Peak-to-Peak Dynamics (PPD) is a particular form of deterministic chaos, where, from a n-th order continuous-time dynamical system, the amplitude and time of occurrence of the next peak of its output variable can be predicted from information concerning at most two previous peaks.

Simple PPD have been discovered by famous metheorologist Edward Lorenz in his pioneering 1963 paper on chaos.. Later, PPD have been noticed by various researchers in some fields, like ecology, bio & electrochemistry, physics and electronics.

E.g., remarkably irregular peaks characterize the dynamics of many plant and animal populations. As such peaks are often associated with undesirable consequences (e.g., pest outbreaks, epidemics, forest fires), the forecast of the intensity of a forthcoming peak is a problem of major concern, and has been shown to be most often predictable from the previous peaks, through the analysis of both, simulations of related mathematical models and of some of the longest and most celebrated ecological time series.

Basics

After a transient, any deterministic, dissipative, nonlinear dynamical system settles on an attractor (an equilibrium, a limit cycle, a torus or, most famously, a strange attractor) and remains there forever if it's not perturbed. Some insights about the attractor can be obtained, even in the absence of a formal mathematical model, if a single variable y of time t has been recorded for a "sufficiently long" period, provided that the system was on the attractor. In particular, you can extract from the record all the peaks (i.e., extreme occurrences) of the variable and plot them one peak against the previous one, sequentially, thus obtaining a set of points called Peak-to-Peak Map (PPM), or sometimes also called next-amplitude map, next-maximum map, or, originally, Lorenz map.

If the dynamical regime is periodic (i.e., the system's attractor is a cycle) and there are k peaks per period, the PPM's composed simply of k distinct points. By contrast, if the regime is quasi-periodic (i.e., the system's attractor is a torus) or chaotic (i.e., the system's attractor is a strange attractor), the points on the PPM are all distinct and sometimes display filiform geometries - the points lie on a closed regular curve when there's quasi-periodicity (corresponding to a slice of a torus), roughly on one or more curves1, when there's low dimensional chaos, and form a cloud-like set when there's high dimensional chaos.

When the PPM is filiform - i.e., when the system's attractor is low-dimensional -, the intensity of the forthcoming peak and its time of occurrence can be predicted with remarkable accuracy from the current peak, when we say the PPD are simple, or from the current + the previous peaks, when we say the PPD are complex (in other words, the forecast of the next peak based solely on the current peak would be ambiguous).

When the set of extracted extreme occurrences is filiform, it can be approximated by a set of curves called Peak-to-Peak Skeleton (PPS), described by a k-value function that can be interpreted as follows: Given a certain peak, the next one's approximately one of the k elements of the function set. When k=1 the PPS is called simple, as well as the underlying PPD. In case the PPD are complex, the extra information needed to accomplish the task (i.e., to forecast the next peak) is a "surrogate" information of the previous peak (i.e., not the previous peak itself), namely knowing if the previous peak was "small" or "large". E.g, in a specific study on plankton-fish interactions it has been shown that the next peak of "young of the year" planktivorous fish, that systematically occurs every year during the summer, can be forecasted on the basis of the current peak + the month during which the previous peak occurred (i.e., the exact date of the occurrence of the previous peak's not needed).

Practical Uses

Once PPD have been identified, the underlying dynamical system (often attached to a model composed by n differential equations) can be described by a very simple reduced order model involving only the peaks of the variable of concern or the corresponding occurrence times.

Identification of attractors

The identification of the attractor of a dynamical system from the observations of a single variable is a problem of major concern, which is usually solved through relatively complex techniques. By contrast, peak-to-peak analysis (i.e., the determination of the PPM associated with a recorded time series) is an almost trivial task that can be even performed by hand. This is also an effective tool for discovering if the dynamics within the attractor can be described by a one-dimensional map. I.e., before performing any analysis of a recorded time series, it's worth extracting its PPM to check if it's really justified to proceed further with more sophisticated (i.e., costly) methods.

Next peak forecast

Although, in theory, forecasting the output peak and the time of occurrence of this peak use equivalent schemes, in practice there's no real equivalence, because the peak amplitude can be known for higher or lower precision than the time of occurrence. E.g., in many ecosystems the time of occurrence of recurrent extreme episodes is well known, while the severity of each episode can be hardly evaluated.

Controlling PPD

In many real-world problems, the peaks of the output variable are associated with high costs, so that it's natural to refer to a reduced order model and formulate an optimal control problem involving only the peaks, their times of occurrences, and the control efforts. Actually, when the peaks are the most crucial episodes, the only available information on the past history of the system are often the times of occurrence of the last peak, fortunately exactly what's needed to forecast the next peak or to exert the control action suggested by the reduced order model.

Challenges

The problem that certainly deserves more attention is the identifiability of PPD from the all-around, real-world, "noisy" field/experimental data, since the structure of the PPM is sensitive to high frequency measurement errors. Of course, this can be somewhat compensated by "filtering" the data before extracting the peaks. General and effective counter-measures to this problem seems to be hard to discover, but the challenge's worth trying. Alternatively, the study of particular cases has been illuminating and suggested how to circumvent particular obstacles. Meanwhile, it seems appropriate to make a combined use of data and models, where a naïve way of proceeding is to use the supporting model only for a qualitative consistency check and/or for a calibration of the simulated PPM with the data. This is certainly justified if the aim is to build an operational forecasting technique based only on information on past peaks.

1 These PPMs would be thinner if the measurement error, inevitably present during experiments/recordings, would have been smaller.

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