\alpha y(x^2+y^2) , \end{matrix} \] where \(\alpha>0\) is a parameter. Transforming to radial coordinates, we see that the periodic orbit lies on a circle with unit radius for any \(\alpha>0\ :\) \[ \frac{d r}{dt} = \alpha r(1-r^2), \qquad \frac{d \theta}{dt} = 1 . \] This periodic orbit is a stable limit cycle for \(\alpha>0\) and unstable limit cycle for \(\alpha<0\ .\) When \(\alpha=0\ ,\) the system above has infinite number of periodic orbits and no limit cycles.

Periodic Orbit for a Map

Figure 2: A periodic orbit for a map.

A periodic orbit with period \(k\) for a map \[ x_{i+1} = g(x_i), \qquad x \in \mathbb{R}^n, \qquad n \ge 1 \] is the set of \(k\) distinct points \(\{p_j = g^j(p_0)| j=0,\cdots,k-1\}\) with \(g^k(p_0) = p_0\) (Guckenheimer and Holmes, 1983). Here \(g^k\) represents the composition of \(g\) with itself \(k\) times. The smallest positive value of \(k\) for which this equality holds is the period of the orbit. An example of a periodic orbit for a map is shown in the figure.

Existence (or Non-Existence) of Periodic Orbits

It is sometimes possible to prove analytically that a periodic orbit exists or cannot exist for a dynamical system using the following techniques. Several of these apply for an autonomous planar vector field \[ \frac{d x}{dt} = F(x,y), \qquad \frac{dy}{dt} = G(x,y), \qquad (x,y) \in \mathbb{R}^2. \]

Index Theory

For an autonomous planar vector field, index theory can be used to show that (Guckenheimer and Holmes, 1983):

Inside the region enclosed by a periodic orbit there must be at least one equilibrium, i.e., a point where \(F(x,y)=G(x,y)=0\ .\) If there is only one, it must be a sink, source, or center. If all equilibria inside the periodic orbit are hyperbolic, then there must be an odd number \(2 m + 1 \ ,\) of which \(m\) are saddles and \(m+1\) are sinks or sources.

This can be useful for showing that a periodic orbit does not exist in a region of phase space: if the appropriate equilibria are not present, a periodic orbit cannot exist.

Dulac's Criterion

For an autonomous planar vector field, Dulac's criterion states (Guckenheimer and Holmes, 1983):

Let \(B(x,y)\) be a scalar function defined on a simply connected region \(D \subset \mathbb{R}^2\) (so that \(D\) has no holes in it). If \(\frac{\partial (B F)}{\partial x} + \frac{\partial (B G)}{\partial y}\) is not identically zero and does not change sign in \(D\ ,\) then there are no periodic orbits lying entirely in \(D\ .\)

Bendixson's Criterion

Dulac's criterion is a generalization of Bendixson's criterion, which corresponds to \(B(x,y) = 1\) in the above result. These criteria can be useful for showing that a periodic orbit does not exist in a region of phase space.

Poincare-Bendixson Theorem

For an autonomous planar vector field, the Poincare-Bendixson Theorem implies (Guckenheimer and Holmes, 1983):

If a trajectory enters and does not leave a closed and bounded region of phase space which contains no equilibria, then the trajectory must approach a periodic orbit as \(t \rightarrow \infty\ .\)

This can sometimes be used to establish the existence of a (stable) periodic orbit for a planar vector field.

Lienard Systems

For nonlinear oscillators satisfying Lienard's equation \[ \frac{d^2 x}{dt^2} + F \left(x,\frac{dx}{dt} \right) \frac{dx}{dt} + G(x) = 0, \qquad x \in \mathbb{R}, \] the existence of a unique, stable limit cycle can be established under appropriate general hypotheses on \(F\) and \(G\ .\) For example, the damping coefficient \(F\) must be negative near the phase space origin \(x = \frac{dx}{dt} = 0\) so trajectories near the origin spiral outwards, and \(F\) must be positive far away from the origin, so that trajectories far from the origin spiral inwards. For a detailed discussion, see Jordan and Smith (1977).

Fast-Slow Planar Systems

For a fast-slow autonomous planar vector field \[ \frac{dx}{dt} = F(x,y), \qquad \frac{dy}{dt} = \epsilon G(x,y), \qquad (x,y) \in \mathbb{R}^2, \qquad \epsilon << 1, \] simple geometrical nullcline analysis can suggest the existence of a relaxation oscillation, a special type of periodic orbit (Keener and Sneyd, 1998). The Poincare-Bendixson theorem can be used to prove the existence of a periodic orbit in some cases, but this does not establish that the orbit is a relaxation oscillation. Rigorous results for relaxation oscillations are given in Grasman (1987) and Mishchenko et al. (1994); these make use of geometric singular perturbation theory and go beyond the planar case. Fast-slow systems can also have special periodic orbit solutions called canards, although these are not robust to perturbations in planar systems.

Hilbert's 16th Problem

In 1900, David Hilbert famously posed 23 problems at the International Congress of Mathematicians in Paris. His 16th problem involves determining the number and location of limit cycles for an autonomous planar vector field for which both \(F\) and \(G\) are real polynomials of degree \(N\ .\) At present, this problem has not been solved, but much progress has been made in the last 100+ years. For example, it has been shown that the number of limit cycles for such a system is finite. This and many other results are summarized in Ilyashenko (2002).

Gradient Flows

An autonomous vector field is called a gradient flow if it can be rewritten as \[ \frac{d x}{dt} = -\nabla V(x), \qquad V:\mathbb{R}^n \rightarrow \mathbb{R}, \] where the minus sign is included by convention, so that \(V(x)\) is a Liapunov function for the system. Periodic orbits cannot exist for gradient flows (Guckenheimer and Holmes, 1983).

Averaging for Non-autonomous Vector Fields

Sometimes a non-autonomous vector field with a small parameter (including weakly nonlinear forced oscillations) can be rewritten in a form which allows the method of averaging (over time) to be applied to understand its dynamics. Most useful for the present discussion is the result that the existence of a hyperbolic equilibrium point of the resulting autonomous equations implies the existence of a periodic orbit (possibly trivial, i.e., an equilibrium point) of the original non-autonomous system, with the same stability properties as the equilibrium point (Guckenheimer and Holmes, 1983).

Finding Periodic Orbits for a Map

From our discussion above, each point \(p_j\) on a period-\(k\) periodic orbit for a map is a fixed point for the map \(g^k\ .\) Thus, one can find points on period-\(k\) periodic orbits by solving the algebraic equation \(g^k(x) = x\) for \(x\ .\) This may locate fixed points and points on periodic orbits with periods less than \(k\ :\) for example, a fixed point with \(g(x) = x\) is also a solution to \(g^k(x) = x\) for any \(k\ .\) Even if the points on periodic orbits cannot be found explicitly, analytical techniques might be used to prove that they must exist.

Numerical Methods for Finding Periodic Orbits

Periodic orbits can sometimes be found for a given vector field using numerical methods. If a periodic orbit is stable, then forward numerical integration of a trajectory with an initial condition in the periodic orbit's basin of attraction will converge to the periodic orbit as \(t \rightarrow \infty\ .\) Other methods can be used to numerically find periodic orbits even if they are unstable. For example, the problem of finding (stable or unstable) periodic orbits for an autonomous vector field can be reformulated so that a variant of the Newton-Raphson algorithm can be applied; one numerically solves \(\phi_T(x) - x = 0\) for \(x\) and \(T\ ,\) where \(\phi_T(x)\) is the location of a trajectory starting at the point \(x\) after a time \(T\) (Parker and Chua, 1989). More robust numerical methods are based on a boundary value problem on the unit interval for the periodic solution \(x(t)=u(t/T) \ :\)

\[ u'-Tf(u)=0, u(0)=u(1), \Psi[u]=0, \]

where \( \Psi \) is a phase condition selecting one periodic solution among infinitely many periodic solutions corresponding to the same periodic orbit but having different initial points (Doedel, Keller, and Kernevez, 1991). This BVP should then be approximated by a proper finite-dimensional discretization (e.g., via orthogonal collocation with piecewise-polynomial functions) and solved for (the discretization of) \( u \) and \( T \ .\)

The Newton-Raphson algorithm (or other root finding methods) can be directly applied to find points on periodic orbits for a map: one just needs to find roots of the equation \(g^k(x) - x = 0\) for the period \(k\) of interest.

Stability of a Periodic Orbit

Figure 3: Poincare map for a vector field.

The stability of a periodic orbit for an autonomous vector field can be calculated by considering the Poincare map which replaces the flow of the \(n\)-dimensional continuous vector field with an \((n-1)\)-dimensional map (Guckenheimer and Holmes, 1983). Specifically, an \((n-1)\)-dimensional surface of section \(\Sigma\) is chosen such that the flow is always transverse to \(\Sigma\) (see figure). Let the successive intersections in a given direction of the solution \(x(t)\) with \(\Sigma\) be denoted by \(x_i\ .\) The Poincare map \[ x_{i+1} = g(x_i) \] determines the \((i+1)\)-th intersection of the trajectory with \(\Sigma\) from the \(i\)-th intersection. A periodic orbit of an autonomous vector field corresponds to a fixed point \(x_f\) of this Poincare map, characterized by \(g(x_f) = x_f\ .\) The linearization of the Poincare map about \(x_f\) is \[ \xi_{i+1} = {\rm D}g(x_f) \xi_i. \] If all eigenvalues of \({\rm D} g\) have modulus less than unity, then \(x_f\) (and thus the corresponding periodic orbit) is asymptotically stable. If any eigenvalues of \({\rm D} g\) have modulus greater than unity, then \(x_f\) (and thus the corresponding periodic orbit) is unstable. The stability properties of a periodic orbit are independent of the cross section \(\Sigma\) (Wiggins 2003). If \( x_f \) is stable then it is an attractor of the Poincare map, and the corresponding periodic orbit is an attractor of the vector field.

Example (continued) (Guckenheimer and Holmes 1983, Strogatz 1994)

For the Example above, the radial line given by \(\theta=0\) is a Poincare section, parameterized by \(r\ .\) The corresponding Poincare map \(r_{i+1} = g(r_i)\) along this section may be found by explicitly integrating the vector field: \[ g(r_i)=\left[ 1+ e^{ - 4 \pi \alpha}(r_i^{-2} -1 ) \right]^{-1/2}, \] with fixed point \(r_f=1\) corresponding the periodic orbit. Linearizing, we find \(g'(r_f)=e^{ - 4 \pi \alpha}\ .\) So, the periodic orbit is stable for any \(\alpha>0\) and is unstable for any \(\alpha<0\ .\)

An alternative way to determine the stability of a periodic orbit is to use Floquet theory, which involves the time-dependent (and \(T\)-periodic) vector field linearized around the periodic orbit. Solutions to these linearized equations are used to define \(n\) Floquet multipliers characterizing the growth or decay of perturbations to the periodic orbit. It can be shown that the \((n-1)\) eigenvalues of \({\rm D}{g}\) are equal to \((n-1)\) of the Floquet multipliers of the periodic orbit; the remaining Floquet multiplier is equal to unity and corresponds to a perturbation along the periodic orbit (Guckenheimer and Holmes, 1983). The determination of Floquet multipliers or the eigenvalues of \({\rm D}{g}\) typically must be done numerically.

Given a point \(x_f\) on the periodic orbit \(\Gamma\) as discussed above, the eigenvalues of the matrix \({\rm D}g(x_f)\) can be used to partition the \((n-1)\)-dimensional subspace \(\Sigma\) into a direct sum of subspaces \(\Sigma^s \oplus \Sigma^c \oplus \Sigma^u\ ,\) corresponding to eigenvalues with modulus less than 1, equal to 1, and greater than 1, respectively. If sections \(\Sigma_x\) are chosen to vary continuously over different base points \(x \in \Gamma\ ,\) then concatenations of the corresponding subspaces \(\Sigma_x^s, \Sigma_x^c, \Sigma_x^u\) form vector bundles over \(\Gamma\ .\) Stable, center, and unstable manifolds of \(\Gamma\) can be defined as graphs over these vector bundles.

For a non-autonomous vector field \(\frac{dx}{dt} = f(x,t)\) with \(f(x,t) = f (x,t+\tau)\) for some \(0<\tau<\infty\ ,\) the calculation of the stability properties of a periodic orbit with period \(T = \frac{p \tau}{q}\ ,\) where \(p\) and \(q\) are integers (see Arnold tongues), can be done by considering a stroboscopic map which takes \[ x(t) \rightarrow x\left(t + \frac{p \tau}{q} \right). \] The stability properties follow from the eigenvalues of this map, as above.

To determine the stability properties of a periodic orbit for a mapping \(x_{i+1} = g(x_i)\ ,\) one can exploit the fact that a point \(p_0\) on a period-\(k\) periodic orbit of the map \(g\) is a fixed point of the map \(g^k\ .\) The stability properties of this fixed point of \(g^k\) are the same as the stability properties of the periodic orbit of the map \(g\) (Guckenheimer and Holmes, 1983).

Bifurcations Involving Periodic Orbits

A bifurcation is a qualitative change in the behavior of a dynamical system as a system parameter is varied. This could involve a change in the stability properties of a periodic orbit, and/or the creation or destruction of one or more periodic orbits. Bifurcation analysis can thus provide another (analytical or numerical) method for establishing the existence or non-existence of a periodic orbit.

Among co-dimension 1 bifurcations of periodic orbits for vector fields are (Guckenheimer and Holmes, 1983; Kuznetsov, 1998):

These bifurcations result in the appearance or disappearance of periodic orbits, depending on the direction in which the bifurcation parameter is varied. The (dis)appearing orbits may be stable or unstable, depending, among other factors, on whether the bifurcations are subcritical or supercritical.

Periodic Orbits and Chaos

As a system parameter is varied, chaos can appear via an infinite sequence of period doubling bifurcations of periodic orbits. This is known as the Feigenbaum phenomenon or the period doubling route to chaos (Ott, 1993). Moreover, a chaotic attractor typically has a dense set of unstable periodic orbits embedded within it. Suitable averages over such periodic orbits can be used to approximate descriptive quantities for chaotic attractors such as Lyapunov exponents and fractal dimensions (Chaos Focus Issue, 1992). Such periodic orbits can sometimes be stabilized (and the chaos thus suppressed) through small manipulations of a system parameter, an approach called controlling chaos (Ott 1993).

References

  • Chaos Focus Issue on Periodic Orbit Theory (1992) Chaos 2:1-158.
  • E. Doedel, H.B. Keller, and J.-P. Kernevez (1991) International Journal of Bifurcation and Chaos, 1:745-772.
  • J. Grasman (1987) Asymptotic Methods for Relaxation Oscillations and Applications. Springer-Verlag, New York.
  • J. Guckenheimer and P. Holmes (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York
  • Yu. Ilyashenko (2002) Centennial history of Hilbert's 16th problem. Bulletin of the American Mathematical Society, 39:301-354
  • D.W. Jordan and P. Smith (1977) Nonlinear Ordinary Differential Equations. Clarendon Press, Oxford
  • J. Keener and J. Sneyd (1998) Mathematical Physiology. Springer-Verlag, New York
  • Yu.A. Kuznetsov (2004) Elements of Applied Bifurcation Theory, Third Edition. Springer-Verlag, New York.
  • E.F. Mishchenko, Yu.S. Kolesov, A.Yu. Kolesov, and N.Kh. Rozov (1994) Asymptotic Methods in Singularly Perturbed Systems. Plenum Publishing Corporation, New York
  • E. Ott (1993) Chaos in Dynamical Systems. Cambridge University Press, Cambridge
  • T.S. Parker and L.O. Chua (1989) Practical Numerical Algorithms for Chaotic Systems. Springer-Verlag, New York
  • S. Strogatz (1994) Nonlinear Dynamics and Chaos. Perseus, Reading
  • S. Wiggins (2003) Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York

Internal references

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See also

Attractor, Bifurcations, Canards, Chaos, Dynamical Systems, Equilibrium, Fixed Point, Isochron, Phase Model, Phase Response Curve, Relaxation Oscillator, Quasiperiodicity, Stability, Unstable Periodic Orbits, Weakly Coupled Oscillators

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