# Chaotic Transitions in Deterministic and Stochastic Dynamical Systems: Applications of Melnikov Processes in Engineering, Physics, and Neuroscience

Emil Simiu
Pages: 240
https://www.jstor.org/stable/j.ctt7zv8n2

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Preface
(pp. xi-xvi)
4. Chapter One Introduction
(pp. 1-8)

This work has two main objectives: (1) to present the Melnikov method as a unified theoretical framework for the study of transitions and chaos in a wide class of deterministic and stochastic nonlinear planar systems, and (2) to demonstrate the method’s usefulness in applications, particularly for stochastic systems. Our interest in the Melnikov method is motivated by its capability to provide criteria and information on the occurrence of transitions and chaotic behavior in a wide variety of systems in engineering, physics, and the life sciences.

To illustrate the type of problem to which the Melnikov method is applicable we consider...

5. ### PART 1. FUNDAMENTALS

• Chapter Two Transitions in Deterministic Systems and the Melnikov Function
(pp. 11-50)

The main purpose of this chapter is to show that the necessary condition for the occurrence of transitions in a planar multistable deterministic system with dissipation and excitation is that the system’s Melnikov function have simple zeros. To define the Melnikov function and derive its expression we must first introduce a number of simple definitions and results from the theory of nonlinear dynamical systems. We then study the particular case of unper-turbed systems, that is, of integrable excitation- and dissipation-free systems. This leads us to the introduction of two special sets, the stable manifold and the unstable manifold, which will...

• Chapter Three Chaos in Deterministic Systems and the Melnikov Function
(pp. 51-75)

In Chapter 2 we considered deterministic planar systems capable of having a Melnikov function, and showed that the necessary condition for the occurrence of motions with transitions—motions of the type illustrated in Fig. 1.2c—is that their Melnikov function have simple zeros. In this chapter we establish that motions with transitions that occur if the Melnikov function has simple zeros are chaotic. Chaos implies, among others, three properties of the motion. First, the motion is sensitive to initial conditions. This entails motion unpredictability, since initial conditions can be ascertained with only limited precision. To the unavoidable errors in their...

• Chapter Four Stochastic Processes
(pp. 76-97)

Probability theory is a mathematical model for the description and interpretation of phenomena represented by variables, referred to asrandomorstochastic variables, that show statistical variability.¹Stochastic processesare defined as collections of infinitely many functions of time {y(t)}, such that the values of the functionsy(t) at any specified time constitute a stochastic variable.² In this chapter we present basic elements of the theory of stochastic processes used in Chapter 5 for the development of the stochastic Melnikov approach. Basic elements of probability theory are reviewed in Appendix A5.

Section 4.1 presents basic definitions and results of the...

• Chapter Five Chaotic Transitions in Stochastic Dynamical Systems and the Melnikov Process
(pp. 98-126)

The purpose of this chapter is to develop thestochastic Melnikov method, that is, to extend the Melnikov method for the case of stochastic near-integrable multistable systems.

It was shown in Chapter 3 that deterministic near-integrable planar multistable systems can have irregular steady-state motions sensitive to initial conditions and exhibiting chaotic transitions between preferred regions of phase space (Figs. 3.12, 3.15, and 3.17; see also Fig. 1.1c). Similar motions with chaotic transitions occur in near-integrable multistable planar systems with stochastic excitation (by definition these include systems with a combination of stochastic and deterministic excitations). Experiments on the electronic device known...

6. ### PART 2. APPLICATIONS

• Chapter Six Vessel Capsizing
(pp. 129-133)

This chapter describes an application of the stochastic Melnikov method in naval architecture, based on a model developed by Falzarano et al. (1992) and results obtained by Hsieh, Troesch, and Shaw (1994). In this model the vessel is safe as long as the angle of roll (rotation about the longitudinal axis of the vessel) due to forces induced by random waves does not exceed a critical value.

Section 6.1 describes the model for the vessel dynamics. Section 6.2 presents a numerical example for a specific type of vessel.

The equation of rolling motion of the vessel represents the balance of...

• Chapter Seven Open-Loop Control of Escapes in Stochastically Excited Systems
(pp. 134-143)

The performance of certain nonlinear stochastic systems is deemed acceptable if, during a specified time interval, the systems have sufficiently low probabilities of escape from a preferred region of phase space. An example is the motion of a vessel subjected to wave loading (Chapter 6). Given a design sea state, the vessel’s motion must have an acceptably small probability of escape from the safe region of phase space. If that probability is too large, its reduction must be achieved by redesigning the system and/or resorting to a suitable control strategy.

In this chapter we describe the basic principle of an...

• Chapter Eight Stochastic Resonance
(pp. 144-155)

In this chapter we briefly review the stochastic resonance phenomenon, for which we provide an interpretation in chaotic dynamics terms. We use the Melnikov method to (1) assess the role of the noise spectrum in stochastic resonance, and (2) extend the definition of stochastic resonance, that is, show that stochastic resonance can be induced not only by increasing the stochastic excitation of the system but, alternatively, by adding a deterministic excitation. We present results of numerical simulations which show the usefulness of the Melnikov approach to stochastic resonance.

In Section 8.1 we define stochastic resonance and briefly describe its underlying...

• Chapter Nine Cutoff Frequency of Experimentally Generated Noise for a First-Order Dynamical System
(pp. 156-158)

For stochastic systems with rates of escape that must be determined experimentally it is necessary to generate experimental noise with a sufficiently high cutoff frequency, so that the effect on the experimental results of suppressing higher-frequency noise components is negligibly small. In this chapter we show that the Melnikov method can be used to calculate an appropriate cutoff frequency for the first-order system$\dot{x}(t)=V'(x)+\xi(t), (9.1.1)$whereV(x) is a multiwell potential,ξ(t) is a process satisfying the equation written formally as$\dot\xi(t)=-\alpha\xi(t)+\alpha\surd{D}w(t), t \geq 0, \xi(0)=0, (9.1.2)$α > 0,D> 0, andw(t) denotes white Gaussian noise with autocovariance δ(t). It can be shown that the...

• Chapter Ten Snap-Through of Transversely Excited Buckled Column
(pp. 159-166)

This section presents a structural/mechanical engineering application of the Melnikov method. We seek to obtain criteria for the occurrence of stochastically induced transitions in a spatially extended dynamical system (i.e., a system governed by a partial differential equation with space and time coordinates). The system we consider is a buckled column with continuous mass, subjected to a transverse, continuously distributed force that varies randomly with time. The force may be due, for example, to seismic motion, pressures induced by air flow turbulence, or effects arising in hydrodynamical systems.

In the absence of axial and transverse loading the axis of the...

• Chapter Eleven Wind-Induced Along-Shore Currents over a Corrugated Ocean Floor
(pp. 167-177)

This chapter presents an oceanographic application of the Melnikov method. We consider a simple model of mesoscale (20–500 km scale) wind-induced along-shore ocean flow over a continental margin. The ocean bottom has variable topography that slopes linearly offshore (i.e., in the direction normal to the shoreline), and along-shore sinusoidal corrugations whose amplitude vanishes at the shoreline (Fig. 11.1). The model was developed and analyzed by Allen et al. (1991) for the case of forcing by harmonically fluctuating stresses induced at the water’s surface by along-shore fluctuating wind. It was extended for the case of stochastic forcing by Simiu (1996)....

• Chapter Twelve The Auditory Nerve Fiber as a Chaotic Dynamical System
(pp. 178-190)

The auditory nerve fiber is a natural device of interest to both neurophysiologists and signal processing engineers. Its dynamics consists of random low-amplitude motions from which escapes occur at irregular intervals. The escapes are referred to asfirings, and are associated with random, high-amplitude bursts calledspikes. A simulated time history of such motions is shown in Fig. 12.1.

In this chapter we review results of three sets of experiments on the behavior of the auditory nerve fiber. The results strongly suggest the modeling of the fiber as a chaotic, one-degree-of-freedom dissipative system with an asymmetrical double-well potential, and are...

7. Appendix A1 Derivation of Expression for the Melnikov Function
(pp. 191-192)
8. Appendix A2 Construction of Phase Space Slice through Stable and Unstable Manifolds
(pp. 193-198)
9. Appendix A3 Topological Conjugacy
(pp. 199-200)
10. Appendix A4 Properties of Space ${\Sigma}_{2}$
(pp. 201-202)
11. Appendix A5 Elements of Probability Theory
(pp. 203-210)
12. Appendix A6 Mean Upcrossing Rate $\tau_{u}^{-1}$ for Gaussian Processes
(pp. 211-212)
13. Appendix A7 Mean Escape Rate $\tau_{\epsilon}^{-1}$ for Systems Excited by White Noise
(pp. 213-214)
14. References
(pp. 215-220)
15. Index
(pp. 221-227)