Narrow Results

In this article using an analytical method called Fishing principle we obtain the region of parameters, where the existence of a homoclinic orbit to a zero saddle equilibrium in the Lorenz-like system is proved. For a qualitative description of the different types of homoclinic bifurcations, a numerical analysis of the obtained region of parameters is organized, which leads to the discovery of new bifurcation scenarios.

This paper investigates the effects of the unsteady nonlinear aerodynamic, plunge/pitch cubic nonlinearities, flap free-play nonlinearity, and coupled nonlinear aeroelasticity on the dynamics of the three-dimensional blade section. The dynamic stall model is developed based on the unsteady Wagner aerodynamics. Coupling the developed nonlinear aerodynamic model and nonlinear elasticity model results in the nonlinear aeroelastic model. The nonlinear aeroelastic equation of motion is converted into a state-space form. The resulting nonlinear state-space equation of motion is simulated by a standard Runge–Kutta algorithm in MATLAB. The proposed model is validated against test data of distinct two- and three-degrees-of-freedom studies and is compared to the ONERA model. Bifurcation diagrams show that there is distinct airspeed, in which the system experiences limit cycle oscillations (LCOs) or chaos. Both hysteresis air loads and structural nonlinearity make the system unstable at airspeed less than linear flutter speed. The nonlinearity of the structure causes supercritical pitchfork bifurcation. Elastic-aerodynamic nonlinearity interaction causes sub-supercritical bifurcation at the lower airspeed and chaotic motion at a higher airspeed. Furthermore, the effects of the initial condition on the response of the nonlinear aero-servo-elastic system are investigated by the Lyapunov exponent method.

There are few explicit examples in the literature of vector fields exhibiting observable chaos that may be proved analytically. This paper reports numerical experiments performed for an explicit two-parameter family of $𝕊𝕆(2)\oplus {\mathbb{Z}}_2$-symmetric vector fields whose organizing center exhibits an attracting heteroclinic network linking two saddle-foci. Each vector field in the family is the restriction to ${𝕊}^3$ of a polynomial vector field in ${ℝ}^4$. We investigate global bifurcations due to symmetry-breaking and we detect strange attractors via a mechanism called Torus-Breakdown. We explain how an attracting torus gets destroyed by following the changes in the unstable manifold of a saddle-focus.

Although a complete understanding of the corresponding bifurcation diagram and the mechanisms underlying the dynamical changes is out of reach, we uncover complex patterns for the symmetric family under analysis, using a combination of theoretical tools and computer simulations. This article suggests a route to obtain rotational horseshoes and strange attractors; additionally, we make an attempt to elucidate some of the bifurcations involved in an Arnold tongue.

In nonlinear dynamics, the study of chaotic systems has attracted the attention of many researchers around the world due to the exciting and peculiar properties of such systems. In this regard, the present paper introduces a new system with a self-excited strange attractor and its twin strange repeller. The unique characteristic of the presented system is that the system variables are all in their quadratic forms; therefore, the proposed system is called a fully-quadratic system. This paper also elaborates on the study of the bifurcation diagram, the interpretation of Lyapunov exponents, the representation of basin of attraction, and the calculation of connecting curves as the employed method for investigating the system’s dynamics. The investigation of 2D bifurcation diagrams and Lyapunov exponents indicated in this paper can better recognize the system’s dynamics since they are plotted considering simultaneous changes of two parameters. Moreover, the connecting curves of the proposed system are calculated.

The system’s connecting curves help identify the system’s different behaviors by providing general information about the nature of the flows.

In this work, we study, from a numerical point of view, the dynamics of a specific hybrid map which is that of the kicked van der Pol system. The dynamics of this system is generated by a two-stage procedure: the first stage is the time-$\tau$ map of the vector field associated with the van der Pol equation, and the second stage is a translation. We propose a numerical method for computing local (un)stable manifolds of a given fixed point, which leads to high order polynomial parameterization of the embedding. Such a representation of the dynamics on the manifold is obtained in terms of a simple conjugacy relation and by constructing two contracting operators. We illustrate our techniques by plotting (for specific values of the parameters) both stable and unstable manifolds displaying homoclinic intersection. Furthermore, our numerical study reveals the presence of a strange attractor included in the closure of the unstable manifold of the fixed point.

To construct symmetrical patterns on the unit sphere from the planar iterative function systems (IFSs), we present a method of constructing IFSs with $D_3$ symmetry which is composed of three-fold rotational symmetries together with reflections. An algorithm is developed to generate strange attractors with $D_3$ symmetry on a triangular face and then project it onto the surface of the unit sphere to form aesthetics patterns with spherical symmetry. As an illustrative example, we consider the regular inscribed icosahedron in the unit sphere which contains 20 triangular faces. This method is valid to randomly generate aesthetic spherical patterns using planar IFSs.

To generate exotic fractals, we investigate the construction of nonlinear iterated function system (IFS) using the complex mapping family $f(z)=z^n+c$ ($\left|n\right|\ge 2,3,\dots$). A set of $c$-values is chosen from the period-1 bulb of the Mandelbrot set, so that each mapping has an attracting fixed point in the dynamic plane. Computer experiments show that a set of arbitrarily chosen $c$-values may not be able to generate a fractal. We prove a sufficient condition that if the $c$-values are chosen from a specific region related to a circle in the period-1 bulb, the nonlinear IFS with such complex mappings is able to generate exotic fractal. Furthermore, if the set of $c$-values possesses a specific symmetry in the Mandelbrot set, then the fractal also exhibits the same symmetry. We present a method of generating aesthetic fractals with $Z_{n-1}$ or $D_{n-1}$ symmetry for $n\ge 2$ and with $Z_{\left|n\right|+1}$ or $D_{\left|n\right|+1}$ symmetry for $n\le -2$.

Predicting foreign exchange rates and stock market indices have been a well researched topic in the field of financial engineering. However, most methods suffer from serious drawback due to the inherent uncertainty in the data acquisition process. Here, we have analyzed the very nature of the time series data from a pure dynamic system point of view and explored the deterministic chaotic characteristic in it. In this research, the concept of chaos has been analyzed thoroughly and the relationships among chaos, stability and order have been explained with respect to the concept of time. A method of predicting time series data based on deterministic dynamically system has been presented in this monograph. The present research revolves around the concepts of embedding and fuzzy reconstruction. In this regard, the necessary and sufficient condition for this reconstruction of the state space of the dynamic system in a multi-dimensional Euclidean space has been substantiated in accordance to Theory of embedding. Finally, a fuzzy reconstruction method based on fuzzy multiple regression analysis method has been used to predict the foreign exchange rates with accuracy.

This paper examines the dynamics of a nonlinear semi-active suspension system using a quarter-car model moving over rough road profiles. The bifurcation analysis of the nonlinear dynamical behavior of this system is performed. Codimension-two bifurcation and homoclinic orbits can be discovered in this system. When the external force of a road profile was added to this system as a parameter with a certain range of values, a strange attractor can be found using the numerical simulation. Finally, the Lyapunov exponent is adopted to identify the onset of chaotic motion and verify the bifurcation analysis.

It is well known that the celebrated Lorenz system has an attractor such that every orbit ends inside a certain ellipsoid in forward time. We complement this result by a new phenomenon and by a new interpretation. We show that "infinity" is a global repeller for a set of parameters wider than that usually treated. We construct in a compacted space, a unit sphere that serves as the image of an ideal set at infinity. This sphere is shown to be the union of a family of periodic solutions. Each periodic solution is viewed as a limit cycle, or an isolated periodic orbit when restricted to a certain plane. The unconventional compactification is used.

This chapter is an introduction to complexity theory (encompassing chaos — a subset of complexity), a nascent domain, although, it possesses a historical root. Some fundamental properties of chaos/complexity (including complexity mindset, nonlinearity, interconnectedness, interdependency, far-from-equilibrium, butterfly effect, determinism/in-determinism, unpredictability, bifurcation, deterministic chaotic dynamic, complex dynamic, complex adaptive dynamic, dissipation, basin of attraction, attractor, chaotic attractor, strange attractor, phase space, rugged landscape, red queen race, holism, self-organization, self-transcending constructions, scale invariance, historical dependency, constructionist hypothesis and emergence), and its development are briefly examined. In particular, the similarities (sensitive dependence on initial conditions, unpredictability) and differences between deterministic chaotic systems (DCS) and complex adaptive systems (CAS) are analyzed. The edge of emergence (2^nd critical value, a new concept) is also conceived to provide a more comprehensive explanation of the complex adaptive dynamic (CAD) and emergence. Subsequently, a simplified system spectrum is introduced to illustrate the attributes, and summarize the relationships of the various categories of common systems.

Next, the recognition that human organizations are nonlinear living systems (high finite dimensionality CAS) with adaptive and thinking agents is examined. This new comprehension indicates that a re-calibration in thinking is essential. In the human world, high levels of human intelligence/consciousness (the latent impetus that is fundamentally stability-centric) drives a redefined human adaptive and evolution dynamic encompassing better potentials of self-organization or self-transcending constructions, autocatalysis, circular causation, localized spaces/networks, hysteresis, futuristic, and emergent of new order (involving a multi-layer structure and dynamic) — vividly indicating that intelligence/consciousness-centric is extremely vital. Simultaneously, complexity associated properties/characteristics in human organizations must be better scrutinized and exploited — that is, establishing appropriate complexity-intelligence linkages is a significant necessity. In this respect, nurturing of the intelligence mindset and developing the associated paradigmatic shift is inevitable.

A distinct attempt (the basic strategic approach) of the new intelligence mindset is to organize around human intrinsic intelligence — intense intelligence-intelligence linkages that exploits human intelligence/consciousness sources individually and collectively by focusing on intelligence/consciousness-centricity, complexity-centricity, network-centricity, complexity-intelligence linkages, collective intelligence, org-consciousness, complex networks, spaces of complexity (better risk management <=> new opportunities <=> higher sustainability) and prepares for punctuation points (better crisis management <=> collectively more intelligent <=> higher resilience/sustainability) concurrently — illustrating the significance of self-organizing capability and emergence-intelligence capacity. The conceptual development introduced will serve as the basic foundation of the intelligent organization (IO) theory.

The first section of this chapter is an introduction to relativistic complexity (a significant component of the intelligent organization theory). The presence of intense intelligence/consciousness-centricity and 3^rd order stability-centricity in the human world renders complexity relativistic. The impact of the human mental space is so tremendous that complexity is in the mind of the beholder, and predictability becomes highly subjective. In this situation, the state of relativistic static equilibrium may be beneficial. Certain spaces of complexity appear as spaces of relativistic order with surface patterns becoming more apparent. Such spaces must be creatively explored and exploited (higher exploratory capacity) leading to a more advanced level of intelligence advantage. In this respect, effective self-transcending constructions, high self-organizing capacity and emergence-intelligence capacity are significant attributes that the new leadership and governance system in intelligent human organizations must exploit. Holistically, the two strategies focus on concurrent exploitation of intelligence/consciousness-centricity and relative complexity, and optimizing the more comprehensive contributions of the integrated deliberate and emergent strategy.

Many issues/problems that present human organizations (nations, political systems, communities, business organizations, markets) are encountering due to accelerating changes (mindset, thinking, values, perceptions, expectations, redefined boundaries and high interactive dynamics) that cannot be well-managed with traditional knowledge and hierarchical practices are affecting governance and governance systems. Fundamentally, governance deals with power, interest, and conflict. The traditional governance systems are hierarchical, highly directed, controlled and managed, and the relational aspect has not been allocated sufficient priority resulting in extensive disparities. In the current complex dynamical and high interdependency environment, its weaknesses and constraints are highly apparent. The latter includes ‘space-time compression’; incoherency in thinking, values, perceptions, and expectations between the leadership and the other agents; diversification in stakeholders’ needs not accommodated; and constraints of current governance theories. Thus, a new theory that provides a more ‘realistic’ foundation is essential for deeper contemplation.

Primarily, recognizing the inherent strengths of human agents and the fundamental constraints/weaknesses of human organizations is a key foundation towards better adaptation, leadership, governance, resilience and sustainability. In all human organizations, the agents are intrinsic intense intelligence/consciousness sources that could easily transform their behavioral schemata. This observation contradicts the Newtonian/design paradigm, as the organizational dynamic of human agents is complex, nonlinear, constantly/continuously changing with limited predictability. In addition, human agents are self-centric, self-powered, stability-centric, independent and interdependent, network-centric and self-organizing due to high awareness. In this situation, high self-organizing capacity and emergence-intelligence capacity are new niches. However, this phenomenon can create new opportunities, innovation, and elevates competiveness; or destruction.

In particular, effective leadership and governance are spontaneously emerging key requirements in all human groupings — a primary trait for human collective survival. Historically, many organizations disintegrated because of the weaknesses in leadership and governance. Currently, with more knowledge-intensive and higher participative new agents (self-powered intrinsic leadership) possessing modified attributes that are dissimilar from the older generations (also due to the deeper integration of the economic, social, political, and environmental perspective), reduces consensus and collaboration, and renders governance and leadership even more nonlinear or dysfunctional. In particular, the traditional governance systems of more organizations are manifesting their constraints and incompetency, including incoherency due to new values and cultural pressure, and the wider spread of self-organizing networks. The emergent of informal networks is a more commonly observed phenomenon worldwide. Apparently, a deeper comprehension on the diminishing effect of the traditional organizational thinking (political, social, economic), governance capacity, precise strategic planning, decision making, hierarchical structure, communications and engagement, empowerment leadership, management, operations, and the highly nonlinear relational parameter is essential. Apparently, new principles of governance must emerge (intelligent human organization > thinking system + feeling system).

The new paradigmatic path of the intelligence governance strategy that exploits intelligence/consciousness-centricity, complexity-centricity, and network-centricity concurrently, introduces a new basic strategic path towards better adaptive governance and acceptance governance. The latter focuses on integrating self-powered self-organizing governance, reducing direct governance, and increasing e-governance and network-centric governance as a new necessity. In this case, the merits of adopting the intelligence leadership strategic approach simultaneously are more apparent. Hence, the new governance focal points must include more and better interconnected actors, the critical ability of self-organizing communications (supported by mobile/social media development), immersion of leadership nodes in networks (better exploitation of e-governance), increasing coherency of complex networks (exploiting interdependency of network of networks, and better network management), and elevating self-transcending constructions capability (higher self-organizing governance capacity and emergence-intelligence capacity) that better facilitates emergence through multi-level and ‘multi-lateral’ dynamics (complex adaptive networks <=> intelligent networks). Thus, the intelligence governance strategy emphasizes that mass lateral collectivity (acceptance governance) rather than selective enforced hierarchical empowerment as the more constructive approach in the present contact. In particular, the stabilityinducing role of leaders and institutions are critical. Apparently, optimizing the ‘everybody is in charge’ phenomenon (whenever necessary) is a more viable option.

This chapter introduces some significant changes taking place in the world that are affecting all categories of human organizations, as well as all individuals. The appropriateness, importance and exploitation of certain properties of Chaos and Complexity Theory are also examined briefly. Human systems are recognized as complex adaptive systems. In particular, the fact that the edge of chaos is an unexplored space embedded with new opportunities is highlighted. This observation and recognition indicates that a recalibration of understanding is essential. A change in era and mindset is also inevitable. The new era is the intelligence era. The primary focus of the new mindset is to organize around human intrinsic intelligence.