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Dynamics with Chaos and Fractals

Part of the Nonlinear Systems and Complexity Book Series (volume 29)

DOI: 10.1007/978-3-030-35854-9

ISBN: 978-3-030-35853-2

Publisher: Springer International Publishing - Springer Journals

Jan 2, 2020

1 - 10 of 12 Chapters

LitStream

Chapter 1Introduction

Akhmet, Marat; Fen, Mehmet Onur; Alejaily, Ejaily Milad

Pages: 1 –14

[In this chapter, the short outline of the conceptual content of the book is provided. Chaos and fractals as well as their connections and interrelations in terms of unpredictability, replication of chaos, fractals geometry, dynamics through fractal mappings, and abstract self-similarity are...
LitStream

Chapter 2The Unpredictable Point and Poincaré Chaos

Akhmet, Marat; Fen, Mehmet Onur; Alejaily, Ejaily Milad

Pages: 15 –23

[This chapter is an introductory one for the book. It is revealed that a special kind of Poisson stable point, which we call an unpredictable point, gives rise to the existence of chaos in the quasi-minimal set. The existing definitions of chaos are formulated in sets of motions. This is the...
LitStream

Chapter 3Unpredictability in Bebutov Dynamics

Akhmet, Marat; Fen, Mehmet Onur; Alejaily, Ejaily Milad

Pages: 25 –40

[In this chapter, an unpredictable solution is considered as a generator of the chaos in a quasilinear system by means of Bebutov dynamical system. The results can be easily extended to different types of differential equations. An example of an unpredictable function is provided. A proper...
LitStream

Chapter 4Nonlinear Unpredictable Perturbations

Akhmet, Marat; Fen, Mehmet Onur; Alejaily, Ejaily Milad

Pages: 41 –56

[The results of this chapter are continuation of the research of Poincaré chaos initiated in Chaps. 2 and 3. We focus on the construction of an unpredictable function, continuous on the real axis. This is the first time that perturbations depend nonlinearly on unpredictable functions.]
LitStream

Chapter 5Unpredictability in Topological Dynamics

Akhmet, Marat; Fen, Mehmet Onur; Alejaily, Ejaily Milad

Pages: 57 –79

[In this chapter, the topology of uniform convergence on compact sets is applied to define unpredictable functions. The unpredictable sequence is defined as a specific unpredictable function on the set of integers. The definitions are convenient to be verified as solutions of differential and...
LitStream

Chapter 6Unpredictable Solutions of Hyperbolic Linear Equations

Akhmet, Marat; Fen, Mehmet Onur; Alejaily, Ejaily Milad

Pages: 81 –95

[In this chapter, the existence and uniqueness of unpredictable solutions in the dynamics of non-homogeneous linear systems of differential and discrete equations are investigated. The hyperbolic cases are under discussion. The presence of unpredictable solutions confirms the existence of...
LitStream

Chapter 7Strongly Unpredictable Solutions

Akhmet, Marat; Fen, Mehmet Onur; Alejaily, Ejaily Milad

Pages: 97 –108

[This chapter is concerned with the existence and uniqueness of asymptotically stable unpredictable solutions for quasilinear differential equations. Two principal novelties are in the basis of the research. The first one is that all coordinates of a solution are unpredictable functions. That is,...
LitStream

Chapter 8Li–Yorke Chaos in Hybrid Systems on a Time Scale

Akhmet, Marat; Fen, Mehmet Onur; Alejaily, Ejaily Milad

Pages: 109 –124

[In this chapter, we apply the method of replication of chaos introduced and developed in our studies. We rigorously prove the presence of chaos in dynamic equations on time scales by using the reduction technique to impulsive differential equations. The results are based on the Li–Yorke...
LitStream

Chapter 9Homoclinic and Heteroclinic Motions in Economic Models

Akhmet, Marat; Fen, Mehmet Onur; Alejaily, Ejaily Milad

Pages: 125 –137

[In this chapter, the technique of replication of chaos is utilized to prove the presence of homoclinic and heteroclinic motions in the dynamics of economic models perturbed with exogenous shocks. An illustrative example based on the Kaldor model of the aggregate economy is presented.]
LitStream

Chapter 10Global Weather and Climate in the Light of El Niño-Southern Oscillation

Akhmet, Marat; Fen, Mehmet Onur; Alejaily, Ejaily Milad

Pages: 139 –172

[In this chapter, we study the chaotic behavior of hydrosphere and its influence on global weather and climate. We give mathematical arguments for the sea surface temperature (SST) to be unpredictable over the global ocean. The impact of SST variability on global climate is clear during global...

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