Sergej Čelikovský

ON A GENERALIZED LORENZ CANONICAL FORM OF CHAOTIC SYSTEMS

This paper shows that a large class of systems, introduced in [Celikovský & Vaněcek, 1994; Vaněcek & Celikovský, 1996] as the so-called generalized Lorenz system, are state-equivalent to a special canonical form that covers a broader class of chaotic systems. This canonical form, called generalized Lorenz canonical form hereafter, generalizes the one introduced and analyzed in [Celikovský & Vaněcek, 1994; Vaněcek & Celikovský, 1996], and also covers the so-called Chen system, recently introduced in [Chen & Ueta, 1999; Ueta & Chen, 2000]. Thus, this new generalized Lorenz canonical form contains as special cases the original Lorenz system, the generalized Lorenz system, and the Chen system, so that a comparison of the structures between two essential types of chaotic systems becomes possible. The most important property of the new canonical form is the parametrization that has precisely a single scalar parameter useful for chaos tuning, which has promising potential in future engineering chaos design. Some...

Robust synchronization of a class of chaotic networks

This paper studies synchronization of a dynamical complex network consisting of nodes being generalized Lorenz chaotic systems and connections created with transmitted synchronizing signals. The focus is on the robustness of the network synchronization with respect to its topology. The robustness is analyzed theoretically for the case of two nodes with two-sided (bidirectional) connections, and numerically for various cases with large numbers of nodes. It is shown that, unless a certain minimal coherent topology is present in the network, synchronization is always preserved. While for a minimal network where synchronization is global, the resulting synchrony reduces to semi-global if redundant connections are added.

Hyperchaotic encryption based on multi-scroll piecewise linear systems

A hyperchaotic multi-scroll piecewise linear system in R4 is binarized to generate a pseudo-random sequence which encrypt a grayscale image via symmetric-key algorithm. The sequence is analyzed throughout statistical tests according to the National Institute of Standards and Technology (NIST) specifications. The scrolls of the system are the result of a switching law that changes between the saddle hyperbolic equilibria of piecewise linear systems with eigenvalues as follows: two negative real and one pair of complex conjugate eigenvalues with positive real part. Thus, the encryption quality is evaluated depending on the variation of the number of scrolls.

On the observer design problem for continuous-time switched linear systems with unknown switchings

Abstract The observer design problem for Switched Linear Systems ( SLS ) subject to an unknown switching signal is addressed in this work. Based on known observability results for SLS , an appropriate SLS observer is proposed and its convergence is analysed showing that the corresponding estimates converge in finite-time to the SLS state. More precisely, the observers of the continuous state evolution and the observers of the switching signal are investigated and their convergence studied separately. The main tool to analyse the observability is the well-known geometric concept of ( A , B )-invariant subspaces. The developed SLS observers are then applied to construct synchronized chaotic generators based on the SLS with chaotic behavior. Finally, an example of a non-trivial chaotic SLS and its detailed analysis are presented to illustrate the achieved results.

Advanced LMI based analysis and design for Acrobot walking

This article aims to further improve previously developed design for Acrobot walking based on partial exact feedback linearisation of order 3. Namely, such an exact system transformation leads to an almost linear system where error dynamics along trajectory to be tracked is a 4-dimensional linear time-varying system having three time-varying entries only, the remaining entries being either zero or one. In such a way, exponentially stable tracking can be obtained by quadratically stabilising a linear system with polytopic uncertainty. The current improvement is based on applying linear matrix inequalities (LMI) methods to solve this problem numerically. This careful analysis significantly improves previously known approaches. Numerical simulations of Acrobot walking based on the above-mentioned LMI design are demonstrated as well.

DESYNCHRONIZATION CHAOS SHIFT KEYING METHOD BASED ON THE ERROR SECOND DERIVATIVE AND ITS SECURITY ANALYSIS

This paper describes the chaos shift keying method based on the second derivative desynchronization error and provides its security analysis, especially against the attacks by power and return map analysis. Desynchronization chaos shift keying method (DECSK) uses methods to detect the correct bit by detecting the wrong bit. Various modifications are possible, here the method using sharp increase of error in the second derivative of synchronizing signal is used. The proposed method requires very reasonable amount of data to encrypt and time to decrypt one bit. Basically, to encrypt one bit, only one iteration (i.e. only one real number of six valid digits) is needed. At the same time, thanks to the desynchronization detection based on the synchronization error second derivative, almost 100% of the carrying chaotic signal can be used. The security of the proposed method can be systematically investigated showing its good resistance against typical decryption attacks. More detailed analysis is devoted to its analysis via power and return map analysis. Conclusion is that the DECSK method cannot be broken by the above two methods which together with other arguments developed there serves as a good basis for the DECSK security.

Position Feedback Tracking of the Acrobot Walking-Like Trajectory Based on the Reduced Velocity Observer

Abstract This paper provides the algorithm how to exponentially track any trajectory of the Acrobot based on the knowledge of two position variables only. Namely, it aims to design a special reduced observer for the Acrobot angular velocities based on the respective angles measurements and then to combine this observer with the earlier developed full state feedback controllers. For the Acrobot with point feet it is not an easy task to measure the angle between its stance leg and the surface. This paper suggests to measure instead of this angle a certain distance using a laser beam sensor and then to compute the stance angle from that distance. For this purpose, the optical laser distance sensor was selected, purchased and analyzed. Remarkably, the corresponding computations maintain basically the same order of precision for the computed angle as achieved for that measured distance. The stability of both the reduced observer and the overall output feedback tracking is demonstrated as well. Both of them are based on the Acrobots model partial exact feedback linearization of order 3. The developed algorithm is then applied to the position feedback tracking of the walking-like trajectory of the Acrobot during a swing phase of a single step. Besides the theoretical analysis, the approach is demonstrated via numerical simulations.

Triangular Forms, Local Controllability and Stabilizability of Single-Input Nonlinear Systems

Abstract A rather complete treatment of the problem of the local state equivalence of single-input nonlinear system to several types of triangular form systems and its relation with local controllability and stabilizability is presented here. A set of necessary and sufficient condition for the small time local controllability and continuous static state feedback stabilizability of triangular form system is obtained. Geometric version of these results is provided as well. Various examples illustrate the results obtained.

LMI based design for the Acrobot walking

Abstract This paper aims to further improve previously developed design for Acrobot walking based on partial exact feedback linearization of order 3. Namely, such an exact system transformation leads to an almost linear system where error dynamics along trajectory to be tracked is a 4-dimensional linear time-varying system having 3 time-varying entries only, the remaining entries being either zero or one. In such a way, exponentially stable tracking can be obtained by quadratically stabilizing a linear system with polytopic uncertainty. The current improvement is based on applying LMI methods to solve this problem numerically. This careful analysis significantly improves previously known approaches. Numerical simulations of Acrobot walking based on the above mentioned LMI design are demonstrated as well.

Message Embedded Chaotic Masking Synchronization Scheme Based on the Generalized Lorenz System and Its Security Analysis

This paper focuses on the design of the novel chaotic masking scheme via message embedded synchronization. A general class of the systems allowing the message embedded synchronization is presented here, moreover, it is shown that the generalized Lorenz system belongs to this class. Furthermore, the secure encryption scheme based on the message embedded synchronization is proposed. This scheme injects the embedded message into the dynamics of the transmitter as well, ensuring thereby synchronization with theoretically zero synchronization error. To ensure the security, the embedded message is a sum of the message and arbitrary bounded function of the internal transmitter states that is independent of the scalar synchronization signal. The hexadecimal alphabet will be used to form a ciphertext making chaotic dynamics of the transmitter even more complicated in comparison with the transmitter influenced just by the binary step-like function. All mentioned results and their security are tested and demonstrated by numerical experiments.