Narrow Results

We distinguish two types of stickiness in systems of two degrees of freedom: (a) stickiness around an island of stability, and (b) stickiness in chaos, along the unstable asymptotic curves of unstable periodic orbits. In fact, there are asymptotic curves of unstable orbits near the outer boundary of an island that remain close to the island for some time, and then extend to large distances into the surrounding chaotic sea. But later the asymptotic curves return close to the island and contribute to the overall stickiness that produces dark regions around the islands and dark lines extending far from the islands. We have studied these effects in the standard map with a rather large nonlinearity K = 5, and we emphasized the role of the asymptotic curves U, S from the central orbit O (x = 0.5, y = 0), that surround two large islands O₁ and O′₁, and the asymptotic curves U₊U_-S₊S_- from the simplest unstable orbit around the island O₁. This is the orbit 4/9 that has 9 points around the island O₁ and 9 more points around the symmetric island O′₁. The asymptotic curves produce stickiness in the positive time direction (U, U₊, U_-) and in the negative time direction (S, S₊, S_-). The asymptotic curves U₊, S₊ are closer to the island O₁ and make many oscillations before reaching the chaotic sea. The curves U_-, S_- are further away from the island O₁ and escape faster. Nevertheless all curves return many times close to O₁ and contribute to the stickiness near this island. The overall stickiness effects of U₊, U_- are very similar and the stickiness effects along S₊, S_- are also very similar. However, the stickiness in the forward time direction, along U₊, U_-, is very different from the stickiness in the opposite time direction along S₊, S_-. We calculated the finite time LCN (Lyapunov characteristic number) χ(t), which is initially smaller for U₊, S₊ than for U_-, S_-. However, after a long time all the values of χ(t) in the chaotic zone approach the same final value LCN = lim_{t → ∞} χ(t). The stretching number (LCN for one iteration only) varies along an asymptotic curve going through minima at the turning points of the asymptotic curve. We calculated the escape times (initial stickiness times) for many initial points outside but close to the island O₁. The lines that separate the regions of the fast from the slow escape time follow the shape of the asymptotic curves S₊, S_-. We explained this phenomenon by noting that lines close to S₊ on its inner side (closer to O₁) approach a point of the orbit 4/9, say P₁, and then follow the oscillations of the asymptotic curve U₊, and escape after a rather long time, while the curves outside S₊ after their approach to P₁ follow the shape of the asymptotic curves U_- and escape fast into the chaotic sea. All these curves return near the original arcs of U₊, U_- and contribute to the overall stickiness close to U₊, U_-. The isodensity curves follow the shape of the curves U₊, U_- and the maxima of density are along U₊, U_-. For a rather long time, the stickiness effects along U₊, U_- are very pronounced. However, after much longer times (about 1000 iterations) the overall stickiness effects are reduced and the distribution of points in the chaotic sea outside the islands tends to be uniform. The stickiness along the asymptotic curve U of the orbit O is very similar to the stickiness along the asymptotic curves U₊, U_- of the orbit 4/9. This is related to the fact that the asymptotic curves of O and 4/9 are connected by heteroclinic orbits. However, the main reason for this similarity is the fact that the asymptotic curves U, U₊, U_- cannot intersect but follow each other.

Stickiness refers to chaotic orbits that stay in a particular region for a long time before escaping. For example, stickiness appears near the borders of an island of stability in the phase space of a 2-D dynamical system. This is pronounced when the KAM tori surrounding the island are destroyed and become cantori (see [Contopoulos, 2002]). We find the time scale of stickiness along the unstable asymptotic curves of unstable periodic orbits around an island of stability, that depends on several factors: (a) the largest eigenvalue |λ| of the asymptotic curve. If λ > 0 the orbits on the unstable asymptotic manifold in one direction (fast direction) escape faster than the orbits in the opposite direction (slow direction) (b) the distance from the last KAM curve or from the main cantorus (the cantorus with the smallest gaps) (c) the size of the gaps of the main cantorus and (d) the other cantori, islands and asymptotic curves. The most important factor is the size of the gaps of the main cantorus. Then we find when the various KAM curves are destroyed. The distance of the last KAM curve from the center of an island gives the size of the island. When the central periodic orbit becomes unstable, chaos is also formed around it, limited by a first KAM curve. Between the first and the last KAM curves there are still closed invariant curves. The sizes of the islands as functions of the perturbation, have abrupt changes at resonances. These functions have some universal features but also some differences. A new type of stickiness appears near the unstable asymptotic curves of unstable periodic orbits that extend far into the large chaotic sea. Such a stickiness lasts for long times, increasing the density of points close to the unstable asymptotic curves. However after a much longer time, the density becomes almost equal everywhere outside the islands of stability. We consider also stickiness near the asymptotic curves from new periodic orbits, and stickiness in Anosov systems and near totally unstable orbits. In systems that allow escapes to infinity the stickiness delays the escapes. An important astrophysical application is the case of barred-spiral galaxies. The spiral arms outside corotation consist mainly of sticky chaotic orbits. Stickiness keeps the spiral forms for times longer than a Hubble time, but after a much longer time most of the chaotic orbits escape to infinity.

In this work, we use the finite-time Lyapunov exponent (FTLE) for a version of the potential well model with an oscillating bottom. This model is described by a conservative two-dimensional mapping that exhibits a mixed phase space, namely it contains a sea of chaos and the regions of stability. Via FTLE, we show that throughout its orbit, an initial condition in the chaotic sea can become trapped in a certain region. We construct and analyze phase spaces for different ranges of Lyapunov exponents in finite time, highlighting these traps. For smaller and larger values of FTLE, we identified a very regular phase space and a phase space with more chaos. By means of low FTLE values, we observe some small regions in the phase space that present regularities. Therefore, we demonstrate that FTLE can be useful to identify the stickiness effect in a time-dependent potential well.

Marketing managers often use consumer attitude metrics such as awareness, consideration, and preference as performance indicators because they represent their brand’s health and are readily connected to marketing activity. However, this does not mean that financially focused executives know how such metrics translate into sales performance, which would allow them to make beneficial marketing mix decisions. We propose four criteria — potential, responsiveness, stickiness, and sales conversion — that determine the connection between marketing actions, attitudinal metrics, and sales outcomes.

We test our approach with a rich dataset of four-weekly marketing actions, attitude metrics, and sales for several consumer brands in four categories over a seven-year period. The results quantify how marketing actions affect sales performance through their differential impact on attitudinal metrics, as captured by our proposed criteria. We find that marketing—attitude and attitude—sales relationships are predominantly stable over time but differ substantially across brands and product categories. We also establish that combining marketing and attitudinal metrics criteria improves the prediction of brand sales performance, often substantially so. Based on these insights, we provide specific recommendations on improving the marketing mix for different brands, and we validate them in a holdout sample. For managers and researchers alike, our criteria offer a verifiable explanation for differences in marketing elasticities and an actionable connection between marketing and financial performance metrics.